Morten Hjorth-Jensen
7206 lines
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Applied Data Analysis and Machine Learning
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Review of Statistics with Resampling Techniques and Linear Algebra
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1. Elements of Probability Theory and Statistical Data Analysis
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2. Linear Algebra, Handling of Arrays and more Python Features
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From Regression to Support Vector Machines
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3. Linear Regression
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4. Ridge and Lasso Regression
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5. Resampling Methods
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6. Logistic Regression
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7. Optimization, the central part of any Machine Learning algortithm
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8. Support Vector Machines, overarching aims
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<p aria-level="2" class="caption" role="heading">
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Decision Trees, Ensemble Methods and Boosting
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9. Decision trees, overarching aims
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10. Ensemble Methods: From a Single Tree to Many Trees and Extreme Boosting, Meet the Jungle of Methods
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Dimensionality Reduction
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11. Basic ideas of the Principal Component Analysis (PCA)
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12. Clustering and Unsupervised Learning
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Deep Learning Methods
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13. Neural networks
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14. Building a Feed Forward Neural Network
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15. Solving Differential Equations with Deep Learning
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16. Convolutional Neural Networks
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17. Recurrent neural networks: Overarching view
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Weekly material, notes and exercises
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Exercises week 34
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Week 34: Introduction to the course, Logistics and Practicalities
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Exercises week 35
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Week 35: From Ordinary Linear Regression to Ridge and Lasso Regression
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Exercises week 36
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Week 36: Linear Regression and Statistical interpretations
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Exercises week 37
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Week 37: Statistical interpretations and Resampling Methods
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Exercises week 38
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Week 38: Logistic Regression and Optimization
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Exercises week 39
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Week 39: Optimization and Gradient Methods
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Week 40: Gradient descent methods (continued) and start Neural networks
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Exercises week 41
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Week 41 Neural networks and constructing a neural network code
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Exercises week 42
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Week 42 Constructing a Neural Network code with examples
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<a class="reference internal" href="additionweek42.html">
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Exercises Week 42: Logistic Regression and Optimization, reminders from week 38 and week 40
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Week 43: Deep Learning: Constructing a Neural Network code and solving differential equations
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Exercises week 43
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Projects
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Project 1 on Machine Learning, deadline October 7 (midnight), 2024
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Project 2 on Machine Learning, deadline November 4 (Midnight)
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Exercises and lab session week 43
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</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#using-automatic-differentiation">
|
||
Using Automatic differentiation
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#back-propagation-and-automatic-differentiation">
|
||
Back propagation and automatic differentiation
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#lecture-monday-october-21">
|
||
Lecture Monday October 21
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#setting-up-the-back-propagation-algorithm-and-algorithm-for-a-feed-forward-nn-initalizations">
|
||
Setting up the back propagation algorithm and algorithm for a feed forward NN, initalizations
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#setting-up-the-back-propagation-algorithm-part-1">
|
||
Setting up the back propagation algorithm, part 1
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#setting-up-the-back-propagation-algorithm-part-2">
|
||
Setting up the back propagation algorithm, part 2
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#setting-up-the-back-propagation-algorithm-part-3">
|
||
Setting up the Back propagation algorithm, part 3
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#updating-the-gradients">
|
||
Updating the gradients
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#activation-functions">
|
||
Activation functions
|
||
</a>
|
||
<ul class="nav section-nav flex-column">
|
||
<li class="toc-h3 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#activation-functions-examples">
|
||
Activation functions, examples
|
||
</a>
|
||
</li>
|
||
</ul>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#the-relu-function-family">
|
||
The RELU function family
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#elu-function">
|
||
ELU function
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#which-activation-function-should-we-use">
|
||
Which activation function should we use?
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#more-on-activation-functions-output-layers">
|
||
More on activation functions, output layers
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#setting-up-a-multi-layer-perceptron-model-for-classification">
|
||
Setting up a Multi-layer perceptron model for classification
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#defining-the-cost-function">
|
||
Defining the cost function
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#example-binary-classification-problem">
|
||
Example: binary classification problem
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#the-softmax-function">
|
||
The Softmax function
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#developing-a-code-for-doing-neural-networks-with-back-propagation">
|
||
Developing a code for doing neural networks with back propagation
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#collect-and-pre-process-data">
|
||
Collect and pre-process data
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#train-and-test-datasets">
|
||
Train and test datasets
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#define-model-and-architecture">
|
||
Define model and architecture
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#layers">
|
||
Layers
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#weights-and-biases">
|
||
Weights and biases
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#feed-forward-pass">
|
||
Feed-forward pass
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#matrix-multiplications">
|
||
Matrix multiplications
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#choose-cost-function-and-optimizer">
|
||
Choose cost function and optimizer
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#optimizing-the-cost-function">
|
||
Optimizing the cost function
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#regularization">
|
||
Regularization
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#matrix-multiplication">
|
||
Matrix multiplication
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#improving-performance">
|
||
Improving performance
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#full-object-oriented-implementation">
|
||
Full object-oriented implementation
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#evaluate-model-performance-on-test-data">
|
||
Evaluate model performance on test data
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#adjust-hyperparameters">
|
||
Adjust hyperparameters
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#visualization">
|
||
Visualization
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#scikit-learn-implementation">
|
||
scikit-learn implementation
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#id1">
|
||
Visualization
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#building-neural-networks-in-tensorflow-and-keras">
|
||
Building neural networks in Tensorflow and Keras
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#tensorflow">
|
||
Tensorflow
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#using-keras">
|
||
Using Keras
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#id2">
|
||
Collect and pre-process data
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#the-breast-cancer-data-now-with-keras">
|
||
The Breast Cancer Data, now with Keras
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#building-a-neural-network-code">
|
||
Building a neural network code
|
||
</a>
|
||
<ul class="nav section-nav flex-column">
|
||
<li class="toc-h3 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#learning-rate-methods">
|
||
Learning rate methods
|
||
</a>
|
||
</li>
|
||
<li class="toc-h3 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#usage-of-the-above-learning-rate-schedulers">
|
||
Usage of the above learning rate schedulers
|
||
</a>
|
||
</li>
|
||
<li class="toc-h3 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#cost-functions">
|
||
Cost functions
|
||
</a>
|
||
</li>
|
||
<li class="toc-h3 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#id3">
|
||
Activation functions
|
||
</a>
|
||
</li>
|
||
<li class="toc-h3 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#the-neural-network">
|
||
The Neural Network
|
||
</a>
|
||
</li>
|
||
<li class="toc-h3 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#multiclass-classification">
|
||
Multiclass classification
|
||
</a>
|
||
</li>
|
||
</ul>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#testing-the-xor-gate-and-other-gates">
|
||
Testing the XOR gate and other gates
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#solving-differential-equations-with-deep-learning">
|
||
Solving differential equations with Deep Learning
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#ordinary-differential-equations-first">
|
||
Ordinary Differential Equations first
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#the-trial-solution">
|
||
The trial solution
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#minimization-process">
|
||
Minimization process
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#minimizing-the-cost-function-using-gradient-descent-and-automatic-differentiation">
|
||
Minimizing the cost function using gradient descent and automatic differentiation
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#example-exponential-decay">
|
||
Example: Exponential decay
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#the-function-to-solve-for">
|
||
The function to solve for
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#id4">
|
||
The trial solution
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#setup-of-network">
|
||
Setup of Network
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#reformulating-the-problem">
|
||
Reformulating the problem
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#more-technicalities">
|
||
More technicalities
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#more-details">
|
||
More details
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#a-possible-implementation-of-a-neural-network">
|
||
A possible implementation of a neural network
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#technicalities">
|
||
Technicalities
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#final-technicalities-i">
|
||
Final technicalities I
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#final-technicalities-ii">
|
||
Final technicalities II
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#final-technicalities-iii">
|
||
Final technicalities III
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#final-technicalities-iv">
|
||
Final technicalities IV
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#back-propagation">
|
||
Back propagation
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#gradient-descent">
|
||
Gradient descent
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#the-code-for-solving-the-ode">
|
||
The code for solving the ODE
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#the-network-with-one-input-layer-specified-number-of-hidden-layers-and-one-output-layer">
|
||
The network with one input layer, specified number of hidden layers, and one output layer
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#example-population-growth">
|
||
Example: Population growth
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#setting-up-the-problem">
|
||
Setting up the problem
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#id5">
|
||
The trial solution
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#the-program-using-autograd">
|
||
The program using Autograd
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#using-forward-euler-to-solve-the-ode">
|
||
Using forward Euler to solve the ODE
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#example-solving-the-one-dimensional-poisson-equation">
|
||
Example: Solving the one dimensional Poisson equation
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#the-specific-equation-to-solve-for">
|
||
The specific equation to solve for
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#solving-the-equation-using-autograd">
|
||
Solving the equation using Autograd
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#comparing-with-a-numerical-scheme">
|
||
Comparing with a numerical scheme
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#setting-up-the-code">
|
||
Setting up the code
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#partial-differential-equations">
|
||
Partial Differential Equations
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#type-of-problem">
|
||
Type of problem
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#network-requirements">
|
||
Network requirements
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#id6">
|
||
More details
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#example-the-diffusion-equation">
|
||
Example: The diffusion equation
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#defining-the-problem">
|
||
Defining the problem
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#setting-up-the-network-using-autograd">
|
||
Setting up the network using Autograd
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#setting-up-the-network-using-autograd-the-trial-solution">
|
||
Setting up the network using Autograd; The trial solution
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#why-the-jacobian">
|
||
Why the jacobian?
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#setting-up-the-network-using-autograd-the-full-program">
|
||
Setting up the network using Autograd; The full program
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#example-solving-the-wave-equation-with-neural-networks">
|
||
Example: Solving the wave equation with Neural Networks
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#the-problem-to-solve-for">
|
||
The problem to solve for
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#id7">
|
||
The trial solution
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#the-analytical-solution">
|
||
The analytical solution
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#solving-the-wave-equation-the-full-program-using-autograd">
|
||
Solving the wave equation - the full program using Autograd
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#resources-on-differential-equations-and-deep-learning">
|
||
Resources on differential equations and deep learning
|
||
</a>
|
||
</li>
|
||
</ul>
|
||
|
||
</nav>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<div id="main-content" class="row">
|
||
<div class="col-12 col-md-9 pl-md-3 pr-md-0">
|
||
<!-- Table of contents that is only displayed when printing the page -->
|
||
<div id="jb-print-docs-body" class="onlyprint">
|
||
<h1>Week 43: Deep Learning: Constructing a Neural Network code and solving differential equations</h1>
|
||
<!-- Table of contents -->
|
||
<div id="print-main-content">
|
||
<div id="jb-print-toc">
|
||
|
||
<div>
|
||
<h2> Contents </h2>
|
||
</div>
|
||
<nav aria-label="Page">
|
||
<ul class="visible nav section-nav flex-column">
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#plans-for-week-43">
|
||
Plans for week 43
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#exercises-and-lab-session-week-43">
|
||
Exercises and lab session week 43
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#mathematics-of-deep-learning">
|
||
Mathematics of deep learning
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#reminder-on-books-with-hands-on-material-and-codes">
|
||
Reminder on books with hands-on material and codes
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#reading-recommendations">
|
||
Reading recommendations
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#using-automatic-differentiation">
|
||
Using Automatic differentiation
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#back-propagation-and-automatic-differentiation">
|
||
Back propagation and automatic differentiation
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#lecture-monday-october-21">
|
||
Lecture Monday October 21
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#setting-up-the-back-propagation-algorithm-and-algorithm-for-a-feed-forward-nn-initalizations">
|
||
Setting up the back propagation algorithm and algorithm for a feed forward NN, initalizations
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#setting-up-the-back-propagation-algorithm-part-1">
|
||
Setting up the back propagation algorithm, part 1
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#setting-up-the-back-propagation-algorithm-part-2">
|
||
Setting up the back propagation algorithm, part 2
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#setting-up-the-back-propagation-algorithm-part-3">
|
||
Setting up the Back propagation algorithm, part 3
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#updating-the-gradients">
|
||
Updating the gradients
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#activation-functions">
|
||
Activation functions
|
||
</a>
|
||
<ul class="nav section-nav flex-column">
|
||
<li class="toc-h3 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#activation-functions-examples">
|
||
Activation functions, examples
|
||
</a>
|
||
</li>
|
||
</ul>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#the-relu-function-family">
|
||
The RELU function family
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#elu-function">
|
||
ELU function
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#which-activation-function-should-we-use">
|
||
Which activation function should we use?
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#more-on-activation-functions-output-layers">
|
||
More on activation functions, output layers
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#setting-up-a-multi-layer-perceptron-model-for-classification">
|
||
Setting up a Multi-layer perceptron model for classification
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#defining-the-cost-function">
|
||
Defining the cost function
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#example-binary-classification-problem">
|
||
Example: binary classification problem
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#the-softmax-function">
|
||
The Softmax function
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#developing-a-code-for-doing-neural-networks-with-back-propagation">
|
||
Developing a code for doing neural networks with back propagation
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#collect-and-pre-process-data">
|
||
Collect and pre-process data
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#train-and-test-datasets">
|
||
Train and test datasets
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#define-model-and-architecture">
|
||
Define model and architecture
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#layers">
|
||
Layers
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#weights-and-biases">
|
||
Weights and biases
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#feed-forward-pass">
|
||
Feed-forward pass
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#matrix-multiplications">
|
||
Matrix multiplications
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#choose-cost-function-and-optimizer">
|
||
Choose cost function and optimizer
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#optimizing-the-cost-function">
|
||
Optimizing the cost function
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#regularization">
|
||
Regularization
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#matrix-multiplication">
|
||
Matrix multiplication
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#improving-performance">
|
||
Improving performance
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#full-object-oriented-implementation">
|
||
Full object-oriented implementation
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#evaluate-model-performance-on-test-data">
|
||
Evaluate model performance on test data
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#adjust-hyperparameters">
|
||
Adjust hyperparameters
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#visualization">
|
||
Visualization
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#scikit-learn-implementation">
|
||
scikit-learn implementation
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#id1">
|
||
Visualization
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#building-neural-networks-in-tensorflow-and-keras">
|
||
Building neural networks in Tensorflow and Keras
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#tensorflow">
|
||
Tensorflow
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#using-keras">
|
||
Using Keras
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#id2">
|
||
Collect and pre-process data
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#the-breast-cancer-data-now-with-keras">
|
||
The Breast Cancer Data, now with Keras
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#building-a-neural-network-code">
|
||
Building a neural network code
|
||
</a>
|
||
<ul class="nav section-nav flex-column">
|
||
<li class="toc-h3 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#learning-rate-methods">
|
||
Learning rate methods
|
||
</a>
|
||
</li>
|
||
<li class="toc-h3 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#usage-of-the-above-learning-rate-schedulers">
|
||
Usage of the above learning rate schedulers
|
||
</a>
|
||
</li>
|
||
<li class="toc-h3 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#cost-functions">
|
||
Cost functions
|
||
</a>
|
||
</li>
|
||
<li class="toc-h3 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#id3">
|
||
Activation functions
|
||
</a>
|
||
</li>
|
||
<li class="toc-h3 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#the-neural-network">
|
||
The Neural Network
|
||
</a>
|
||
</li>
|
||
<li class="toc-h3 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#multiclass-classification">
|
||
Multiclass classification
|
||
</a>
|
||
</li>
|
||
</ul>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#testing-the-xor-gate-and-other-gates">
|
||
Testing the XOR gate and other gates
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#solving-differential-equations-with-deep-learning">
|
||
Solving differential equations with Deep Learning
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#ordinary-differential-equations-first">
|
||
Ordinary Differential Equations first
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#the-trial-solution">
|
||
The trial solution
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#minimization-process">
|
||
Minimization process
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#minimizing-the-cost-function-using-gradient-descent-and-automatic-differentiation">
|
||
Minimizing the cost function using gradient descent and automatic differentiation
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#example-exponential-decay">
|
||
Example: Exponential decay
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#the-function-to-solve-for">
|
||
The function to solve for
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#id4">
|
||
The trial solution
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#setup-of-network">
|
||
Setup of Network
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#reformulating-the-problem">
|
||
Reformulating the problem
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#more-technicalities">
|
||
More technicalities
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#more-details">
|
||
More details
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#a-possible-implementation-of-a-neural-network">
|
||
A possible implementation of a neural network
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#technicalities">
|
||
Technicalities
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#final-technicalities-i">
|
||
Final technicalities I
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#final-technicalities-ii">
|
||
Final technicalities II
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#final-technicalities-iii">
|
||
Final technicalities III
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#final-technicalities-iv">
|
||
Final technicalities IV
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#back-propagation">
|
||
Back propagation
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#gradient-descent">
|
||
Gradient descent
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#the-code-for-solving-the-ode">
|
||
The code for solving the ODE
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#the-network-with-one-input-layer-specified-number-of-hidden-layers-and-one-output-layer">
|
||
The network with one input layer, specified number of hidden layers, and one output layer
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#example-population-growth">
|
||
Example: Population growth
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#setting-up-the-problem">
|
||
Setting up the problem
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#id5">
|
||
The trial solution
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#the-program-using-autograd">
|
||
The program using Autograd
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#using-forward-euler-to-solve-the-ode">
|
||
Using forward Euler to solve the ODE
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#example-solving-the-one-dimensional-poisson-equation">
|
||
Example: Solving the one dimensional Poisson equation
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#the-specific-equation-to-solve-for">
|
||
The specific equation to solve for
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#solving-the-equation-using-autograd">
|
||
Solving the equation using Autograd
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#comparing-with-a-numerical-scheme">
|
||
Comparing with a numerical scheme
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#setting-up-the-code">
|
||
Setting up the code
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#partial-differential-equations">
|
||
Partial Differential Equations
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#type-of-problem">
|
||
Type of problem
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#network-requirements">
|
||
Network requirements
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#id6">
|
||
More details
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#example-the-diffusion-equation">
|
||
Example: The diffusion equation
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#defining-the-problem">
|
||
Defining the problem
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#setting-up-the-network-using-autograd">
|
||
Setting up the network using Autograd
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#setting-up-the-network-using-autograd-the-trial-solution">
|
||
Setting up the network using Autograd; The trial solution
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#why-the-jacobian">
|
||
Why the jacobian?
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#setting-up-the-network-using-autograd-the-full-program">
|
||
Setting up the network using Autograd; The full program
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#example-solving-the-wave-equation-with-neural-networks">
|
||
Example: Solving the wave equation with Neural Networks
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#the-problem-to-solve-for">
|
||
The problem to solve for
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#id7">
|
||
The trial solution
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#the-analytical-solution">
|
||
The analytical solution
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#solving-the-wave-equation-the-full-program-using-autograd">
|
||
Solving the wave equation - the full program using Autograd
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#resources-on-differential-equations-and-deep-learning">
|
||
Resources on differential equations and deep learning
|
||
</a>
|
||
</li>
|
||
</ul>
|
||
|
||
</nav>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
|
||
<div>
|
||
|
||
<!-- HTML file automatically generated from DocOnce source (https://github.com/doconce/doconce/)
|
||
doconce format html week43.do.txt --no_mako -->
|
||
<!-- dom:TITLE: Week 43: Deep Learning: Constructing a Neural Network code and solving differential equations --><div class="tex2jax_ignore mathjax_ignore section" id="week-43-deep-learning-constructing-a-neural-network-code-and-solving-differential-equations">
|
||
<h1>Week 43: Deep Learning: Constructing a Neural Network code and solving differential equations<a class="headerlink" href="#week-43-deep-learning-constructing-a-neural-network-code-and-solving-differential-equations" title="Permalink to this headline">¶</a></h1>
|
||
<p><strong>Morten Hjorth-Jensen</strong>, Department of Physics, University of Oslo and Department of Physics and Astronomy and Facility for Rare Isotope Beams, Michigan State University</p>
|
||
<p>Date: <strong>October 21, 2024</strong></p>
|
||
<div class="section" id="plans-for-week-43">
|
||
<h2>Plans for week 43<a class="headerlink" href="#plans-for-week-43" title="Permalink to this headline">¶</a></h2>
|
||
<p><strong>Material for the lecture on Monday October 21, 2024.</strong></p>
|
||
<ul class="simple">
|
||
<li><p>Building our own Feed-forward Neural Network with intro to Tensorflow</p></li>
|
||
<li><p>Solving differential equations with Neural Networks</p></li>
|
||
<li><p>“Video of lecture at <a class="reference external" href="https://youtu.be/vkBNTn-MLqs">https://youtu.be/vkBNTn-MLqs</a></p></li>
|
||
</ul>
|
||
<!-- * [Whiteboard notes](https://github.com/CompPhysics/MachineLearning/blob/master/doc/HandWrittenNotes/2024/NotesOct21.pdf) --></div>
|
||
<div class="section" id="exercises-and-lab-session-week-43">
|
||
<h2>Exercises and lab session week 43<a class="headerlink" href="#exercises-and-lab-session-week-43" title="Permalink to this headline">¶</a></h2>
|
||
<p><strong>Lab sessions on Tuesday and Wednesday.</strong></p>
|
||
<ul class="simple">
|
||
<li><p>Exercise on writing your own neural network code</p></li>
|
||
<li><p>The exercises this week will be continued next week as well</p></li>
|
||
<li><p>Discussion of project 2</p></li>
|
||
</ul>
|
||
</div>
|
||
<div class="section" id="mathematics-of-deep-learning">
|
||
<h2>Mathematics of deep learning<a class="headerlink" href="#mathematics-of-deep-learning" title="Permalink to this headline">¶</a></h2>
|
||
<p><strong>Two recent books online.</strong></p>
|
||
<ol class="simple">
|
||
<li><p>The Modern Mathematics of Deep Learning, by Julius Berner, Philipp Grohs, Gitta Kutyniok, Philipp Petersen at <a class="reference external" href="https://arxiv.org/abs/2105.04026">https://arxiv.org/abs/2105.04026</a>, published as <a class="reference external" href="https://doi.org/10.1017/9781009025096.002">Mathematical Aspects of Deep Learning, pp. 1-111. Cambridge University Press, 2022</a></p></li>
|
||
<li><p>Mathematical Introduction to Deep Learning: Methods, Implementations, and Theory, Arnulf Jentzen, Benno Kuckuck, Philippe von Wurstemberger at <a class="reference external" href="https://doi.org/10.48550/arXiv.2310.20360">https://doi.org/10.48550/arXiv.2310.20360</a></p></li>
|
||
</ol>
|
||
</div>
|
||
<div class="section" id="reminder-on-books-with-hands-on-material-and-codes">
|
||
<h2>Reminder on books with hands-on material and codes<a class="headerlink" href="#reminder-on-books-with-hands-on-material-and-codes" title="Permalink to this headline">¶</a></h2>
|
||
<ul class="simple">
|
||
<li><p>Sebastian Rashcka et al, Machine learning with Scikit-Learn and PyTorch at <a class="reference external" href="https://sebastianraschka.com/blog/2022/ml-pytorch-book.html">https://sebastianraschka.com/blog/2022/ml-pytorch-book.html</a></p></li>
|
||
</ul>
|
||
</div>
|
||
<div class="section" id="reading-recommendations">
|
||
<h2>Reading recommendations<a class="headerlink" href="#reading-recommendations" title="Permalink to this headline">¶</a></h2>
|
||
<ol class="simple">
|
||
<li><p>Rashkca et al., chapter 11, jupyter-notebook sent separately, from GitHub site at <a class="reference external" href="https://github.com/rasbt/machine-learning-book">https://github.com/rasbt/machine-learning-book</a>. See also chapters 12 and 13 on using Pytorch to make a Neural network code.</p></li>
|
||
<li><p>Goodfellow et al, chapter 6 and 7 contain most of the neural network background.</p></li>
|
||
</ol>
|
||
</div>
|
||
<div class="section" id="using-automatic-differentiation">
|
||
<h2>Using Automatic differentiation<a class="headerlink" href="#using-automatic-differentiation" title="Permalink to this headline">¶</a></h2>
|
||
<p>In our discussions of ordinary differential equations and neural network codes
|
||
we will also study the usage of Autograd, see for example <a class="reference external" href="https://www.youtube.com/watch?v=fRf4l5qaX1M&amp;ab_channel=AlexSmola">https://www.youtube.com/watch?v=fRf4l5qaX1M&amp;ab_channel=AlexSmola</a> in computing gradients for deep learning. For the documentation of Autograd and examples see the lectures slides from <a class="reference external" href="https://compphysics.github.io/MachineLearning/doc/pub/week39/html/week39.html">week 39</a> and the Autograd documentation at <a class="reference external" href="https://github.com/HIPS/autograd">https://github.com/HIPS/autograd</a>.</p>
|
||
</div>
|
||
<div class="section" id="back-propagation-and-automatic-differentiation">
|
||
<h2>Back propagation and automatic differentiation<a class="headerlink" href="#back-propagation-and-automatic-differentiation" title="Permalink to this headline">¶</a></h2>
|
||
<p>For more details on the back propagation algorithm and automatic differentiation see</p>
|
||
<ol class="simple">
|
||
<li><p><a class="reference external" href="https://www.jmlr.org/papers/volume18/17-468/17-468.pdf">https://www.jmlr.org/papers/volume18/17-468/17-468.pdf</a></p></li>
|
||
<li><p><a class="reference external" href="https://deepimaging.github.io/lectures/lecture_11_Backpropagation.pdf">https://deepimaging.github.io/lectures/lecture_11_Backpropagation.pdf</a></p></li>
|
||
<li><p>Slides 12-44 at <a class="reference external" href="http://cs231n.stanford.edu/slides/2017/cs231n_2017_lecture4.pdf">http://cs231n.stanford.edu/slides/2017/cs231n_2017_lecture4.pdf</a></p></li>
|
||
</ol>
|
||
</div>
|
||
<div class="section" id="lecture-monday-october-21">
|
||
<h2>Lecture Monday October 21<a class="headerlink" href="#lecture-monday-october-21" title="Permalink to this headline">¶</a></h2>
|
||
</div>
|
||
<div class="section" id="setting-up-the-back-propagation-algorithm-and-algorithm-for-a-feed-forward-nn-initalizations">
|
||
<h2>Setting up the back propagation algorithm and algorithm for a feed forward NN, initalizations<a class="headerlink" href="#setting-up-the-back-propagation-algorithm-and-algorithm-for-a-feed-forward-nn-initalizations" title="Permalink to this headline">¶</a></h2>
|
||
<p>This is a reminder from where we ended last week.</p>
|
||
<p><strong>The architecture (our model).</strong></p>
|
||
<ol class="simple">
|
||
<li><p>Set up your inputs and outputs (scalars, vectors, matrices or higher-order arrays)</p></li>
|
||
<li><p>Define the number of hidden layers and hidden nodes</p></li>
|
||
<li><p>Define activation functions for hidden layers and output layers</p></li>
|
||
<li><p>Define optimizer (plan learning rate, momentum, ADAgrad, RMSprop, ADAM etc) and array of initial learning rates</p></li>
|
||
<li><p>Define cost function and possible regularization terms with hyperparameters</p></li>
|
||
<li><p>Initialize weights and biases</p></li>
|
||
<li><p>Fix number of iterations for the feed forward part and back propagation part</p></li>
|
||
</ol>
|
||
</div>
|
||
<div class="section" id="setting-up-the-back-propagation-algorithm-part-1">
|
||
<h2>Setting up the back propagation algorithm, part 1<a class="headerlink" href="#setting-up-the-back-propagation-algorithm-part-1" title="Permalink to this headline">¶</a></h2>
|
||
<p>Let us write this out in the form of an algorithm.</p>
|
||
<p><strong>First</strong>, we set up the input data <span class="math notranslate nohighlight">\(\boldsymbol{x}\)</span> and the activations
|
||
<span class="math notranslate nohighlight">\(\boldsymbol{z}_1\)</span> of the input layer and compute the activation function and
|
||
the pertinent outputs <span class="math notranslate nohighlight">\(\boldsymbol{a}^1\)</span>.</p>
|
||
<p><strong>Secondly</strong>, we perform then the feed forward till we reach the output
|
||
layer and compute all <span class="math notranslate nohighlight">\(\boldsymbol{z}_l\)</span> of the input layer and compute the
|
||
activation function and the pertinent outputs <span class="math notranslate nohighlight">\(\boldsymbol{a}^l\)</span> for
|
||
<span class="math notranslate nohighlight">\(l=1,2,3,\dots,L\)</span>.</p>
|
||
<p><strong>Notation</strong>: The first hidden layer has <span class="math notranslate nohighlight">\(l=1\)</span> as label and the final output layer has <span class="math notranslate nohighlight">\(l=L\)</span>.</p>
|
||
</div>
|
||
<div class="section" id="setting-up-the-back-propagation-algorithm-part-2">
|
||
<h2>Setting up the back propagation algorithm, part 2<a class="headerlink" href="#setting-up-the-back-propagation-algorithm-part-2" title="Permalink to this headline">¶</a></h2>
|
||
<p>Thereafter we compute the ouput error <span class="math notranslate nohighlight">\(\boldsymbol{\delta}^L\)</span> by computing all</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\delta_j^L = \sigma'(z_j^L)\frac{\partial {\cal C}}{\partial (a_j^L)}.
|
||
\]</div>
|
||
<p>Then we compute the back propagate error for each <span class="math notranslate nohighlight">\(l=L-1,L-2,\dots,1\)</span> as</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\delta_j^l = \sum_k \delta_k^{l+1}w_{kj}^{l+1}\sigma'(z_j^l).
|
||
\]</div>
|
||
</div>
|
||
<div class="section" id="setting-up-the-back-propagation-algorithm-part-3">
|
||
<h2>Setting up the Back propagation algorithm, part 3<a class="headerlink" href="#setting-up-the-back-propagation-algorithm-part-3" title="Permalink to this headline">¶</a></h2>
|
||
<p>Finally, we update the weights and the biases using gradient descent
|
||
for each <span class="math notranslate nohighlight">\(l=L-1,L-2,\dots,1\)</span> (the first hidden layer) and update the weights and biases
|
||
according to the rules</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
w_{ij}^l\leftarrow = w_{ij}^l- \eta \delta_j^la_i^{l-1},
|
||
\]</div>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
b_j^l \leftarrow b_j^l-\eta \frac{\partial {\cal C}}{\partial b_j^l}=b_j^l-\eta \delta_j^l,
|
||
\]</div>
|
||
<p>with <span class="math notranslate nohighlight">\(\eta\)</span> being the learning rate.</p>
|
||
</div>
|
||
<div class="section" id="updating-the-gradients">
|
||
<h2>Updating the gradients<a class="headerlink" href="#updating-the-gradients" title="Permalink to this headline">¶</a></h2>
|
||
<p>With the back propagate error for each <span class="math notranslate nohighlight">\(l=L-1,L-2,\dots,1\)</span> as</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\delta_j^l = \sum_k \delta_k^{l+1}w_{kj}^{l+1}\sigma'(z_j^l),
|
||
\]</div>
|
||
<p>we update the weights and the biases using gradient descent for each <span class="math notranslate nohighlight">\(l=L-1,L-2,\dots,1\)</span> and update the weights and biases according to the rules</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
w_{ij}^l\leftarrow = w_{ij}^l- \eta \delta_j^la_i^{l-1},
|
||
\]</div>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
b_j^l \leftarrow b_j^l-\eta \frac{\partial {\cal C}}{\partial b_j^l}=b_j^l-\eta \delta_j^l,
|
||
\]</div>
|
||
</div>
|
||
<div class="section" id="activation-functions">
|
||
<h2>Activation functions<a class="headerlink" href="#activation-functions" title="Permalink to this headline">¶</a></h2>
|
||
<p>A property that characterizes a neural network, other than its
|
||
connectivity, is the choice of activation function(s). The following
|
||
restrictions are imposed on an activation function for an FFNN to
|
||
fulfill the universal approximation theorem</p>
|
||
<ul class="simple">
|
||
<li><p>Non-constant</p></li>
|
||
<li><p>Bounded</p></li>
|
||
<li><p>Monotonically-increasing</p></li>
|
||
<li><p>Continuous</p></li>
|
||
</ul>
|
||
<div class="section" id="activation-functions-examples">
|
||
<h3>Activation functions, examples<a class="headerlink" href="#activation-functions-examples" title="Permalink to this headline">¶</a></h3>
|
||
<p>Typical examples are the logistic <em>Sigmoid</em></p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\sigma(x) = \frac{1}{1 + e^{-x}},
|
||
\]</div>
|
||
<p>and the <em>hyperbolic tangent</em> function</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\sigma(x) = \tanh(x)
|
||
\]</div>
|
||
</div>
|
||
</div>
|
||
<div class="section" id="the-relu-function-family">
|
||
<h2>The RELU function family<a class="headerlink" href="#the-relu-function-family" title="Permalink to this headline">¶</a></h2>
|
||
<p>The ReLU activation function suffers from a problem known as the dying
|
||
ReLUs: during training, some neurons effectively die, meaning they
|
||
stop outputting anything other than 0.</p>
|
||
<p>In some cases, you may find that half of your network’s neurons are
|
||
dead, especially if you used a large learning rate. During training,
|
||
if a neuron’s weights get updated such that the weighted sum of the
|
||
neuron’s inputs is negative, it will start outputting 0. When this
|
||
happen, the neuron is unlikely to come back to life since the gradient
|
||
of the ReLU function is 0 when its input is negative.</p>
|
||
</div>
|
||
<div class="section" id="elu-function">
|
||
<h2>ELU function<a class="headerlink" href="#elu-function" title="Permalink to this headline">¶</a></h2>
|
||
<p>To solve this problem, nowadays practitioners use a variant of the
|
||
ReLU function, such as the leaky ReLU discussed above or the so-called
|
||
exponential linear unit (ELU) function</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[\begin{split}
|
||
ELU(z) = \left\{\begin{array}{cc} \alpha\left( \exp{(z)}-1\right) & z < 0,\\ z & z \ge 0.\end{array}\right.
|
||
\end{split}\]</div>
|
||
</div>
|
||
<div class="section" id="which-activation-function-should-we-use">
|
||
<h2>Which activation function should we use?<a class="headerlink" href="#which-activation-function-should-we-use" title="Permalink to this headline">¶</a></h2>
|
||
<p>In general it seems that the ELU activation function is better than
|
||
the leaky ReLU function (and its variants), which is better than
|
||
ReLU. ReLU performs better than <span class="math notranslate nohighlight">\(\tanh\)</span> which in turn performs better
|
||
than the logistic function.</p>
|
||
<p>If runtime performance is an issue, then you may opt for the leaky
|
||
ReLU function over the ELU function If you don’t want to tweak yet
|
||
another hyperparameter, you may just use the default <span class="math notranslate nohighlight">\(\alpha\)</span> of
|
||
<span class="math notranslate nohighlight">\(0.01\)</span> for the leaky ReLU, and <span class="math notranslate nohighlight">\(1\)</span> for ELU. If you have spare time and
|
||
computing power, you can use cross-validation or bootstrap to evaluate
|
||
other activation functions.</p>
|
||
</div>
|
||
<div class="section" id="more-on-activation-functions-output-layers">
|
||
<h2>More on activation functions, output layers<a class="headerlink" href="#more-on-activation-functions-output-layers" title="Permalink to this headline">¶</a></h2>
|
||
<p>In most cases you can use the ReLU activation function in the hidden
|
||
layers (or one of its variants).</p>
|
||
<p>It is a bit faster to compute than other activation functions, and the
|
||
gradient descent optimization does in general not get stuck.</p>
|
||
<p><strong>For the output layer:</strong></p>
|
||
<ul class="simple">
|
||
<li><p>For classification the softmax activation function is generally a good choice for classification tasks (when the classes are mutually exclusive).</p></li>
|
||
<li><p>For regression tasks, you can simply use no activation function at all.</p></li>
|
||
</ul>
|
||
</div>
|
||
<div class="section" id="setting-up-a-multi-layer-perceptron-model-for-classification">
|
||
<h2>Setting up a Multi-layer perceptron model for classification<a class="headerlink" href="#setting-up-a-multi-layer-perceptron-model-for-classification" title="Permalink to this headline">¶</a></h2>
|
||
<p>We are now gong to develop an example based on the MNIST data
|
||
base. This is a classification problem and we need to use our
|
||
cross-entropy function we discussed in connection with logistic
|
||
regression. The cross-entropy defines our cost function for the
|
||
classificaton problems with neural networks.</p>
|
||
<p>In binary classification with two classes <span class="math notranslate nohighlight">\((0, 1)\)</span> we define the
|
||
logistic/sigmoid function as the probability that a particular input
|
||
is in class <span class="math notranslate nohighlight">\(0\)</span> or <span class="math notranslate nohighlight">\(1\)</span>. This is possible because the logistic
|
||
function takes any input from the real numbers and inputs a number
|
||
between 0 and 1, and can therefore be interpreted as a probability. It
|
||
also has other nice properties, such as a derivative that is simple to
|
||
calculate.</p>
|
||
<p>For an input <span class="math notranslate nohighlight">\(\boldsymbol{a}\)</span> from the hidden layer, the probability that the input <span class="math notranslate nohighlight">\(\boldsymbol{x}\)</span>
|
||
is in class 0 or 1 is just. We let <span class="math notranslate nohighlight">\(\theta\)</span> represent the unknown weights and biases to be adjusted by our equations). The variable <span class="math notranslate nohighlight">\(x\)</span>
|
||
represents our activation values <span class="math notranslate nohighlight">\(z\)</span>. We have</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
P(y = 0 \mid \boldsymbol{x}, \boldsymbol{\theta}) = \frac{1}{1 + \exp{(- \boldsymbol{x}})} ,
|
||
\]</div>
|
||
<p>and</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
P(y = 1 \mid \boldsymbol{x}, \boldsymbol{\theta}) = 1 - P(y = 0 \mid \boldsymbol{x}, \boldsymbol{\theta}) ,
|
||
\]</div>
|
||
<p>where <span class="math notranslate nohighlight">\(y \in \{0, 1\}\)</span> and <span class="math notranslate nohighlight">\(\boldsymbol{\theta}\)</span> represents the weights and biases
|
||
of our network.</p>
|
||
</div>
|
||
<div class="section" id="defining-the-cost-function">
|
||
<h2>Defining the cost function<a class="headerlink" href="#defining-the-cost-function" title="Permalink to this headline">¶</a></h2>
|
||
<p>Our cost function is given as (see the Logistic regression lectures)</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\mathcal{C}(\boldsymbol{\theta}) = - \ln P(\mathcal{D} \mid \boldsymbol{\theta}) = - \sum_{i=1}^n
|
||
y_i \ln[P(y_i = 0)] + (1 - y_i) \ln [1 - P(y_i = 0)] = \sum_{i=1}^n \mathcal{L}_i(\boldsymbol{\theta}) .
|
||
\]</div>
|
||
<p>This last equality means that we can interpret our <em>cost</em> function as a sum over the <em>loss</em> function
|
||
for each point in the dataset <span class="math notranslate nohighlight">\(\mathcal{L}_i(\boldsymbol{\theta})\)</span>.<br />
|
||
The negative sign is just so that we can think about our algorithm as minimizing a positive number, rather
|
||
than maximizing a negative number.</p>
|
||
<p>In <em>multiclass</em> classification it is common to treat each integer label as a so called <em>one-hot</em> vector:</p>
|
||
<p><span class="math notranslate nohighlight">\(y = 5 \quad \rightarrow \quad \boldsymbol{y} = (0, 0, 0, 0, 0, 1, 0, 0, 0, 0) ,\)</span> and</p>
|
||
<p><span class="math notranslate nohighlight">\(y = 1 \quad \rightarrow \quad \boldsymbol{y} = (0, 1, 0, 0, 0, 0, 0, 0, 0, 0) ,\)</span></p>
|
||
<p>i.e. a binary bit string of length <span class="math notranslate nohighlight">\(C\)</span>, where <span class="math notranslate nohighlight">\(C = 10\)</span> is the number of classes in the MNIST dataset (numbers from <span class="math notranslate nohighlight">\(0\)</span> to <span class="math notranslate nohighlight">\(9\)</span>)..</p>
|
||
<p>If <span class="math notranslate nohighlight">\(\boldsymbol{x}_i\)</span> is the <span class="math notranslate nohighlight">\(i\)</span>-th input (image), <span class="math notranslate nohighlight">\(y_{ic}\)</span> refers to the <span class="math notranslate nohighlight">\(c\)</span>-th component of the <span class="math notranslate nohighlight">\(i\)</span>-th
|
||
output vector <span class="math notranslate nohighlight">\(\boldsymbol{y}_i\)</span>.<br />
|
||
The probability of <span class="math notranslate nohighlight">\(\boldsymbol{x}_i\)</span> being in class <span class="math notranslate nohighlight">\(c\)</span> will be given by the softmax function:</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
P(y_{ic} = 1 \mid \boldsymbol{x}_i, \boldsymbol{\theta}) = \frac{\exp{((\boldsymbol{a}_i^{hidden})^T \boldsymbol{w}_c)}}
|
||
{\sum_{c'=0}^{C-1} \exp{((\boldsymbol{a}_i^{hidden})^T \boldsymbol{w}_{c'})}} ,
|
||
\]</div>
|
||
<p>which reduces to the logistic function in the binary case.<br />
|
||
The likelihood of this <span class="math notranslate nohighlight">\(C\)</span>-class classifier
|
||
is now given as:</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
P(\mathcal{D} \mid \boldsymbol{\theta}) = \prod_{i=1}^n \prod_{c=0}^{C-1} [P(y_{ic} = 1)]^{y_{ic}} .
|
||
\]</div>
|
||
<p>Again we take the negative log-likelihood to define our cost function:</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\mathcal{C}(\boldsymbol{\theta}) = - \log{P(\mathcal{D} \mid \boldsymbol{\theta})}.
|
||
\]</div>
|
||
<p>See the logistic regression lectures for a full definition of the cost function.</p>
|
||
<p>The back propagation equations need now only a small change, namely the definition of a new cost function. We are thus ready to use the same equations as before!</p>
|
||
</div>
|
||
<div class="section" id="example-binary-classification-problem">
|
||
<h2>Example: binary classification problem<a class="headerlink" href="#example-binary-classification-problem" title="Permalink to this headline">¶</a></h2>
|
||
<p>As an example of the above, relevant for project 2 as well, let us consider a binary class. As discussed in our logistic regression lectures, we defined a cost function in terms of the parameters <span class="math notranslate nohighlight">\(\beta\)</span> as</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\mathcal{C}(\boldsymbol{\beta}) = - \sum_{i=1}^n \left(y_i\log{p(y_i \vert x_i,\boldsymbol{\beta})}+(1-y_i)\log{1-p(y_i \vert x_i,\boldsymbol{\beta})}\right),
|
||
\]</div>
|
||
<p>where we had defined the logistic (sigmoid) function</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
p(y_i =1\vert x_i,\boldsymbol{\beta})=\frac{\exp{(\beta_0+\beta_1 x_i)}}{1+\exp{(\beta_0+\beta_1 x_i)}},
|
||
\]</div>
|
||
<p>and</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
p(y_i =0\vert x_i,\boldsymbol{\beta})=1-p(y_i =1\vert x_i,\boldsymbol{\beta}).
|
||
\]</div>
|
||
<p>The parameters <span class="math notranslate nohighlight">\(\boldsymbol{\beta}\)</span> were defined using a minimization method like gradient descent or Newton-Raphson’s method.</p>
|
||
<p>Now we replace <span class="math notranslate nohighlight">\(x_i\)</span> with the activation <span class="math notranslate nohighlight">\(z_i^l\)</span> for a given layer <span class="math notranslate nohighlight">\(l\)</span> and the outputs as <span class="math notranslate nohighlight">\(y_i=a_i^l=f(z_i^l)\)</span>, with <span class="math notranslate nohighlight">\(z_i^l\)</span> now being a function of the weights <span class="math notranslate nohighlight">\(w_{ij}^l\)</span> and biases <span class="math notranslate nohighlight">\(b_i^l\)</span>.
|
||
We have then</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
a_i^l = y_i = \frac{\exp{(z_i^l)}}{1+\exp{(z_i^l)}},
|
||
\]</div>
|
||
<p>with</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
z_i^l = \sum_{j}w_{ij}^l a_j^{l-1}+b_i^l,
|
||
\]</div>
|
||
<p>where the superscript <span class="math notranslate nohighlight">\(l-1\)</span> indicates that these are the outputs from layer <span class="math notranslate nohighlight">\(l-1\)</span>.
|
||
Our cost function at the final layer <span class="math notranslate nohighlight">\(l=L\)</span> is now</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\mathcal{C}(\boldsymbol{W}) = - \sum_{i=1}^n \left(t_i\log{a_i^L}+(1-t_i)\log{(1-a_i^L)}\right),
|
||
\]</div>
|
||
<p>where we have defined the targets <span class="math notranslate nohighlight">\(t_i\)</span>. The derivatives of the cost function with respect to the output <span class="math notranslate nohighlight">\(a_i^L\)</span> are then easily calculated and we get</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\frac{\partial \mathcal{C}(\boldsymbol{W})}{\partial a_i^L} = \frac{a_i^L-t_i}{a_i^L(1-a_i^L)}.
|
||
\]</div>
|
||
<p>In case we use another activation function than the logistic one, we need to evaluate other derivatives.</p>
|
||
</div>
|
||
<div class="section" id="the-softmax-function">
|
||
<h2>The Softmax function<a class="headerlink" href="#the-softmax-function" title="Permalink to this headline">¶</a></h2>
|
||
<p>In case we employ the more general case given by the Softmax equation, we need to evaluate the derivative of the activation function with respect to the activation <span class="math notranslate nohighlight">\(z_i^l\)</span>, that is we need</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\frac{\partial f(z_i^l)}{\partial w_{jk}^l} =
|
||
\frac{\partial f(z_i^l)}{\partial z_j^l} \frac{\partial z_j^l}{\partial w_{jk}^l}= \frac{\partial f(z_i^l)}{\partial z_j^l}a_k^{l-1}.
|
||
\]</div>
|
||
<p>For the Softmax function we have</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
f(z_i^l) = \frac{\exp{(z_i^l)}}{\sum_{m=1}^K\exp{(z_m^l)}}.
|
||
\]</div>
|
||
<p>Its derivative with respect to <span class="math notranslate nohighlight">\(z_j^l\)</span> gives</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\frac{\partial f(z_i^l)}{\partial z_j^l}= f(z_i^l)\left(\delta_{ij}-f(z_j^l)\right),
|
||
\]</div>
|
||
<p>which in case of the simply binary model reduces to having <span class="math notranslate nohighlight">\(i=j\)</span>.</p>
|
||
</div>
|
||
<div class="section" id="developing-a-code-for-doing-neural-networks-with-back-propagation">
|
||
<h2>Developing a code for doing neural networks with back propagation<a class="headerlink" href="#developing-a-code-for-doing-neural-networks-with-back-propagation" title="Permalink to this headline">¶</a></h2>
|
||
<p>One can identify a set of key steps when using neural networks to solve supervised learning problems:</p>
|
||
<ol class="simple">
|
||
<li><p>Collect and pre-process data</p></li>
|
||
<li><p>Define model and architecture</p></li>
|
||
<li><p>Choose cost function and optimizer</p></li>
|
||
<li><p>Train the model</p></li>
|
||
<li><p>Evaluate model performance on test data</p></li>
|
||
<li><p>Adjust hyperparameters (if necessary, network architecture)</p></li>
|
||
</ol>
|
||
</div>
|
||
<div class="section" id="collect-and-pre-process-data">
|
||
<h2>Collect and pre-process data<a class="headerlink" href="#collect-and-pre-process-data" title="Permalink to this headline">¶</a></h2>
|
||
<p>Here we will be using the MNIST dataset, which is readily available through the <strong>scikit-learn</strong>
|
||
package. You may also find it for example <a class="reference external" href="http://yann.lecun.com/exdb/mnist/">here</a>.<br />
|
||
The <em>MNIST</em> (Modified National Institute of Standards and Technology) database is a large database
|
||
of handwritten digits that is commonly used for training various image processing systems.<br />
|
||
The MNIST dataset consists of 70 000 images of size <span class="math notranslate nohighlight">\(28\times 28\)</span> pixels, each labeled from 0 to 9.<br />
|
||
The scikit-learn dataset we will use consists of a selection of 1797 images of size <span class="math notranslate nohighlight">\(8\times 8\)</span> collected and processed from this database.</p>
|
||
<p>To feed data into a feed-forward neural network we need to represent
|
||
the inputs as a design/feature matrix <span class="math notranslate nohighlight">\(X = (n_{inputs}, n_{features})\)</span>. Each
|
||
row represents an <em>input</em>, in this case a handwritten digit, and
|
||
each column represents a <em>feature</em>, in this case a pixel. The
|
||
correct answers, also known as <em>labels</em> or <em>targets</em> are
|
||
represented as a 1D array of integers
|
||
<span class="math notranslate nohighlight">\(Y = (n_{inputs}) = (5, 3, 1, 8,...)\)</span>.</p>
|
||
<p>As an example, say we want to build a neural network using supervised learning to predict Body-Mass Index (BMI) from
|
||
measurements of height (in m)<br />
|
||
and weight (in kg). If we have measurements of 5 people the design/feature matrix could be for example:</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[\begin{split} X = \begin{bmatrix}
|
||
1.85 & 81\\
|
||
1.71 & 65\\
|
||
1.95 & 103\\
|
||
1.55 & 42\\
|
||
1.63 & 56
|
||
\end{bmatrix} ,\end{split}\]</div>
|
||
<p>and the targets would be:</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[ Y = (23.7, 22.2, 27.1, 17.5, 21.1) \]</div>
|
||
<p>Since each input image is a 2D matrix, we need to flatten the image
|
||
(i.e. “unravel” the 2D matrix into a 1D array) to turn the data into a
|
||
design/feature matrix. This means we lose all spatial information in the
|
||
image, such as locality and translational invariance. More complicated
|
||
architectures such as Convolutional Neural Networks can take advantage
|
||
of such information, and are most commonly applied when analyzing
|
||
images.</p>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="o">%</span><span class="k">matplotlib</span> inline
|
||
|
||
<span class="c1"># import necessary packages</span>
|
||
<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
|
||
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
|
||
<span class="kn">from</span> <span class="nn">sklearn</span> <span class="kn">import</span> <span class="n">datasets</span>
|
||
|
||
|
||
<span class="c1"># ensure the same random numbers appear every time</span>
|
||
<span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">seed</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
|
||
|
||
<span class="c1"># display images in notebook</span>
|
||
<span class="o">%</span><span class="k">matplotlib</span> inline
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">rcParams</span><span class="p">[</span><span class="s1">'figure.figsize'</span><span class="p">]</span> <span class="o">=</span> <span class="p">(</span><span class="mi">12</span><span class="p">,</span><span class="mi">12</span><span class="p">)</span>
|
||
|
||
|
||
<span class="c1"># download MNIST dataset</span>
|
||
<span class="n">digits</span> <span class="o">=</span> <span class="n">datasets</span><span class="o">.</span><span class="n">load_digits</span><span class="p">()</span>
|
||
|
||
<span class="c1"># define inputs and labels</span>
|
||
<span class="n">inputs</span> <span class="o">=</span> <span class="n">digits</span><span class="o">.</span><span class="n">images</span>
|
||
<span class="n">labels</span> <span class="o">=</span> <span class="n">digits</span><span class="o">.</span><span class="n">target</span>
|
||
|
||
<span class="nb">print</span><span class="p">(</span><span class="s2">"inputs = (n_inputs, pixel_width, pixel_height) = "</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">inputs</span><span class="o">.</span><span class="n">shape</span><span class="p">))</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="s2">"labels = (n_inputs) = "</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">labels</span><span class="o">.</span><span class="n">shape</span><span class="p">))</span>
|
||
|
||
|
||
<span class="c1"># flatten the image</span>
|
||
<span class="c1"># the value -1 means dimension is inferred from the remaining dimensions: 8x8 = 64</span>
|
||
<span class="n">n_inputs</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">inputs</span><span class="p">)</span>
|
||
<span class="n">inputs</span> <span class="o">=</span> <span class="n">inputs</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">n_inputs</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">)</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="s2">"X = (n_inputs, n_features) = "</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">inputs</span><span class="o">.</span><span class="n">shape</span><span class="p">))</span>
|
||
|
||
|
||
<span class="c1"># choose some random images to display</span>
|
||
<span class="n">indices</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">n_inputs</span><span class="p">)</span>
|
||
<span class="n">random_indices</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">choice</span><span class="p">(</span><span class="n">indices</span><span class="p">,</span> <span class="n">size</span><span class="o">=</span><span class="mi">5</span><span class="p">)</span>
|
||
|
||
<span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">image</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">digits</span><span class="o">.</span><span class="n">images</span><span class="p">[</span><span class="n">random_indices</span><span class="p">]):</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="n">i</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">axis</span><span class="p">(</span><span class="s1">'off'</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">imshow</span><span class="p">(</span><span class="n">image</span><span class="p">,</span> <span class="n">cmap</span><span class="o">=</span><span class="n">plt</span><span class="o">.</span><span class="n">cm</span><span class="o">.</span><span class="n">gray_r</span><span class="p">,</span> <span class="n">interpolation</span><span class="o">=</span><span class="s1">'nearest'</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">"Label: </span><span class="si">%d</span><span class="s2">"</span> <span class="o">%</span> <span class="n">digits</span><span class="o">.</span><span class="n">target</span><span class="p">[</span><span class="n">random_indices</span><span class="p">[</span><span class="n">i</span><span class="p">]])</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
<div class="cell_output docutils container">
|
||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>inputs = (n_inputs, pixel_width, pixel_height) = (1797, 8, 8)
|
||
labels = (n_inputs) = (1797,)
|
||
X = (n_inputs, n_features) = (1797, 64)
|
||
</pre></div>
|
||
</div>
|
||
<img alt="_images/week43_73_1.png" src="_images/week43_73_1.png" />
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<div class="section" id="train-and-test-datasets">
|
||
<h2>Train and test datasets<a class="headerlink" href="#train-and-test-datasets" title="Permalink to this headline">¶</a></h2>
|
||
<p>Performing analysis before partitioning the dataset is a major error, that can lead to incorrect conclusions.</p>
|
||
<p>We will reserve <span class="math notranslate nohighlight">\(80 \%\)</span> of our dataset for training and <span class="math notranslate nohighlight">\(20 \%\)</span> for testing.</p>
|
||
<p>It is important that the train and test datasets are drawn randomly from our dataset, to ensure
|
||
no bias in the sampling.<br />
|
||
Say you are taking measurements of weather data to predict the weather in the coming 5 days.
|
||
You don’t want to train your model on measurements taken from the hours 00.00 to 12.00, and then test it on data
|
||
collected from 12.00 to 24.00.</p>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">sklearn.model_selection</span> <span class="kn">import</span> <span class="n">train_test_split</span>
|
||
|
||
<span class="c1"># one-liner from scikit-learn library</span>
|
||
<span class="n">train_size</span> <span class="o">=</span> <span class="mf">0.8</span>
|
||
<span class="n">test_size</span> <span class="o">=</span> <span class="mi">1</span> <span class="o">-</span> <span class="n">train_size</span>
|
||
<span class="n">X_train</span><span class="p">,</span> <span class="n">X_test</span><span class="p">,</span> <span class="n">Y_train</span><span class="p">,</span> <span class="n">Y_test</span> <span class="o">=</span> <span class="n">train_test_split</span><span class="p">(</span><span class="n">inputs</span><span class="p">,</span> <span class="n">labels</span><span class="p">,</span> <span class="n">train_size</span><span class="o">=</span><span class="n">train_size</span><span class="p">,</span>
|
||
<span class="n">test_size</span><span class="o">=</span><span class="n">test_size</span><span class="p">)</span>
|
||
|
||
<span class="c1"># equivalently in numpy</span>
|
||
<span class="k">def</span> <span class="nf">train_test_split_numpy</span><span class="p">(</span><span class="n">inputs</span><span class="p">,</span> <span class="n">labels</span><span class="p">,</span> <span class="n">train_size</span><span class="p">,</span> <span class="n">test_size</span><span class="p">):</span>
|
||
<span class="n">n_inputs</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">inputs</span><span class="p">)</span>
|
||
<span class="n">inputs_shuffled</span> <span class="o">=</span> <span class="n">inputs</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span>
|
||
<span class="n">labels_shuffled</span> <span class="o">=</span> <span class="n">labels</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span>
|
||
|
||
<span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">shuffle</span><span class="p">(</span><span class="n">inputs_shuffled</span><span class="p">)</span>
|
||
<span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">shuffle</span><span class="p">(</span><span class="n">labels_shuffled</span><span class="p">)</span>
|
||
|
||
<span class="n">train_end</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">n_inputs</span><span class="o">*</span><span class="n">train_size</span><span class="p">)</span>
|
||
<span class="n">X_train</span><span class="p">,</span> <span class="n">X_test</span> <span class="o">=</span> <span class="n">inputs_shuffled</span><span class="p">[:</span><span class="n">train_end</span><span class="p">],</span> <span class="n">inputs_shuffled</span><span class="p">[</span><span class="n">train_end</span><span class="p">:]</span>
|
||
<span class="n">Y_train</span><span class="p">,</span> <span class="n">Y_test</span> <span class="o">=</span> <span class="n">labels_shuffled</span><span class="p">[:</span><span class="n">train_end</span><span class="p">],</span> <span class="n">labels_shuffled</span><span class="p">[</span><span class="n">train_end</span><span class="p">:]</span>
|
||
|
||
<span class="k">return</span> <span class="n">X_train</span><span class="p">,</span> <span class="n">X_test</span><span class="p">,</span> <span class="n">Y_train</span><span class="p">,</span> <span class="n">Y_test</span>
|
||
|
||
<span class="c1">#X_train, X_test, Y_train, Y_test = train_test_split_numpy(inputs, labels, train_size, test_size)</span>
|
||
|
||
<span class="nb">print</span><span class="p">(</span><span class="s2">"Number of training images: "</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">X_train</span><span class="p">)))</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="s2">"Number of test images: "</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">X_test</span><span class="p">)))</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
<div class="cell_output docutils container">
|
||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Number of training images: 1437
|
||
Number of test images: 360
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<div class="section" id="define-model-and-architecture">
|
||
<h2>Define model and architecture<a class="headerlink" href="#define-model-and-architecture" title="Permalink to this headline">¶</a></h2>
|
||
<p>Our simple feed-forward neural network will consist of an <em>input</em> layer, a single <em>hidden</em> layer and an <em>output</em> layer. The activation <span class="math notranslate nohighlight">\(y\)</span> of each neuron is a weighted sum of inputs, passed through an activation function. In case of the simple perceptron model we have</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[ z = \sum_{i=1}^n w_i a_i ,\]</div>
|
||
<div class="math notranslate nohighlight">
|
||
\[ y = f(z) ,\]</div>
|
||
<p>where <span class="math notranslate nohighlight">\(f\)</span> is the activation function, <span class="math notranslate nohighlight">\(a_i\)</span> represents input from neuron <span class="math notranslate nohighlight">\(i\)</span> in the preceding layer
|
||
and <span class="math notranslate nohighlight">\(w_i\)</span> is the weight to input <span class="math notranslate nohighlight">\(i\)</span>.<br />
|
||
The activation of the neurons in the input layer is just the features (e.g. a pixel value).</p>
|
||
<p>The simplest activation function for a neuron is the <em>Heaviside</em> function:</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[\begin{split} f(z) =
|
||
\begin{cases}
|
||
1, & z > 0\\
|
||
0, & \text{otherwise}
|
||
\end{cases}
|
||
\end{split}\]</div>
|
||
<p>A feed-forward neural network with this activation is known as a <em>perceptron</em>.<br />
|
||
For a binary classifier (i.e. two classes, 0 or 1, dog or not-dog) we can also use this in our output layer.<br />
|
||
This activation can be generalized to <span class="math notranslate nohighlight">\(k\)</span> classes (using e.g. the <em>one-against-all</em> strategy),
|
||
and we call these architectures <em>multiclass perceptrons</em>.</p>
|
||
<p>However, it is now common to use the terms Single Layer Perceptron (SLP) (1 hidden layer) and<br />
|
||
Multilayer Perceptron (MLP) (2 or more hidden layers) to refer to feed-forward neural networks with any activation function.</p>
|
||
<p>Typical choices for activation functions include the sigmoid function, hyperbolic tangent, and Rectified Linear Unit (ReLU).<br />
|
||
We will be using the sigmoid function <span class="math notranslate nohighlight">\(\sigma(x)\)</span>:</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[ f(x) = \sigma(x) = \frac{1}{1 + e^{-x}} ,\]</div>
|
||
<p>which is inspired by probability theory (see logistic regression) and was most commonly used until about 2011. See the discussion below concerning other activation functions.</p>
|
||
</div>
|
||
<div class="section" id="layers">
|
||
<h2>Layers<a class="headerlink" href="#layers" title="Permalink to this headline">¶</a></h2>
|
||
<ul class="simple">
|
||
<li><p>Input</p></li>
|
||
</ul>
|
||
<p>Since each input image has 8x8 = 64 pixels or features, we have an input layer of 64 neurons.</p>
|
||
<ul class="simple">
|
||
<li><p>Hidden layer</p></li>
|
||
</ul>
|
||
<p>We will use 50 neurons in the hidden layer receiving input from the neurons in the input layer.<br />
|
||
Since each neuron in the hidden layer is connected to the 64 inputs we have 64x50 = 3200 weights to the hidden layer.</p>
|
||
<ul class="simple">
|
||
<li><p>Output</p></li>
|
||
</ul>
|
||
<p>If we were building a binary classifier, it would be sufficient with a single neuron in the output layer,
|
||
which could output 0 or 1 according to the Heaviside function. This would be an example of a <em>hard</em> classifier, meaning it outputs the class of the input directly. However, if we are dealing with noisy data it is often beneficial to use a <em>soft</em> classifier, which outputs the probability of being in class 0 or 1.</p>
|
||
<p>For a soft binary classifier, we could use a single neuron and interpret the output as either being the probability of being in class 0 or the probability of being in class 1. Alternatively we could use 2 neurons, and interpret each neuron as the probability of being in each class.</p>
|
||
<p>Since we are doing multiclass classification, with 10 categories, it is natural to use 10 neurons in the output layer. We number the neurons <span class="math notranslate nohighlight">\(j = 0,1,...,9\)</span>. The activation of each output neuron <span class="math notranslate nohighlight">\(j\)</span> will be according to the <em>softmax</em> function:</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[ P(\text{class $j$} \mid \text{input $\boldsymbol{a}$}) = \frac{\exp{(\boldsymbol{a}^T \boldsymbol{w}_j)}}
|
||
{\sum_{c=0}^{9} \exp{(\boldsymbol{a}^T \boldsymbol{w}_c)}} ,\]</div>
|
||
<p>i.e. each neuron <span class="math notranslate nohighlight">\(j\)</span> outputs the probability of being in class <span class="math notranslate nohighlight">\(j\)</span> given an input from the hidden layer <span class="math notranslate nohighlight">\(\boldsymbol{a}\)</span>, with <span class="math notranslate nohighlight">\(\boldsymbol{w}_j\)</span> the weights of neuron <span class="math notranslate nohighlight">\(j\)</span> to the inputs.<br />
|
||
The denominator is a normalization factor to ensure the outputs (probabilities) sum up to 1.<br />
|
||
The exponent is just the weighted sum of inputs as before:</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[ z_j = \sum_{i=1}^n w_ {ij} a_i+b_j.\]</div>
|
||
<p>Since each neuron in the output layer is connected to the 50 inputs from the hidden layer we have 50x10 = 500
|
||
weights to the output layer.</p>
|
||
</div>
|
||
<div class="section" id="weights-and-biases">
|
||
<h2>Weights and biases<a class="headerlink" href="#weights-and-biases" title="Permalink to this headline">¶</a></h2>
|
||
<p>Typically weights are initialized with small values distributed around zero, drawn from a uniform
|
||
or normal distribution. Setting all weights to zero means all neurons give the same output, making the network useless.</p>
|
||
<p>Adding a bias value to the weighted sum of inputs allows the neural network to represent a greater range
|
||
of values. Without it, any input with the value 0 will be mapped to zero (before being passed through the activation). The bias unit has an output of 1, and a weight to each neuron <span class="math notranslate nohighlight">\(j\)</span>, <span class="math notranslate nohighlight">\(b_j\)</span>:</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[ z_j = \sum_{i=1}^n w_ {ij} a_i + b_j.\]</div>
|
||
<p>The bias weights <span class="math notranslate nohighlight">\(\boldsymbol{b}\)</span> are often initialized to zero, but a small value like <span class="math notranslate nohighlight">\(0.01\)</span> ensures all neurons have some output which can be backpropagated in the first training cycle.</p>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># building our neural network</span>
|
||
|
||
<span class="n">n_inputs</span><span class="p">,</span> <span class="n">n_features</span> <span class="o">=</span> <span class="n">X_train</span><span class="o">.</span><span class="n">shape</span>
|
||
<span class="n">n_hidden_neurons</span> <span class="o">=</span> <span class="mi">50</span>
|
||
<span class="n">n_categories</span> <span class="o">=</span> <span class="mi">10</span>
|
||
|
||
<span class="c1"># we make the weights normally distributed using numpy.random.randn</span>
|
||
|
||
<span class="c1"># weights and bias in the hidden layer</span>
|
||
<span class="n">hidden_weights</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="n">n_features</span><span class="p">,</span> <span class="n">n_hidden_neurons</span><span class="p">)</span>
|
||
<span class="n">hidden_bias</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">n_hidden_neurons</span><span class="p">)</span> <span class="o">+</span> <span class="mf">0.01</span>
|
||
|
||
<span class="c1"># weights and bias in the output layer</span>
|
||
<span class="n">output_weights</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="n">n_hidden_neurons</span><span class="p">,</span> <span class="n">n_categories</span><span class="p">)</span>
|
||
<span class="n">output_bias</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">n_categories</span><span class="p">)</span> <span class="o">+</span> <span class="mf">0.01</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<div class="section" id="feed-forward-pass">
|
||
<h2>Feed-forward pass<a class="headerlink" href="#feed-forward-pass" title="Permalink to this headline">¶</a></h2>
|
||
<p>Denote <span class="math notranslate nohighlight">\(F\)</span> the number of features, <span class="math notranslate nohighlight">\(H\)</span> the number of hidden neurons and <span class="math notranslate nohighlight">\(C\)</span> the number of categories.<br />
|
||
For each input image we calculate a weighted sum of input features (pixel values) to each neuron <span class="math notranslate nohighlight">\(j\)</span> in the hidden layer <span class="math notranslate nohighlight">\(l\)</span>:</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[ z_{j}^{l} = \sum_{i=1}^{F} w_{ij}^{l} x_i + b_{j}^{l},\]</div>
|
||
<p>this is then passed through our activation function</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[ a_{j}^{l} = f(z_{j}^{l}) .\]</div>
|
||
<p>We calculate a weighted sum of inputs (activations in the hidden layer) to each neuron <span class="math notranslate nohighlight">\(j\)</span> in the output layer:</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[ z_{j}^{L} = \sum_{i=1}^{H} w_{ij}^{L} a_{i}^{l} + b_{j}^{L}.\]</div>
|
||
<p>Finally we calculate the output of neuron <span class="math notranslate nohighlight">\(j\)</span> in the output layer using the softmax function:</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[ a_{j}^{L} = \frac{\exp{(z_j^{L})}}
|
||
{\sum_{c=0}^{C-1} \exp{(z_c^{L})}} .\]</div>
|
||
</div>
|
||
<div class="section" id="matrix-multiplications">
|
||
<h2>Matrix multiplications<a class="headerlink" href="#matrix-multiplications" title="Permalink to this headline">¶</a></h2>
|
||
<p>Since our data has the dimensions <span class="math notranslate nohighlight">\(X = (n_{inputs}, n_{features})\)</span> and our weights to the hidden
|
||
layer have the dimensions<br />
|
||
<span class="math notranslate nohighlight">\(W_{hidden} = (n_{features}, n_{hidden})\)</span>,
|
||
we can easily feed the network all our training data in one go by taking the matrix product</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[ X W^{h} = (n_{inputs}, n_{hidden}),\]</div>
|
||
<p>and obtain a matrix that holds the weighted sum of inputs to the hidden layer
|
||
for each input image and each hidden neuron.<br />
|
||
We also add the bias to obtain a matrix of weighted sums to the hidden layer <span class="math notranslate nohighlight">\(Z^{h}\)</span>:</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[ \boldsymbol{z}^{l} = \boldsymbol{X} \boldsymbol{W}^{l} + \boldsymbol{b}^{l} ,\]</div>
|
||
<p>meaning the same bias (1D array with size equal number of hidden neurons) is added to each input image.<br />
|
||
This is then passed through the activation:</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[ \boldsymbol{a}^{l} = f(\boldsymbol{z}^l) .\]</div>
|
||
<p>This is fed to the output layer:</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[ \boldsymbol{z}^{L} = \boldsymbol{a}^{L} \boldsymbol{W}^{L} + \boldsymbol{b}^{L} .\]</div>
|
||
<p>Finally we receive our output values for each image and each category by passing it through the softmax function:</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[ output = softmax (\boldsymbol{z}^{L}) = (n_{inputs}, n_{categories}) .\]</div>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># setup the feed-forward pass, subscript h = hidden layer</span>
|
||
|
||
<span class="k">def</span> <span class="nf">sigmoid</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
|
||
<span class="k">return</span> <span class="mi">1</span><span class="o">/</span><span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">x</span><span class="p">))</span>
|
||
|
||
<span class="k">def</span> <span class="nf">feed_forward</span><span class="p">(</span><span class="n">X</span><span class="p">):</span>
|
||
<span class="c1"># weighted sum of inputs to the hidden layer</span>
|
||
<span class="n">z_h</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">matmul</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">hidden_weights</span><span class="p">)</span> <span class="o">+</span> <span class="n">hidden_bias</span>
|
||
<span class="c1"># activation in the hidden layer</span>
|
||
<span class="n">a_h</span> <span class="o">=</span> <span class="n">sigmoid</span><span class="p">(</span><span class="n">z_h</span><span class="p">)</span>
|
||
|
||
<span class="c1"># weighted sum of inputs to the output layer</span>
|
||
<span class="n">z_o</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">matmul</span><span class="p">(</span><span class="n">a_h</span><span class="p">,</span> <span class="n">output_weights</span><span class="p">)</span> <span class="o">+</span> <span class="n">output_bias</span>
|
||
<span class="c1"># softmax output</span>
|
||
<span class="c1"># axis 0 holds each input and axis 1 the probabilities of each category</span>
|
||
<span class="n">exp_term</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="n">z_o</span><span class="p">)</span>
|
||
<span class="n">probabilities</span> <span class="o">=</span> <span class="n">exp_term</span> <span class="o">/</span> <span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">exp_term</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">keepdims</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
|
||
|
||
<span class="k">return</span> <span class="n">probabilities</span>
|
||
|
||
<span class="n">probabilities</span> <span class="o">=</span> <span class="n">feed_forward</span><span class="p">(</span><span class="n">X_train</span><span class="p">)</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="s2">"probabilities = (n_inputs, n_categories) = "</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">probabilities</span><span class="o">.</span><span class="n">shape</span><span class="p">))</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="s2">"probability that image 0 is in category 0,1,2,...,9 = </span><span class="se">\n</span><span class="s2">"</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">probabilities</span><span class="p">[</span><span class="mi">0</span><span class="p">]))</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="s2">"probabilities sum up to: "</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">probabilities</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">sum</span><span class="p">()))</span>
|
||
<span class="nb">print</span><span class="p">()</span>
|
||
|
||
<span class="c1"># we obtain a prediction by taking the class with the highest likelihood</span>
|
||
<span class="k">def</span> <span class="nf">predict</span><span class="p">(</span><span class="n">X</span><span class="p">):</span>
|
||
<span class="n">probabilities</span> <span class="o">=</span> <span class="n">feed_forward</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
|
||
<span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">argmax</span><span class="p">(</span><span class="n">probabilities</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
|
||
|
||
<span class="n">predictions</span> <span class="o">=</span> <span class="n">predict</span><span class="p">(</span><span class="n">X_train</span><span class="p">)</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="s2">"predictions = (n_inputs) = "</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">predictions</span><span class="o">.</span><span class="n">shape</span><span class="p">))</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="s2">"prediction for image 0: "</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">predictions</span><span class="p">[</span><span class="mi">0</span><span class="p">]))</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="s2">"correct label for image 0: "</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">Y_train</span><span class="p">[</span><span class="mi">0</span><span class="p">]))</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
<div class="cell_output docutils container">
|
||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>probabilities = (n_inputs, n_categories) = (1437, 10)
|
||
probability that image 0 is in category 0,1,2,...,9 =
|
||
[5.41511965e-04 2.17174962e-03 8.84355903e-03 1.44970586e-03
|
||
1.10378326e-04 5.08318298e-09 2.03256632e-04 1.92507116e-03
|
||
9.84443254e-01 3.11507992e-04]
|
||
probabilities sum up to: 1.0
|
||
|
||
predictions = (n_inputs) = (1437,)
|
||
prediction for image 0: 8
|
||
correct label for image 0: 6
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<div class="section" id="choose-cost-function-and-optimizer">
|
||
<h2>Choose cost function and optimizer<a class="headerlink" href="#choose-cost-function-and-optimizer" title="Permalink to this headline">¶</a></h2>
|
||
<p>To measure how well our neural network is doing we need to introduce a cost function.<br />
|
||
We will call the function that gives the error of a single sample output the <em>loss</em> function, and the function
|
||
that gives the total error of our network across all samples the <em>cost</em> function.
|
||
A typical choice for multiclass classification is the <em>cross-entropy</em> loss, also known as the negative log likelihood.</p>
|
||
<p>In <em>multiclass</em> classification it is common to treat each integer label as a so called <em>one-hot</em> vector:</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[ y = 5 \quad \rightarrow \quad \boldsymbol{y} = (0, 0, 0, 0, 0, 1, 0, 0, 0, 0) ,\]</div>
|
||
<div class="math notranslate nohighlight">
|
||
\[ y = 1 \quad \rightarrow \quad \boldsymbol{y} = (0, 1, 0, 0, 0, 0, 0, 0, 0, 0) ,\]</div>
|
||
<p>i.e. a binary bit string of length <span class="math notranslate nohighlight">\(C\)</span>, where <span class="math notranslate nohighlight">\(C = 10\)</span> is the number of classes in the MNIST dataset.</p>
|
||
<p>Let <span class="math notranslate nohighlight">\(y_{ic}\)</span> denote the <span class="math notranslate nohighlight">\(c\)</span>-th component of the <span class="math notranslate nohighlight">\(i\)</span>-th one-hot vector.<br />
|
||
We define the cost function <span class="math notranslate nohighlight">\(\mathcal{C}\)</span> as a sum over the cross-entropy loss for each point <span class="math notranslate nohighlight">\(\boldsymbol{x}_i\)</span> in the dataset.</p>
|
||
<p>In the one-hot representation only one of the terms in the loss function is non-zero, namely the
|
||
probability of the correct category <span class="math notranslate nohighlight">\(c'\)</span><br />
|
||
(i.e. the category <span class="math notranslate nohighlight">\(c'\)</span> such that <span class="math notranslate nohighlight">\(y_{ic'} = 1\)</span>). This means that the cross entropy loss only punishes you for how wrong
|
||
you got the correct label. The probability of category <span class="math notranslate nohighlight">\(c\)</span> is given by the softmax function. The vector <span class="math notranslate nohighlight">\(\boldsymbol{\theta}\)</span> represents the parameters of our network, i.e. all the weights and biases.</p>
|
||
</div>
|
||
<div class="section" id="optimizing-the-cost-function">
|
||
<h2>Optimizing the cost function<a class="headerlink" href="#optimizing-the-cost-function" title="Permalink to this headline">¶</a></h2>
|
||
<p>The network is trained by finding the weights and biases that minimize the cost function. One of the most widely used classes of methods is <em>gradient descent</em> and its generalizations. The idea behind gradient descent
|
||
is simply to adjust the weights in the direction where the gradient of the cost function is large and negative. This ensures we flow toward a <em>local</em> minimum of the cost function.<br />
|
||
Each parameter <span class="math notranslate nohighlight">\(\theta\)</span> is iteratively adjusted according to the rule</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[ \theta_{i+1} = \theta_i - \eta \nabla \mathcal{C}(\theta_i) ,\]</div>
|
||
<p>where <span class="math notranslate nohighlight">\(\eta\)</span> is known as the <em>learning rate</em>, which controls how big a step we take towards the minimum.<br />
|
||
This update can be repeated for any number of iterations, or until we are satisfied with the result.</p>
|
||
<p>A simple and effective improvement is a variant called <em>Batch Gradient Descent</em>.<br />
|
||
Instead of calculating the gradient on the whole dataset, we calculate an approximation of the gradient
|
||
on a subset of the data called a <em>minibatch</em>.<br />
|
||
If there are <span class="math notranslate nohighlight">\(N\)</span> data points and we have a minibatch size of <span class="math notranslate nohighlight">\(M\)</span>, the total number of batches
|
||
is <span class="math notranslate nohighlight">\(N/M\)</span>.<br />
|
||
We denote each minibatch <span class="math notranslate nohighlight">\(B_k\)</span>, with <span class="math notranslate nohighlight">\(k = 1, 2,...,N/M\)</span>. The gradient then becomes:</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[ \nabla \mathcal{C}(\theta) = \frac{1}{N} \sum_{i=1}^N \nabla \mathcal{L}_i(\theta) \quad \rightarrow \quad
|
||
\frac{1}{M} \sum_{i \in B_k} \nabla \mathcal{L}_i(\theta) ,\]</div>
|
||
<p>i.e. instead of averaging the loss over the entire dataset, we average over a minibatch.</p>
|
||
<p>This has two important benefits:</p>
|
||
<ol class="simple">
|
||
<li><p>Introducing stochasticity decreases the chance that the algorithm becomes stuck in a local minima.</p></li>
|
||
<li><p>It significantly speeds up the calculation, since we do not have to use the entire dataset to calculate the gradient.</p></li>
|
||
</ol>
|
||
<p>The various optmization methods, with codes and algorithms, are discussed in our lectures on <a class="reference external" href="https://compphysics.github.io/MachineLearning/doc/pub/Splines/html/Splines-bs.html">Gradient descent approaches</a>.</p>
|
||
</div>
|
||
<div class="section" id="regularization">
|
||
<h2>Regularization<a class="headerlink" href="#regularization" title="Permalink to this headline">¶</a></h2>
|
||
<p>It is common to add an extra term to the cost function, proportional
|
||
to the size of the weights. This is equivalent to constraining the
|
||
size of the weights, so that they do not grow out of control.
|
||
Constraining the size of the weights means that the weights cannot
|
||
grow arbitrarily large to fit the training data, and in this way
|
||
reduces <em>overfitting</em>.</p>
|
||
<p>We will measure the size of the weights using the so called <em>L2-norm</em>, meaning our cost function becomes:</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[ \mathcal{C}(\theta) = \frac{1}{N} \sum_{i=1}^N \mathcal{L}_i(\theta) \quad \rightarrow \quad
|
||
\frac{1}{N} \sum_{i=1}^N \mathcal{L}_i(\theta) + \lambda \lvert \lvert \boldsymbol{w} \rvert \rvert_2^2
|
||
= \frac{1}{N} \sum_{i=1}^N \mathcal{L}(\theta) + \lambda \sum_{ij} w_{ij}^2,\]</div>
|
||
<p>i.e. we sum up all the weights squared. The factor <span class="math notranslate nohighlight">\(\lambda\)</span> is known as a regularization parameter.</p>
|
||
<p>In order to train the model, we need to calculate the derivative of
|
||
the cost function with respect to every bias and weight in the
|
||
network. In total our network has <span class="math notranslate nohighlight">\((64 + 1)\times 50=3250\)</span> weights in
|
||
the hidden layer and <span class="math notranslate nohighlight">\((50 + 1)\times 10=510\)</span> weights to the output
|
||
layer (<span class="math notranslate nohighlight">\(+1\)</span> for the bias), and the gradient must be calculated for
|
||
every parameter. We use the <em>backpropagation</em> algorithm discussed
|
||
above. This is a clever use of the chain rule that allows us to
|
||
calculate the gradient efficently.</p>
|
||
</div>
|
||
<div class="section" id="matrix-multiplication">
|
||
<h2>Matrix multiplication<a class="headerlink" href="#matrix-multiplication" title="Permalink to this headline">¶</a></h2>
|
||
<p>To more efficently train our network these equations are implemented using matrix operations.<br />
|
||
The error in the output layer is calculated simply as, with <span class="math notranslate nohighlight">\(\boldsymbol{t}\)</span> being our targets,</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[ \delta_L = \boldsymbol{t} - \boldsymbol{y} = (n_{inputs}, n_{categories}) .\]</div>
|
||
<p>The gradient for the output weights is calculated as</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[ \nabla W_{L} = \boldsymbol{a}^T \delta_L = (n_{hidden}, n_{categories}) ,\]</div>
|
||
<p>where <span class="math notranslate nohighlight">\(\boldsymbol{a} = (n_{inputs}, n_{hidden})\)</span>. This simply means that we are summing up the gradients for each input.<br />
|
||
Since we are going backwards we have to transpose the activation matrix.</p>
|
||
<p>The gradient with respect to the output bias is then</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[ \nabla \boldsymbol{b}_{L} = \sum_{i=1}^{n_{inputs}} \delta_L = (n_{categories}) .\]</div>
|
||
<p>The error in the hidden layer is</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[ \Delta_h = \delta_L W_{L}^T \circ f'(z_{h}) = \delta_L W_{L}^T \circ a_{h} \circ (1 - a_{h}) = (n_{inputs}, n_{hidden}) ,\]</div>
|
||
<p>where <span class="math notranslate nohighlight">\(f'(a_{h})\)</span> is the derivative of the activation in the hidden layer. The matrix products mean
|
||
that we are summing up the products for each neuron in the output layer. The symbol <span class="math notranslate nohighlight">\(\circ\)</span> denotes
|
||
the <em>Hadamard product</em>, meaning element-wise multiplication.</p>
|
||
<p>This again gives us the gradients in the hidden layer:</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[ \nabla W_{h} = X^T \delta_h = (n_{features}, n_{hidden}) ,\]</div>
|
||
<div class="math notranslate nohighlight">
|
||
\[ \nabla b_{h} = \sum_{i=1}^{n_{inputs}} \delta_h = (n_{hidden}) .\]</div>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># to categorical turns our integer vector into a onehot representation</span>
|
||
<span class="kn">from</span> <span class="nn">sklearn.metrics</span> <span class="kn">import</span> <span class="n">accuracy_score</span>
|
||
|
||
<span class="c1"># one-hot in numpy</span>
|
||
<span class="k">def</span> <span class="nf">to_categorical_numpy</span><span class="p">(</span><span class="n">integer_vector</span><span class="p">):</span>
|
||
<span class="n">n_inputs</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">integer_vector</span><span class="p">)</span>
|
||
<span class="n">n_categories</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">max</span><span class="p">(</span><span class="n">integer_vector</span><span class="p">)</span> <span class="o">+</span> <span class="mi">1</span>
|
||
<span class="n">onehot_vector</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">n_inputs</span><span class="p">,</span> <span class="n">n_categories</span><span class="p">))</span>
|
||
<span class="n">onehot_vector</span><span class="p">[</span><span class="nb">range</span><span class="p">(</span><span class="n">n_inputs</span><span class="p">),</span> <span class="n">integer_vector</span><span class="p">]</span> <span class="o">=</span> <span class="mi">1</span>
|
||
|
||
<span class="k">return</span> <span class="n">onehot_vector</span>
|
||
|
||
<span class="c1">#Y_train_onehot, Y_test_onehot = to_categorical(Y_train), to_categorical(Y_test)</span>
|
||
<span class="n">Y_train_onehot</span><span class="p">,</span> <span class="n">Y_test_onehot</span> <span class="o">=</span> <span class="n">to_categorical_numpy</span><span class="p">(</span><span class="n">Y_train</span><span class="p">),</span> <span class="n">to_categorical_numpy</span><span class="p">(</span><span class="n">Y_test</span><span class="p">)</span>
|
||
|
||
<span class="k">def</span> <span class="nf">feed_forward_train</span><span class="p">(</span><span class="n">X</span><span class="p">):</span>
|
||
<span class="c1"># weighted sum of inputs to the hidden layer</span>
|
||
<span class="n">z_h</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">matmul</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">hidden_weights</span><span class="p">)</span> <span class="o">+</span> <span class="n">hidden_bias</span>
|
||
<span class="c1"># activation in the hidden layer</span>
|
||
<span class="n">a_h</span> <span class="o">=</span> <span class="n">sigmoid</span><span class="p">(</span><span class="n">z_h</span><span class="p">)</span>
|
||
|
||
<span class="c1"># weighted sum of inputs to the output layer</span>
|
||
<span class="n">z_o</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">matmul</span><span class="p">(</span><span class="n">a_h</span><span class="p">,</span> <span class="n">output_weights</span><span class="p">)</span> <span class="o">+</span> <span class="n">output_bias</span>
|
||
<span class="c1"># softmax output</span>
|
||
<span class="c1"># axis 0 holds each input and axis 1 the probabilities of each category</span>
|
||
<span class="n">exp_term</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="n">z_o</span><span class="p">)</span>
|
||
<span class="n">probabilities</span> <span class="o">=</span> <span class="n">exp_term</span> <span class="o">/</span> <span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">exp_term</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">keepdims</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
|
||
|
||
<span class="c1"># for backpropagation need activations in hidden and output layers</span>
|
||
<span class="k">return</span> <span class="n">a_h</span><span class="p">,</span> <span class="n">probabilities</span>
|
||
|
||
<span class="k">def</span> <span class="nf">backpropagation</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">Y</span><span class="p">):</span>
|
||
<span class="n">a_h</span><span class="p">,</span> <span class="n">probabilities</span> <span class="o">=</span> <span class="n">feed_forward_train</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
|
||
|
||
<span class="c1"># error in the output layer</span>
|
||
<span class="n">error_output</span> <span class="o">=</span> <span class="n">probabilities</span> <span class="o">-</span> <span class="n">Y</span>
|
||
<span class="c1"># error in the hidden layer</span>
|
||
<span class="n">error_hidden</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">matmul</span><span class="p">(</span><span class="n">error_output</span><span class="p">,</span> <span class="n">output_weights</span><span class="o">.</span><span class="n">T</span><span class="p">)</span> <span class="o">*</span> <span class="n">a_h</span> <span class="o">*</span> <span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">a_h</span><span class="p">)</span>
|
||
|
||
<span class="c1"># gradients for the output layer</span>
|
||
<span class="n">output_weights_gradient</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">matmul</span><span class="p">(</span><span class="n">a_h</span><span class="o">.</span><span class="n">T</span><span class="p">,</span> <span class="n">error_output</span><span class="p">)</span>
|
||
<span class="n">output_bias_gradient</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">error_output</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
|
||
|
||
<span class="c1"># gradient for the hidden layer</span>
|
||
<span class="n">hidden_weights_gradient</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">matmul</span><span class="p">(</span><span class="n">X</span><span class="o">.</span><span class="n">T</span><span class="p">,</span> <span class="n">error_hidden</span><span class="p">)</span>
|
||
<span class="n">hidden_bias_gradient</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">error_hidden</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
|
||
|
||
<span class="k">return</span> <span class="n">output_weights_gradient</span><span class="p">,</span> <span class="n">output_bias_gradient</span><span class="p">,</span> <span class="n">hidden_weights_gradient</span><span class="p">,</span> <span class="n">hidden_bias_gradient</span>
|
||
|
||
<span class="nb">print</span><span class="p">(</span><span class="s2">"Old accuracy on training data: "</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">accuracy_score</span><span class="p">(</span><span class="n">predict</span><span class="p">(</span><span class="n">X_train</span><span class="p">),</span> <span class="n">Y_train</span><span class="p">)))</span>
|
||
|
||
<span class="n">eta</span> <span class="o">=</span> <span class="mf">0.01</span>
|
||
<span class="n">lmbd</span> <span class="o">=</span> <span class="mf">0.01</span>
|
||
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1000</span><span class="p">):</span>
|
||
<span class="c1"># calculate gradients</span>
|
||
<span class="n">dWo</span><span class="p">,</span> <span class="n">dBo</span><span class="p">,</span> <span class="n">dWh</span><span class="p">,</span> <span class="n">dBh</span> <span class="o">=</span> <span class="n">backpropagation</span><span class="p">(</span><span class="n">X_train</span><span class="p">,</span> <span class="n">Y_train_onehot</span><span class="p">)</span>
|
||
|
||
<span class="c1"># regularization term gradients</span>
|
||
<span class="n">dWo</span> <span class="o">+=</span> <span class="n">lmbd</span> <span class="o">*</span> <span class="n">output_weights</span>
|
||
<span class="n">dWh</span> <span class="o">+=</span> <span class="n">lmbd</span> <span class="o">*</span> <span class="n">hidden_weights</span>
|
||
|
||
<span class="c1"># update weights and biases</span>
|
||
<span class="n">output_weights</span> <span class="o">-=</span> <span class="n">eta</span> <span class="o">*</span> <span class="n">dWo</span>
|
||
<span class="n">output_bias</span> <span class="o">-=</span> <span class="n">eta</span> <span class="o">*</span> <span class="n">dBo</span>
|
||
<span class="n">hidden_weights</span> <span class="o">-=</span> <span class="n">eta</span> <span class="o">*</span> <span class="n">dWh</span>
|
||
<span class="n">hidden_bias</span> <span class="o">-=</span> <span class="n">eta</span> <span class="o">*</span> <span class="n">dBh</span>
|
||
|
||
<span class="nb">print</span><span class="p">(</span><span class="s2">"New accuracy on training data: "</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">accuracy_score</span><span class="p">(</span><span class="n">predict</span><span class="p">(</span><span class="n">X_train</span><span class="p">),</span> <span class="n">Y_train</span><span class="p">)))</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
<div class="cell_output docutils container">
|
||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Old accuracy on training data: 0.1440501043841336
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_94373/953065564.py:4: RuntimeWarning: overflow encountered in exp
|
||
return 1/(1 + np.exp(-x))
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>New accuracy on training data: 0.09951287404314545
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<div class="section" id="improving-performance">
|
||
<h2>Improving performance<a class="headerlink" href="#improving-performance" title="Permalink to this headline">¶</a></h2>
|
||
<p>As we can see the network does not seem to be learning at all. It seems to be just guessing the label for each image.<br />
|
||
In order to obtain a network that does something useful, we will have to do a bit more work.</p>
|
||
<p>The choice of <em>hyperparameters</em> such as learning rate and regularization parameter is hugely influential for the performance of the network. Typically a <em>grid-search</em> is performed, wherein we test different hyperparameters separated by orders of magnitude. For example we could test the learning rates <span class="math notranslate nohighlight">\(\eta = 10^{-6}, 10^{-5},...,10^{-1}\)</span> with different regularization parameters <span class="math notranslate nohighlight">\(\lambda = 10^{-6},...,10^{-0}\)</span>.</p>
|
||
<p>Next, we haven’t implemented minibatching yet, which introduces stochasticity and is though to act as an important regularizer on the weights. We call a feed-forward + backward pass with a minibatch an <em>iteration</em>, and a full training period
|
||
going through the entire dataset (<span class="math notranslate nohighlight">\(n/M\)</span> batches) an <em>epoch</em>.</p>
|
||
<p>If this does not improve network performance, you may want to consider altering the network architecture, adding more neurons or hidden layers.<br />
|
||
Andrew Ng goes through some of these considerations in this <a class="reference external" href="https://youtu.be/F1ka6a13S9I">video</a>. You can find a summary of the video <a class="reference external" href="https://kevinzakka.github.io/2016/09/26/applying-deep-learning/">here</a>.</p>
|
||
</div>
|
||
<div class="section" id="full-object-oriented-implementation">
|
||
<h2>Full object-oriented implementation<a class="headerlink" href="#full-object-oriented-implementation" title="Permalink to this headline">¶</a></h2>
|
||
<p>It is very natural to think of the network as an object, with specific instances of the network
|
||
being realizations of this object with different hyperparameters. An implementation using Python classes provides a clean structure and interface, and the full implementation of our neural network is given below.</p>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="k">class</span> <span class="nc">NeuralNetwork</span><span class="p">:</span>
|
||
<span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span>
|
||
<span class="bp">self</span><span class="p">,</span>
|
||
<span class="n">X_data</span><span class="p">,</span>
|
||
<span class="n">Y_data</span><span class="p">,</span>
|
||
<span class="n">n_hidden_neurons</span><span class="o">=</span><span class="mi">50</span><span class="p">,</span>
|
||
<span class="n">n_categories</span><span class="o">=</span><span class="mi">10</span><span class="p">,</span>
|
||
<span class="n">epochs</span><span class="o">=</span><span class="mi">10</span><span class="p">,</span>
|
||
<span class="n">batch_size</span><span class="o">=</span><span class="mi">100</span><span class="p">,</span>
|
||
<span class="n">eta</span><span class="o">=</span><span class="mf">0.1</span><span class="p">,</span>
|
||
<span class="n">lmbd</span><span class="o">=</span><span class="mf">0.0</span><span class="p">):</span>
|
||
|
||
<span class="bp">self</span><span class="o">.</span><span class="n">X_data_full</span> <span class="o">=</span> <span class="n">X_data</span>
|
||
<span class="bp">self</span><span class="o">.</span><span class="n">Y_data_full</span> <span class="o">=</span> <span class="n">Y_data</span>
|
||
|
||
<span class="bp">self</span><span class="o">.</span><span class="n">n_inputs</span> <span class="o">=</span> <span class="n">X_data</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
|
||
<span class="bp">self</span><span class="o">.</span><span class="n">n_features</span> <span class="o">=</span> <span class="n">X_data</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
|
||
<span class="bp">self</span><span class="o">.</span><span class="n">n_hidden_neurons</span> <span class="o">=</span> <span class="n">n_hidden_neurons</span>
|
||
<span class="bp">self</span><span class="o">.</span><span class="n">n_categories</span> <span class="o">=</span> <span class="n">n_categories</span>
|
||
|
||
<span class="bp">self</span><span class="o">.</span><span class="n">epochs</span> <span class="o">=</span> <span class="n">epochs</span>
|
||
<span class="bp">self</span><span class="o">.</span><span class="n">batch_size</span> <span class="o">=</span> <span class="n">batch_size</span>
|
||
<span class="bp">self</span><span class="o">.</span><span class="n">iterations</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">n_inputs</span> <span class="o">//</span> <span class="bp">self</span><span class="o">.</span><span class="n">batch_size</span>
|
||
<span class="bp">self</span><span class="o">.</span><span class="n">eta</span> <span class="o">=</span> <span class="n">eta</span>
|
||
<span class="bp">self</span><span class="o">.</span><span class="n">lmbd</span> <span class="o">=</span> <span class="n">lmbd</span>
|
||
|
||
<span class="bp">self</span><span class="o">.</span><span class="n">create_biases_and_weights</span><span class="p">()</span>
|
||
|
||
<span class="k">def</span> <span class="nf">create_biases_and_weights</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
|
||
<span class="bp">self</span><span class="o">.</span><span class="n">hidden_weights</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">n_features</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">n_hidden_neurons</span><span class="p">)</span>
|
||
<span class="bp">self</span><span class="o">.</span><span class="n">hidden_bias</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">n_hidden_neurons</span><span class="p">)</span> <span class="o">+</span> <span class="mf">0.01</span>
|
||
|
||
<span class="bp">self</span><span class="o">.</span><span class="n">output_weights</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">n_hidden_neurons</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">n_categories</span><span class="p">)</span>
|
||
<span class="bp">self</span><span class="o">.</span><span class="n">output_bias</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">n_categories</span><span class="p">)</span> <span class="o">+</span> <span class="mf">0.01</span>
|
||
|
||
<span class="k">def</span> <span class="nf">feed_forward</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
|
||
<span class="c1"># feed-forward for training</span>
|
||
<span class="bp">self</span><span class="o">.</span><span class="n">z_h</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">matmul</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">X_data</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">hidden_weights</span><span class="p">)</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">hidden_bias</span>
|
||
<span class="bp">self</span><span class="o">.</span><span class="n">a_h</span> <span class="o">=</span> <span class="n">sigmoid</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">z_h</span><span class="p">)</span>
|
||
|
||
<span class="bp">self</span><span class="o">.</span><span class="n">z_o</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">matmul</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">a_h</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">output_weights</span><span class="p">)</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">output_bias</span>
|
||
|
||
<span class="n">exp_term</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">z_o</span><span class="p">)</span>
|
||
<span class="bp">self</span><span class="o">.</span><span class="n">probabilities</span> <span class="o">=</span> <span class="n">exp_term</span> <span class="o">/</span> <span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">exp_term</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">keepdims</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
|
||
|
||
<span class="k">def</span> <span class="nf">feed_forward_out</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">X</span><span class="p">):</span>
|
||
<span class="c1"># feed-forward for output</span>
|
||
<span class="n">z_h</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">matmul</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">hidden_weights</span><span class="p">)</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">hidden_bias</span>
|
||
<span class="n">a_h</span> <span class="o">=</span> <span class="n">sigmoid</span><span class="p">(</span><span class="n">z_h</span><span class="p">)</span>
|
||
|
||
<span class="n">z_o</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">matmul</span><span class="p">(</span><span class="n">a_h</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">output_weights</span><span class="p">)</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">output_bias</span>
|
||
|
||
<span class="n">exp_term</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="n">z_o</span><span class="p">)</span>
|
||
<span class="n">probabilities</span> <span class="o">=</span> <span class="n">exp_term</span> <span class="o">/</span> <span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">exp_term</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">keepdims</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
|
||
<span class="k">return</span> <span class="n">probabilities</span>
|
||
|
||
<span class="k">def</span> <span class="nf">backpropagation</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
|
||
<span class="n">error_output</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">probabilities</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">Y_data</span>
|
||
<span class="n">error_hidden</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">matmul</span><span class="p">(</span><span class="n">error_output</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">output_weights</span><span class="o">.</span><span class="n">T</span><span class="p">)</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">a_h</span> <span class="o">*</span> <span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">a_h</span><span class="p">)</span>
|
||
|
||
<span class="bp">self</span><span class="o">.</span><span class="n">output_weights_gradient</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">matmul</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">a_h</span><span class="o">.</span><span class="n">T</span><span class="p">,</span> <span class="n">error_output</span><span class="p">)</span>
|
||
<span class="bp">self</span><span class="o">.</span><span class="n">output_bias_gradient</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">error_output</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
|
||
|
||
<span class="bp">self</span><span class="o">.</span><span class="n">hidden_weights_gradient</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">matmul</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">X_data</span><span class="o">.</span><span class="n">T</span><span class="p">,</span> <span class="n">error_hidden</span><span class="p">)</span>
|
||
<span class="bp">self</span><span class="o">.</span><span class="n">hidden_bias_gradient</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">error_hidden</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
|
||
|
||
<span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">lmbd</span> <span class="o">></span> <span class="mf">0.0</span><span class="p">:</span>
|
||
<span class="bp">self</span><span class="o">.</span><span class="n">output_weights_gradient</span> <span class="o">+=</span> <span class="bp">self</span><span class="o">.</span><span class="n">lmbd</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">output_weights</span>
|
||
<span class="bp">self</span><span class="o">.</span><span class="n">hidden_weights_gradient</span> <span class="o">+=</span> <span class="bp">self</span><span class="o">.</span><span class="n">lmbd</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">hidden_weights</span>
|
||
|
||
<span class="bp">self</span><span class="o">.</span><span class="n">output_weights</span> <span class="o">-=</span> <span class="bp">self</span><span class="o">.</span><span class="n">eta</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">output_weights_gradient</span>
|
||
<span class="bp">self</span><span class="o">.</span><span class="n">output_bias</span> <span class="o">-=</span> <span class="bp">self</span><span class="o">.</span><span class="n">eta</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">output_bias_gradient</span>
|
||
<span class="bp">self</span><span class="o">.</span><span class="n">hidden_weights</span> <span class="o">-=</span> <span class="bp">self</span><span class="o">.</span><span class="n">eta</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">hidden_weights_gradient</span>
|
||
<span class="bp">self</span><span class="o">.</span><span class="n">hidden_bias</span> <span class="o">-=</span> <span class="bp">self</span><span class="o">.</span><span class="n">eta</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">hidden_bias_gradient</span>
|
||
|
||
<span class="k">def</span> <span class="nf">predict</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">X</span><span class="p">):</span>
|
||
<span class="n">probabilities</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">feed_forward_out</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
|
||
<span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">argmax</span><span class="p">(</span><span class="n">probabilities</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
|
||
|
||
<span class="k">def</span> <span class="nf">predict_probabilities</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">X</span><span class="p">):</span>
|
||
<span class="n">probabilities</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">feed_forward_out</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
|
||
<span class="k">return</span> <span class="n">probabilities</span>
|
||
|
||
<span class="k">def</span> <span class="nf">train</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
|
||
<span class="n">data_indices</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">n_inputs</span><span class="p">)</span>
|
||
|
||
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">epochs</span><span class="p">):</span>
|
||
<span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">iterations</span><span class="p">):</span>
|
||
<span class="c1"># pick datapoints with replacement</span>
|
||
<span class="n">chosen_datapoints</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">choice</span><span class="p">(</span>
|
||
<span class="n">data_indices</span><span class="p">,</span> <span class="n">size</span><span class="o">=</span><span class="bp">self</span><span class="o">.</span><span class="n">batch_size</span><span class="p">,</span> <span class="n">replace</span><span class="o">=</span><span class="kc">False</span>
|
||
<span class="p">)</span>
|
||
|
||
<span class="c1"># minibatch training data</span>
|
||
<span class="bp">self</span><span class="o">.</span><span class="n">X_data</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">X_data_full</span><span class="p">[</span><span class="n">chosen_datapoints</span><span class="p">]</span>
|
||
<span class="bp">self</span><span class="o">.</span><span class="n">Y_data</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">Y_data_full</span><span class="p">[</span><span class="n">chosen_datapoints</span><span class="p">]</span>
|
||
|
||
<span class="bp">self</span><span class="o">.</span><span class="n">feed_forward</span><span class="p">()</span>
|
||
<span class="bp">self</span><span class="o">.</span><span class="n">backpropagation</span><span class="p">()</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<div class="section" id="evaluate-model-performance-on-test-data">
|
||
<h2>Evaluate model performance on test data<a class="headerlink" href="#evaluate-model-performance-on-test-data" title="Permalink to this headline">¶</a></h2>
|
||
<p>To measure the performance of our network we evaluate how well it does it data it has never seen before, i.e. the test data.<br />
|
||
We measure the performance of the network using the <em>accuracy</em> score.<br />
|
||
The accuracy is as you would expect just the number of images correctly labeled divided by the total number of images. A perfect classifier will have an accuracy score of <span class="math notranslate nohighlight">\(1\)</span>.</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[ \text{Accuracy} = \frac{\sum_{i=1}^n I(\tilde{y}_i = y_i)}{n} ,\]</div>
|
||
<p>where <span class="math notranslate nohighlight">\(I\)</span> is the indicator function, <span class="math notranslate nohighlight">\(1\)</span> if <span class="math notranslate nohighlight">\(\tilde{y}_i = y_i\)</span> and <span class="math notranslate nohighlight">\(0\)</span> otherwise.</p>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">epochs</span> <span class="o">=</span> <span class="mi">100</span>
|
||
<span class="n">batch_size</span> <span class="o">=</span> <span class="mi">100</span>
|
||
|
||
<span class="n">dnn</span> <span class="o">=</span> <span class="n">NeuralNetwork</span><span class="p">(</span><span class="n">X_train</span><span class="p">,</span> <span class="n">Y_train_onehot</span><span class="p">,</span> <span class="n">eta</span><span class="o">=</span><span class="n">eta</span><span class="p">,</span> <span class="n">lmbd</span><span class="o">=</span><span class="n">lmbd</span><span class="p">,</span> <span class="n">epochs</span><span class="o">=</span><span class="n">epochs</span><span class="p">,</span> <span class="n">batch_size</span><span class="o">=</span><span class="n">batch_size</span><span class="p">,</span>
|
||
<span class="n">n_hidden_neurons</span><span class="o">=</span><span class="n">n_hidden_neurons</span><span class="p">,</span> <span class="n">n_categories</span><span class="o">=</span><span class="n">n_categories</span><span class="p">)</span>
|
||
<span class="n">dnn</span><span class="o">.</span><span class="n">train</span><span class="p">()</span>
|
||
<span class="n">test_predict</span> <span class="o">=</span> <span class="n">dnn</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X_test</span><span class="p">)</span>
|
||
|
||
<span class="c1"># accuracy score from scikit library</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="s2">"Accuracy score on test set: "</span><span class="p">,</span> <span class="n">accuracy_score</span><span class="p">(</span><span class="n">Y_test</span><span class="p">,</span> <span class="n">test_predict</span><span class="p">))</span>
|
||
|
||
<span class="c1"># equivalent in numpy</span>
|
||
<span class="k">def</span> <span class="nf">accuracy_score_numpy</span><span class="p">(</span><span class="n">Y_test</span><span class="p">,</span> <span class="n">Y_pred</span><span class="p">):</span>
|
||
<span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">Y_test</span> <span class="o">==</span> <span class="n">Y_pred</span><span class="p">)</span> <span class="o">/</span> <span class="nb">len</span><span class="p">(</span><span class="n">Y_test</span><span class="p">)</span>
|
||
|
||
<span class="c1">#print("Accuracy score on test set: ", accuracy_score_numpy(Y_test, test_predict))</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
<div class="cell_output docutils container">
|
||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Accuracy score on test set: 0.9444444444444444
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<div class="section" id="adjust-hyperparameters">
|
||
<h2>Adjust hyperparameters<a class="headerlink" href="#adjust-hyperparameters" title="Permalink to this headline">¶</a></h2>
|
||
<p>We now perform a grid search to find the optimal hyperparameters for the network.<br />
|
||
Note that we are only using 1 layer with 50 neurons, and human performance is estimated to be around <span class="math notranslate nohighlight">\(98\%\)</span> (<span class="math notranslate nohighlight">\(2\%\)</span> error rate).</p>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">eta_vals</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">logspace</span><span class="p">(</span><span class="o">-</span><span class="mi">5</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">7</span><span class="p">)</span>
|
||
<span class="n">lmbd_vals</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">logspace</span><span class="p">(</span><span class="o">-</span><span class="mi">5</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">7</span><span class="p">)</span>
|
||
<span class="c1"># store the models for later use</span>
|
||
<span class="n">DNN_numpy</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="nb">len</span><span class="p">(</span><span class="n">eta_vals</span><span class="p">),</span> <span class="nb">len</span><span class="p">(</span><span class="n">lmbd_vals</span><span class="p">)),</span> <span class="n">dtype</span><span class="o">=</span><span class="nb">object</span><span class="p">)</span>
|
||
|
||
<span class="c1"># grid search</span>
|
||
<span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">eta</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">eta_vals</span><span class="p">):</span>
|
||
<span class="k">for</span> <span class="n">j</span><span class="p">,</span> <span class="n">lmbd</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">lmbd_vals</span><span class="p">):</span>
|
||
<span class="n">dnn</span> <span class="o">=</span> <span class="n">NeuralNetwork</span><span class="p">(</span><span class="n">X_train</span><span class="p">,</span> <span class="n">Y_train_onehot</span><span class="p">,</span> <span class="n">eta</span><span class="o">=</span><span class="n">eta</span><span class="p">,</span> <span class="n">lmbd</span><span class="o">=</span><span class="n">lmbd</span><span class="p">,</span> <span class="n">epochs</span><span class="o">=</span><span class="n">epochs</span><span class="p">,</span> <span class="n">batch_size</span><span class="o">=</span><span class="n">batch_size</span><span class="p">,</span>
|
||
<span class="n">n_hidden_neurons</span><span class="o">=</span><span class="n">n_hidden_neurons</span><span class="p">,</span> <span class="n">n_categories</span><span class="o">=</span><span class="n">n_categories</span><span class="p">)</span>
|
||
<span class="n">dnn</span><span class="o">.</span><span class="n">train</span><span class="p">()</span>
|
||
|
||
<span class="n">DNN_numpy</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">dnn</span>
|
||
|
||
<span class="n">test_predict</span> <span class="o">=</span> <span class="n">dnn</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X_test</span><span class="p">)</span>
|
||
|
||
<span class="nb">print</span><span class="p">(</span><span class="s2">"Learning rate = "</span><span class="p">,</span> <span class="n">eta</span><span class="p">)</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="s2">"Lambda = "</span><span class="p">,</span> <span class="n">lmbd</span><span class="p">)</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="s2">"Accuracy score on test set: "</span><span class="p">,</span> <span class="n">accuracy_score</span><span class="p">(</span><span class="n">Y_test</span><span class="p">,</span> <span class="n">test_predict</span><span class="p">))</span>
|
||
<span class="nb">print</span><span class="p">()</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
<div class="cell_output docutils container">
|
||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 1e-05
|
||
Lambda = 1e-05
|
||
Accuracy score on test set: 0.11666666666666667
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 1e-05
|
||
Lambda = 0.0001
|
||
Accuracy score on test set: 0.20833333333333334
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 1e-05
|
||
Lambda = 0.001
|
||
Accuracy score on test set: 0.12222222222222222
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 1e-05
|
||
Lambda = 0.01
|
||
Accuracy score on test set: 0.14722222222222223
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 1e-05
|
||
Lambda = 0.1
|
||
Accuracy score on test set: 0.17777777777777778
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 1e-05
|
||
Lambda = 1.0
|
||
Accuracy score on test set: 0.16111111111111112
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 1e-05
|
||
Lambda = 10.0
|
||
Accuracy score on test set: 0.20277777777777778
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 0.0001
|
||
Lambda = 1e-05
|
||
Accuracy score on test set: 0.5305555555555556
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 0.0001
|
||
Lambda = 0.0001
|
||
Accuracy score on test set: 0.5944444444444444
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 0.0001
|
||
Lambda = 0.001
|
||
Accuracy score on test set: 0.5888888888888889
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 0.0001
|
||
Lambda = 0.01
|
||
Accuracy score on test set: 0.6111111111111112
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 0.0001
|
||
Lambda = 0.1
|
||
Accuracy score on test set: 0.5222222222222223
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 0.0001
|
||
Lambda = 1.0
|
||
Accuracy score on test set: 0.5555555555555556
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 0.0001
|
||
Lambda = 10.0
|
||
Accuracy score on test set: 0.8055555555555556
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 0.001
|
||
Lambda = 1e-05
|
||
Accuracy score on test set: 0.85
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 0.001
|
||
Lambda = 0.0001
|
||
Accuracy score on test set: 0.85
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 0.001
|
||
Lambda = 0.001
|
||
Accuracy score on test set: 0.875
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 0.001
|
||
Lambda = 0.01
|
||
Accuracy score on test set: 0.8666666666666667
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 0.001
|
||
Lambda = 0.1
|
||
Accuracy score on test set: 0.8638888888888889
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 0.001
|
||
Lambda = 1.0
|
||
Accuracy score on test set: 0.9555555555555556
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 0.001
|
||
Lambda = 10.0
|
||
Accuracy score on test set: 0.925
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 0.01
|
||
Lambda = 1e-05
|
||
Accuracy score on test set: 0.9472222222222222
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 0.01
|
||
Lambda = 0.0001
|
||
Accuracy score on test set: 0.9277777777777778
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 0.01
|
||
Lambda = 0.001
|
||
Accuracy score on test set: 0.9472222222222222
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 0.01
|
||
Lambda = 0.01
|
||
Accuracy score on test set: 0.9305555555555556
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 0.01
|
||
Lambda = 0.1
|
||
Accuracy score on test set: 0.9555555555555556
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 0.01
|
||
Lambda = 1.0
|
||
Accuracy score on test set: 0.7694444444444445
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 0.01
|
||
Lambda = 10.0
|
||
Accuracy score on test set: 0.19166666666666668
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_94373/953065564.py:4: RuntimeWarning: overflow encountered in exp
|
||
return 1/(1 + np.exp(-x))
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 0.1
|
||
Lambda = 1e-05
|
||
Accuracy score on test set: 0.10555555555555556
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_94373/953065564.py:4: RuntimeWarning: overflow encountered in exp
|
||
return 1/(1 + np.exp(-x))
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 0.1
|
||
Lambda = 0.0001
|
||
Accuracy score on test set: 0.08611111111111111
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_94373/953065564.py:4: RuntimeWarning: overflow encountered in exp
|
||
return 1/(1 + np.exp(-x))
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 0.1
|
||
Lambda = 0.001
|
||
Accuracy score on test set: 0.10555555555555556
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_94373/953065564.py:4: RuntimeWarning: overflow encountered in exp
|
||
return 1/(1 + np.exp(-x))
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 0.1
|
||
Lambda = 0.01
|
||
Accuracy score on test set: 0.08888888888888889
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_94373/953065564.py:4: RuntimeWarning: overflow encountered in exp
|
||
return 1/(1 + np.exp(-x))
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 0.1
|
||
Lambda = 0.1
|
||
Accuracy score on test set: 0.08611111111111111
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_94373/953065564.py:4: RuntimeWarning: overflow encountered in exp
|
||
return 1/(1 + np.exp(-x))
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 0.1
|
||
Lambda = 1.0
|
||
Accuracy score on test set: 0.08888888888888889
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_94373/953065564.py:4: RuntimeWarning: overflow encountered in exp
|
||
return 1/(1 + np.exp(-x))
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 0.1
|
||
Lambda = 10.0
|
||
Accuracy score on test set: 0.09166666666666666
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_94373/953065564.py:4: RuntimeWarning: overflow encountered in exp
|
||
return 1/(1 + np.exp(-x))
|
||
/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_94373/1630775253.py:43: RuntimeWarning: overflow encountered in exp
|
||
exp_term = np.exp(self.z_o)
|
||
/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_94373/1630775253.py:44: RuntimeWarning: invalid value encountered in divide
|
||
self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 1.0
|
||
Lambda = 1e-05
|
||
Accuracy score on test set: 0.07777777777777778
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_94373/953065564.py:4: RuntimeWarning: overflow encountered in exp
|
||
return 1/(1 + np.exp(-x))
|
||
/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_94373/1630775253.py:43: RuntimeWarning: overflow encountered in exp
|
||
exp_term = np.exp(self.z_o)
|
||
/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_94373/1630775253.py:44: RuntimeWarning: invalid value encountered in divide
|
||
self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 1.0
|
||
Lambda = 0.0001
|
||
Accuracy score on test set: 0.07777777777777778
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_94373/953065564.py:4: RuntimeWarning: overflow encountered in exp
|
||
return 1/(1 + np.exp(-x))
|
||
/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_94373/1630775253.py:43: RuntimeWarning: overflow encountered in exp
|
||
exp_term = np.exp(self.z_o)
|
||
/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_94373/1630775253.py:44: RuntimeWarning: invalid value encountered in divide
|
||
self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 1.0
|
||
Lambda = 0.001
|
||
Accuracy score on test set: 0.07777777777777778
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_94373/953065564.py:4: RuntimeWarning: overflow encountered in exp
|
||
return 1/(1 + np.exp(-x))
|
||
/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_94373/1630775253.py:43: RuntimeWarning: overflow encountered in exp
|
||
exp_term = np.exp(self.z_o)
|
||
/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_94373/1630775253.py:44: RuntimeWarning: invalid value encountered in divide
|
||
self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 1.0
|
||
Lambda = 0.01
|
||
Accuracy score on test set: 0.07777777777777778
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_94373/953065564.py:4: RuntimeWarning: overflow encountered in exp
|
||
return 1/(1 + np.exp(-x))
|
||
/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_94373/1630775253.py:43: RuntimeWarning: overflow encountered in exp
|
||
exp_term = np.exp(self.z_o)
|
||
/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_94373/1630775253.py:44: RuntimeWarning: invalid value encountered in divide
|
||
self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 1.0
|
||
Lambda = 0.1
|
||
Accuracy score on test set: 0.07777777777777778
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_94373/953065564.py:4: RuntimeWarning: overflow encountered in exp
|
||
return 1/(1 + np.exp(-x))
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 1.0
|
||
Lambda = 1.0
|
||
Accuracy score on test set: 0.10555555555555556
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_94373/953065564.py:4: RuntimeWarning: overflow encountered in exp
|
||
return 1/(1 + np.exp(-x))
|
||
/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_94373/1630775253.py:43: RuntimeWarning: overflow encountered in exp
|
||
exp_term = np.exp(self.z_o)
|
||
/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_94373/1630775253.py:44: RuntimeWarning: invalid value encountered in divide
|
||
self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 1.0
|
||
Lambda = 10.0
|
||
Accuracy score on test set: 0.07777777777777778
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_94373/953065564.py:4: RuntimeWarning: overflow encountered in exp
|
||
return 1/(1 + np.exp(-x))
|
||
/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_94373/1630775253.py:43: RuntimeWarning: overflow encountered in exp
|
||
exp_term = np.exp(self.z_o)
|
||
/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_94373/1630775253.py:44: RuntimeWarning: invalid value encountered in divide
|
||
self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 10.0
|
||
Lambda = 1e-05
|
||
Accuracy score on test set: 0.07777777777777778
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_94373/953065564.py:4: RuntimeWarning: overflow encountered in exp
|
||
return 1/(1 + np.exp(-x))
|
||
/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_94373/1630775253.py:43: RuntimeWarning: overflow encountered in exp
|
||
exp_term = np.exp(self.z_o)
|
||
/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_94373/1630775253.py:44: RuntimeWarning: invalid value encountered in divide
|
||
self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 10.0
|
||
Lambda = 0.0001
|
||
Accuracy score on test set: 0.07777777777777778
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_94373/953065564.py:4: RuntimeWarning: overflow encountered in exp
|
||
return 1/(1 + np.exp(-x))
|
||
/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_94373/1630775253.py:43: RuntimeWarning: overflow encountered in exp
|
||
exp_term = np.exp(self.z_o)
|
||
/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_94373/1630775253.py:44: RuntimeWarning: invalid value encountered in divide
|
||
self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 10.0
|
||
Lambda = 0.001
|
||
Accuracy score on test set: 0.07777777777777778
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_94373/953065564.py:4: RuntimeWarning: overflow encountered in exp
|
||
return 1/(1 + np.exp(-x))
|
||
/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_94373/1630775253.py:43: RuntimeWarning: overflow encountered in exp
|
||
exp_term = np.exp(self.z_o)
|
||
/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_94373/1630775253.py:44: RuntimeWarning: invalid value encountered in divide
|
||
self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 10.0
|
||
Lambda = 0.01
|
||
Accuracy score on test set: 0.07777777777777778
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_94373/953065564.py:4: RuntimeWarning: overflow encountered in exp
|
||
return 1/(1 + np.exp(-x))
|
||
/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_94373/1630775253.py:43: RuntimeWarning: overflow encountered in exp
|
||
exp_term = np.exp(self.z_o)
|
||
/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_94373/1630775253.py:44: RuntimeWarning: invalid value encountered in divide
|
||
self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 10.0
|
||
Lambda = 0.1
|
||
Accuracy score on test set: 0.07777777777777778
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_94373/953065564.py:4: RuntimeWarning: overflow encountered in exp
|
||
return 1/(1 + np.exp(-x))
|
||
/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_94373/1630775253.py:43: RuntimeWarning: overflow encountered in exp
|
||
exp_term = np.exp(self.z_o)
|
||
/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_94373/1630775253.py:44: RuntimeWarning: invalid value encountered in divide
|
||
self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 10.0
|
||
Lambda = 1.0
|
||
Accuracy score on test set: 0.07777777777777778
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_94373/953065564.py:4: RuntimeWarning: overflow encountered in exp
|
||
return 1/(1 + np.exp(-x))
|
||
/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_94373/1630775253.py:43: RuntimeWarning: overflow encountered in exp
|
||
exp_term = np.exp(self.z_o)
|
||
/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_94373/1630775253.py:44: RuntimeWarning: invalid value encountered in divide
|
||
self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 10.0
|
||
Lambda = 10.0
|
||
Accuracy score on test set: 0.07777777777777778
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<div class="section" id="visualization">
|
||
<h2>Visualization<a class="headerlink" href="#visualization" title="Permalink to this headline">¶</a></h2>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># visual representation of grid search</span>
|
||
<span class="c1"># uses seaborn heatmap, you can also do this with matplotlib imshow</span>
|
||
<span class="kn">import</span> <span class="nn">seaborn</span> <span class="k">as</span> <span class="nn">sns</span>
|
||
|
||
<span class="n">sns</span><span class="o">.</span><span class="n">set</span><span class="p">()</span>
|
||
|
||
<span class="n">train_accuracy</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="nb">len</span><span class="p">(</span><span class="n">eta_vals</span><span class="p">),</span> <span class="nb">len</span><span class="p">(</span><span class="n">lmbd_vals</span><span class="p">)))</span>
|
||
<span class="n">test_accuracy</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="nb">len</span><span class="p">(</span><span class="n">eta_vals</span><span class="p">),</span> <span class="nb">len</span><span class="p">(</span><span class="n">lmbd_vals</span><span class="p">)))</span>
|
||
|
||
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">eta_vals</span><span class="p">)):</span>
|
||
<span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">lmbd_vals</span><span class="p">)):</span>
|
||
<span class="n">dnn</span> <span class="o">=</span> <span class="n">DNN_numpy</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span>
|
||
|
||
<span class="n">train_pred</span> <span class="o">=</span> <span class="n">dnn</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X_train</span><span class="p">)</span>
|
||
<span class="n">test_pred</span> <span class="o">=</span> <span class="n">dnn</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X_test</span><span class="p">)</span>
|
||
|
||
<span class="n">train_accuracy</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">accuracy_score</span><span class="p">(</span><span class="n">Y_train</span><span class="p">,</span> <span class="n">train_pred</span><span class="p">)</span>
|
||
<span class="n">test_accuracy</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">accuracy_score</span><span class="p">(</span><span class="n">Y_test</span><span class="p">,</span> <span class="n">test_pred</span><span class="p">)</span>
|
||
|
||
|
||
<span class="n">fig</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">figsize</span> <span class="o">=</span> <span class="p">(</span><span class="mi">10</span><span class="p">,</span> <span class="mi">10</span><span class="p">))</span>
|
||
<span class="n">sns</span><span class="o">.</span><span class="n">heatmap</span><span class="p">(</span><span class="n">train_accuracy</span><span class="p">,</span> <span class="n">annot</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span> <span class="n">ax</span><span class="o">=</span><span class="n">ax</span><span class="p">,</span> <span class="n">cmap</span><span class="o">=</span><span class="s2">"viridis"</span><span class="p">)</span>
|
||
<span class="n">ax</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s2">"Training Accuracy"</span><span class="p">)</span>
|
||
<span class="n">ax</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s2">"$\eta$"</span><span class="p">)</span>
|
||
<span class="n">ax</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s2">"$\lambda$"</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
|
||
|
||
<span class="n">fig</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">figsize</span> <span class="o">=</span> <span class="p">(</span><span class="mi">10</span><span class="p">,</span> <span class="mi">10</span><span class="p">))</span>
|
||
<span class="n">sns</span><span class="o">.</span><span class="n">heatmap</span><span class="p">(</span><span class="n">test_accuracy</span><span class="p">,</span> <span class="n">annot</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span> <span class="n">ax</span><span class="o">=</span><span class="n">ax</span><span class="p">,</span> <span class="n">cmap</span><span class="o">=</span><span class="s2">"viridis"</span><span class="p">)</span>
|
||
<span class="n">ax</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s2">"Test Accuracy"</span><span class="p">)</span>
|
||
<span class="n">ax</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s2">"$\eta$"</span><span class="p">)</span>
|
||
<span class="n">ax</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s2">"$\lambda$"</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
<div class="cell_output docutils container">
|
||
<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_94373/953065564.py:4: RuntimeWarning: overflow encountered in exp
|
||
return 1/(1 + np.exp(-x))
|
||
/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_94373/953065564.py:4: RuntimeWarning: overflow encountered in exp
|
||
return 1/(1 + np.exp(-x))
|
||
/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_94373/953065564.py:4: RuntimeWarning: overflow encountered in exp
|
||
return 1/(1 + np.exp(-x))
|
||
/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_94373/953065564.py:4: RuntimeWarning: overflow encountered in exp
|
||
return 1/(1 + np.exp(-x))
|
||
/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_94373/953065564.py:4: RuntimeWarning: overflow encountered in exp
|
||
return 1/(1 + np.exp(-x))
|
||
</pre></div>
|
||
</div>
|
||
<img alt="_images/week43_96_1.png" src="_images/week43_96_1.png" />
|
||
<img alt="_images/week43_96_2.png" src="_images/week43_96_2.png" />
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<div class="section" id="scikit-learn-implementation">
|
||
<h2>scikit-learn implementation<a class="headerlink" href="#scikit-learn-implementation" title="Permalink to this headline">¶</a></h2>
|
||
<p><strong>scikit-learn</strong> focuses more
|
||
on traditional machine learning methods, such as regression,
|
||
clustering, decision trees, etc. As such, it has only two types of
|
||
neural networks: Multi Layer Perceptron outputting continuous values,
|
||
<em>MPLRegressor</em>, and Multi Layer Perceptron outputting labels,
|
||
<em>MLPClassifier</em>. We will see how simple it is to use these classes.</p>
|
||
<p><strong>scikit-learn</strong> implements a few improvements from our neural network,
|
||
such as early stopping, a varying learning rate, different
|
||
optimization methods, etc. We would therefore expect a better
|
||
performance overall.</p>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">sklearn.neural_network</span> <span class="kn">import</span> <span class="n">MLPClassifier</span>
|
||
<span class="c1"># store models for later use</span>
|
||
<span class="n">DNN_scikit</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="nb">len</span><span class="p">(</span><span class="n">eta_vals</span><span class="p">),</span> <span class="nb">len</span><span class="p">(</span><span class="n">lmbd_vals</span><span class="p">)),</span> <span class="n">dtype</span><span class="o">=</span><span class="nb">object</span><span class="p">)</span>
|
||
|
||
<span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">eta</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">eta_vals</span><span class="p">):</span>
|
||
<span class="k">for</span> <span class="n">j</span><span class="p">,</span> <span class="n">lmbd</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">lmbd_vals</span><span class="p">):</span>
|
||
<span class="n">dnn</span> <span class="o">=</span> <span class="n">MLPClassifier</span><span class="p">(</span><span class="n">hidden_layer_sizes</span><span class="o">=</span><span class="p">(</span><span class="n">n_hidden_neurons</span><span class="p">),</span> <span class="n">activation</span><span class="o">=</span><span class="s1">'logistic'</span><span class="p">,</span>
|
||
<span class="n">alpha</span><span class="o">=</span><span class="n">lmbd</span><span class="p">,</span> <span class="n">learning_rate_init</span><span class="o">=</span><span class="n">eta</span><span class="p">,</span> <span class="n">max_iter</span><span class="o">=</span><span class="n">epochs</span><span class="p">)</span>
|
||
<span class="n">dnn</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_train</span><span class="p">,</span> <span class="n">Y_train</span><span class="p">)</span>
|
||
|
||
<span class="n">DNN_scikit</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">dnn</span>
|
||
|
||
<span class="nb">print</span><span class="p">(</span><span class="s2">"Learning rate = "</span><span class="p">,</span> <span class="n">eta</span><span class="p">)</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="s2">"Lambda = "</span><span class="p">,</span> <span class="n">lmbd</span><span class="p">)</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="s2">"Accuracy score on test set: "</span><span class="p">,</span> <span class="n">dnn</span><span class="o">.</span><span class="n">score</span><span class="p">(</span><span class="n">X_test</span><span class="p">,</span> <span class="n">Y_test</span><span class="p">))</span>
|
||
<span class="nb">print</span><span class="p">()</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
<div class="cell_output docutils container">
|
||
<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/Users/mhjensen/miniforge3/envs/myenv/lib/python3.9/site-packages/sklearn/neural_network/_multilayer_perceptron.py:691: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
|
||
warnings.warn(
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 1e-05
|
||
Lambda = 1e-05
|
||
Accuracy score on test set: 0.18333333333333332
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/Users/mhjensen/miniforge3/envs/myenv/lib/python3.9/site-packages/sklearn/neural_network/_multilayer_perceptron.py:691: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
|
||
warnings.warn(
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 1e-05
|
||
Lambda = 0.0001
|
||
Accuracy score on test set: 0.18611111111111112
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/Users/mhjensen/miniforge3/envs/myenv/lib/python3.9/site-packages/sklearn/neural_network/_multilayer_perceptron.py:691: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
|
||
warnings.warn(
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 1e-05
|
||
Lambda = 0.001
|
||
Accuracy score on test set: 0.13055555555555556
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/Users/mhjensen/miniforge3/envs/myenv/lib/python3.9/site-packages/sklearn/neural_network/_multilayer_perceptron.py:691: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
|
||
warnings.warn(
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 1e-05
|
||
Lambda = 0.01
|
||
Accuracy score on test set: 0.24444444444444444
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/Users/mhjensen/miniforge3/envs/myenv/lib/python3.9/site-packages/sklearn/neural_network/_multilayer_perceptron.py:691: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
|
||
warnings.warn(
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 1e-05
|
||
Lambda = 0.1
|
||
Accuracy score on test set: 0.23333333333333334
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/Users/mhjensen/miniforge3/envs/myenv/lib/python3.9/site-packages/sklearn/neural_network/_multilayer_perceptron.py:691: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
|
||
warnings.warn(
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 1e-05
|
||
Lambda = 1.0
|
||
Accuracy score on test set: 0.12777777777777777
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/Users/mhjensen/miniforge3/envs/myenv/lib/python3.9/site-packages/sklearn/neural_network/_multilayer_perceptron.py:691: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
|
||
warnings.warn(
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 1e-05
|
||
Lambda = 10.0
|
||
Accuracy score on test set: 0.1527777777777778
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/Users/mhjensen/miniforge3/envs/myenv/lib/python3.9/site-packages/sklearn/neural_network/_multilayer_perceptron.py:691: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
|
||
warnings.warn(
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 0.0001
|
||
Lambda = 1e-05
|
||
Accuracy score on test set: 0.9111111111111111
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/Users/mhjensen/miniforge3/envs/myenv/lib/python3.9/site-packages/sklearn/neural_network/_multilayer_perceptron.py:691: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
|
||
warnings.warn(
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 0.0001
|
||
Lambda = 0.0001
|
||
Accuracy score on test set: 0.8888888888888888
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/Users/mhjensen/miniforge3/envs/myenv/lib/python3.9/site-packages/sklearn/neural_network/_multilayer_perceptron.py:691: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
|
||
warnings.warn(
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 0.0001
|
||
Lambda = 0.001
|
||
Accuracy score on test set: 0.8722222222222222
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/Users/mhjensen/miniforge3/envs/myenv/lib/python3.9/site-packages/sklearn/neural_network/_multilayer_perceptron.py:691: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
|
||
warnings.warn(
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 0.0001
|
||
Lambda = 0.01
|
||
Accuracy score on test set: 0.8305555555555556
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/Users/mhjensen/miniforge3/envs/myenv/lib/python3.9/site-packages/sklearn/neural_network/_multilayer_perceptron.py:691: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
|
||
warnings.warn(
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 0.0001
|
||
Lambda = 0.1
|
||
Accuracy score on test set: 0.8888888888888888
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/Users/mhjensen/miniforge3/envs/myenv/lib/python3.9/site-packages/sklearn/neural_network/_multilayer_perceptron.py:691: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
|
||
warnings.warn(
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 0.0001
|
||
Lambda = 1.0
|
||
Accuracy score on test set: 0.8805555555555555
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/Users/mhjensen/miniforge3/envs/myenv/lib/python3.9/site-packages/sklearn/neural_network/_multilayer_perceptron.py:691: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
|
||
warnings.warn(
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 0.0001
|
||
Lambda = 10.0
|
||
Accuracy score on test set: 0.8944444444444445
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/Users/mhjensen/miniforge3/envs/myenv/lib/python3.9/site-packages/sklearn/neural_network/_multilayer_perceptron.py:691: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
|
||
warnings.warn(
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 0.001
|
||
Lambda = 1e-05
|
||
Accuracy score on test set: 0.975
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/Users/mhjensen/miniforge3/envs/myenv/lib/python3.9/site-packages/sklearn/neural_network/_multilayer_perceptron.py:691: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
|
||
warnings.warn(
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 0.001
|
||
Lambda = 0.0001
|
||
Accuracy score on test set: 0.9777777777777777
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/Users/mhjensen/miniforge3/envs/myenv/lib/python3.9/site-packages/sklearn/neural_network/_multilayer_perceptron.py:691: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
|
||
warnings.warn(
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 0.001
|
||
Lambda = 0.001
|
||
Accuracy score on test set: 0.9805555555555555
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/Users/mhjensen/miniforge3/envs/myenv/lib/python3.9/site-packages/sklearn/neural_network/_multilayer_perceptron.py:691: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
|
||
warnings.warn(
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 0.001
|
||
Lambda = 0.01
|
||
Accuracy score on test set: 0.9861111111111112
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/Users/mhjensen/miniforge3/envs/myenv/lib/python3.9/site-packages/sklearn/neural_network/_multilayer_perceptron.py:691: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
|
||
warnings.warn(
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 0.001
|
||
Lambda = 0.1
|
||
Accuracy score on test set: 0.9805555555555555
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/Users/mhjensen/miniforge3/envs/myenv/lib/python3.9/site-packages/sklearn/neural_network/_multilayer_perceptron.py:691: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
|
||
warnings.warn(
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 0.001
|
||
Lambda = 1.0
|
||
Accuracy score on test set: 0.9777777777777777
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/Users/mhjensen/miniforge3/envs/myenv/lib/python3.9/site-packages/sklearn/neural_network/_multilayer_perceptron.py:691: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
|
||
warnings.warn(
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 0.001
|
||
Lambda = 10.0
|
||
Accuracy score on test set: 0.9444444444444444
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/Users/mhjensen/miniforge3/envs/myenv/lib/python3.9/site-packages/sklearn/neural_network/_multilayer_perceptron.py:691: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
|
||
warnings.warn(
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 0.01
|
||
Lambda = 1e-05
|
||
Accuracy score on test set: 0.9861111111111112
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 0.01
|
||
Lambda = 0.0001
|
||
Accuracy score on test set: 0.9888888888888889
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 0.01
|
||
Lambda = 0.001
|
||
Accuracy score on test set: 0.9888888888888889
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 0.01
|
||
Lambda = 0.01
|
||
Accuracy score on test set: 0.9861111111111112
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 0.01
|
||
Lambda = 0.1
|
||
Accuracy score on test set: 0.9888888888888889
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 0.01
|
||
Lambda = 1.0
|
||
Accuracy score on test set: 0.9722222222222222
|
||
|
||
Learning rate = 0.01
|
||
Lambda = 10.0
|
||
Accuracy score on test set: 0.9527777777777777
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 0.1
|
||
Lambda = 1e-05
|
||
Accuracy score on test set: 0.9027777777777778
|
||
|
||
Learning rate = 0.1
|
||
Lambda = 0.0001
|
||
Accuracy score on test set: 0.8583333333333333
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 0.1
|
||
Lambda = 0.001
|
||
Accuracy score on test set: 0.8722222222222222
|
||
|
||
Learning rate = 0.1
|
||
Lambda = 0.01
|
||
Accuracy score on test set: 0.9055555555555556
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 0.1
|
||
Lambda = 0.1
|
||
Accuracy score on test set: 0.8805555555555555
|
||
|
||
Learning rate = 0.1
|
||
Lambda = 1.0
|
||
Accuracy score on test set: 0.8722222222222222
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 0.1
|
||
Lambda = 10.0
|
||
Accuracy score on test set: 0.8666666666666667
|
||
|
||
Learning rate = 1.0
|
||
Lambda = 1e-05
|
||
Accuracy score on test set: 0.08611111111111111
|
||
|
||
Learning rate = 1.0
|
||
Lambda = 0.0001
|
||
Accuracy score on test set: 0.10555555555555556
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 1.0
|
||
Lambda = 0.001
|
||
Accuracy score on test set: 0.10555555555555556
|
||
|
||
Learning rate = 1.0
|
||
Lambda = 0.01
|
||
Accuracy score on test set: 0.17777777777777778
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 1.0
|
||
Lambda = 0.1
|
||
Accuracy score on test set: 0.08333333333333333
|
||
|
||
Learning rate = 1.0
|
||
Lambda = 1.0
|
||
Accuracy score on test set: 0.08888888888888889
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 1.0
|
||
Lambda = 10.0
|
||
Accuracy score on test set: 0.09444444444444444
|
||
|
||
Learning rate = 10.0
|
||
Lambda = 1e-05
|
||
Accuracy score on test set: 0.17222222222222222
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 10.0
|
||
Lambda = 0.0001
|
||
Accuracy score on test set: 0.11666666666666667
|
||
|
||
Learning rate = 10.0
|
||
Lambda = 0.001
|
||
Accuracy score on test set: 0.10555555555555556
|
||
|
||
Learning rate = 10.0
|
||
Lambda = 0.01
|
||
Accuracy score on test set: 0.1388888888888889
|
||
|
||
Learning rate = 10.0
|
||
Lambda = 0.1
|
||
Accuracy score on test set: 0.11388888888888889
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 10.0
|
||
Lambda = 1.0
|
||
Accuracy score on test set: 0.10555555555555556
|
||
|
||
Learning rate = 10.0
|
||
Lambda = 10.0
|
||
Accuracy score on test set: 0.09444444444444444
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<div class="section" id="id1">
|
||
<h2>Visualization<a class="headerlink" href="#id1" title="Permalink to this headline">¶</a></h2>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># optional</span>
|
||
<span class="c1"># visual representation of grid search</span>
|
||
<span class="c1"># uses seaborn heatmap, could probably do this in matplotlib</span>
|
||
<span class="kn">import</span> <span class="nn">seaborn</span> <span class="k">as</span> <span class="nn">sns</span>
|
||
|
||
<span class="n">sns</span><span class="o">.</span><span class="n">set</span><span class="p">()</span>
|
||
|
||
<span class="n">train_accuracy</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="nb">len</span><span class="p">(</span><span class="n">eta_vals</span><span class="p">),</span> <span class="nb">len</span><span class="p">(</span><span class="n">lmbd_vals</span><span class="p">)))</span>
|
||
<span class="n">test_accuracy</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="nb">len</span><span class="p">(</span><span class="n">eta_vals</span><span class="p">),</span> <span class="nb">len</span><span class="p">(</span><span class="n">lmbd_vals</span><span class="p">)))</span>
|
||
|
||
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">eta_vals</span><span class="p">)):</span>
|
||
<span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">lmbd_vals</span><span class="p">)):</span>
|
||
<span class="n">dnn</span> <span class="o">=</span> <span class="n">DNN_scikit</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span>
|
||
|
||
<span class="n">train_pred</span> <span class="o">=</span> <span class="n">dnn</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X_train</span><span class="p">)</span>
|
||
<span class="n">test_pred</span> <span class="o">=</span> <span class="n">dnn</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X_test</span><span class="p">)</span>
|
||
|
||
<span class="n">train_accuracy</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">accuracy_score</span><span class="p">(</span><span class="n">Y_train</span><span class="p">,</span> <span class="n">train_pred</span><span class="p">)</span>
|
||
<span class="n">test_accuracy</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">accuracy_score</span><span class="p">(</span><span class="n">Y_test</span><span class="p">,</span> <span class="n">test_pred</span><span class="p">)</span>
|
||
|
||
|
||
<span class="n">fig</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">figsize</span> <span class="o">=</span> <span class="p">(</span><span class="mi">10</span><span class="p">,</span> <span class="mi">10</span><span class="p">))</span>
|
||
<span class="n">sns</span><span class="o">.</span><span class="n">heatmap</span><span class="p">(</span><span class="n">train_accuracy</span><span class="p">,</span> <span class="n">annot</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span> <span class="n">ax</span><span class="o">=</span><span class="n">ax</span><span class="p">,</span> <span class="n">cmap</span><span class="o">=</span><span class="s2">"viridis"</span><span class="p">)</span>
|
||
<span class="n">ax</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s2">"Training Accuracy"</span><span class="p">)</span>
|
||
<span class="n">ax</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s2">"$\eta$"</span><span class="p">)</span>
|
||
<span class="n">ax</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s2">"$\lambda$"</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
|
||
|
||
<span class="n">fig</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">figsize</span> <span class="o">=</span> <span class="p">(</span><span class="mi">10</span><span class="p">,</span> <span class="mi">10</span><span class="p">))</span>
|
||
<span class="n">sns</span><span class="o">.</span><span class="n">heatmap</span><span class="p">(</span><span class="n">test_accuracy</span><span class="p">,</span> <span class="n">annot</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span> <span class="n">ax</span><span class="o">=</span><span class="n">ax</span><span class="p">,</span> <span class="n">cmap</span><span class="o">=</span><span class="s2">"viridis"</span><span class="p">)</span>
|
||
<span class="n">ax</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s2">"Test Accuracy"</span><span class="p">)</span>
|
||
<span class="n">ax</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s2">"$\eta$"</span><span class="p">)</span>
|
||
<span class="n">ax</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s2">"$\lambda$"</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
<div class="cell_output docutils container">
|
||
<img alt="_images/week43_100_0.png" src="_images/week43_100_0.png" />
|
||
<img alt="_images/week43_100_1.png" src="_images/week43_100_1.png" />
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<div class="section" id="building-neural-networks-in-tensorflow-and-keras">
|
||
<h2>Building neural networks in Tensorflow and Keras<a class="headerlink" href="#building-neural-networks-in-tensorflow-and-keras" title="Permalink to this headline">¶</a></h2>
|
||
<p>Now we want to build on the experience gained from our neural network implementation in NumPy and scikit-learn
|
||
and use it to construct a neural network in Tensorflow. Once we have constructed a neural network in NumPy
|
||
and Tensorflow, building one in Keras is really quite trivial, though the performance may suffer.</p>
|
||
<p>In our previous example we used only one hidden layer, and in this we will use two. From this it should be quite
|
||
clear how to build one using an arbitrary number of hidden layers, using data structures such as Python lists or
|
||
NumPy arrays.</p>
|
||
</div>
|
||
<div class="section" id="tensorflow">
|
||
<h2>Tensorflow<a class="headerlink" href="#tensorflow" title="Permalink to this headline">¶</a></h2>
|
||
<p>Tensorflow is an open source library machine learning library
|
||
developed by the Google Brain team for internal use. It was released
|
||
under the Apache 2.0 open source license in November 9, 2015.</p>
|
||
<p>Tensorflow is a computational framework that allows you to construct
|
||
machine learning models at different levels of abstraction, from
|
||
high-level, object-oriented APIs like Keras, down to the C++ kernels
|
||
that Tensorflow is built upon. The higher levels of abstraction are
|
||
simpler to use, but less flexible, and our choice of implementation
|
||
should reflect the problems we are trying to solve.</p>
|
||
<p><a class="reference external" href="https://www.tensorflow.org/guide/graphs">Tensorflow uses</a> so-called graphs to represent your computation
|
||
in terms of the dependencies between individual operations, such that you first build a Tensorflow <em>graph</em>
|
||
to represent your model, and then create a Tensorflow <em>session</em> to run the graph.</p>
|
||
<p>In this guide we will analyze the same data as we did in our NumPy and
|
||
scikit-learn tutorial, gathered from the MNIST database of images. We
|
||
will give an introduction to the lower level Python Application
|
||
Program Interfaces (APIs), and see how we use them to build our graph.
|
||
Then we will build (effectively) the same graph in Keras, to see just
|
||
how simple solving a machine learning problem can be.</p>
|
||
<p>To install tensorflow on Unix/Linux systems, use pip as</p>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">pip3</span> <span class="n">install</span> <span class="n">tensorflow</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
<div class="cell_output docutils container">
|
||
<div class="output traceback highlight-ipythontb notranslate"><div class="highlight"><pre><span></span> <span class="n">Cell</span> <span class="n">In</span><span class="p">[</span><span class="mi">12</span><span class="p">],</span> <span class="n">line</span> <span class="mi">1</span>
|
||
<span class="n">pip3</span> <span class="n">install</span> <span class="n">tensorflow</span>
|
||
<span class="o">^</span>
|
||
<span class="ne">SyntaxError</span>: invalid syntax
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<p>and/or if you use <strong>anaconda</strong>, just write (or install from the graphical user interface)
|
||
(current release of CPU-only TensorFlow)</p>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">conda</span> <span class="n">create</span> <span class="o">-</span><span class="n">n</span> <span class="n">tf</span> <span class="n">tensorflow</span>
|
||
<span class="n">conda</span> <span class="n">activate</span> <span class="n">tf</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<p>To install the current release of GPU TensorFlow</p>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">conda</span> <span class="n">create</span> <span class="o">-</span><span class="n">n</span> <span class="n">tf</span><span class="o">-</span><span class="n">gpu</span> <span class="n">tensorflow</span><span class="o">-</span><span class="n">gpu</span>
|
||
<span class="n">conda</span> <span class="n">activate</span> <span class="n">tf</span><span class="o">-</span><span class="n">gpu</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<div class="section" id="using-keras">
|
||
<h2>Using Keras<a class="headerlink" href="#using-keras" title="Permalink to this headline">¶</a></h2>
|
||
<p>Keras is a high level <a class="reference external" href="https://en.wikipedia.org/wiki/Application_programming_interface">neural network</a>
|
||
that supports Tensorflow, CTNK and Theano as backends.<br />
|
||
If you have Anaconda installed you may run the following command</p>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">conda</span> <span class="n">install</span> <span class="n">keras</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<p>You can look up the <a class="reference external" href="https://keras.io/">instructions here</a> for more information.</p>
|
||
<p>We will to a large extent use <strong>keras</strong> in this course.</p>
|
||
</div>
|
||
<div class="section" id="id2">
|
||
<h2>Collect and pre-process data<a class="headerlink" href="#id2" title="Permalink to this headline">¶</a></h2>
|
||
<p>Let us look again at the MINST data set.</p>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># import necessary packages</span>
|
||
<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
|
||
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
|
||
<span class="kn">import</span> <span class="nn">tensorflow</span> <span class="k">as</span> <span class="nn">tf</span>
|
||
<span class="kn">from</span> <span class="nn">sklearn</span> <span class="kn">import</span> <span class="n">datasets</span>
|
||
|
||
|
||
<span class="c1"># ensure the same random numbers appear every time</span>
|
||
<span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">seed</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
|
||
|
||
<span class="c1"># display images in notebook</span>
|
||
<span class="o">%</span><span class="k">matplotlib</span> inline
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">rcParams</span><span class="p">[</span><span class="s1">'figure.figsize'</span><span class="p">]</span> <span class="o">=</span> <span class="p">(</span><span class="mi">12</span><span class="p">,</span><span class="mi">12</span><span class="p">)</span>
|
||
|
||
|
||
<span class="c1"># download MNIST dataset</span>
|
||
<span class="n">digits</span> <span class="o">=</span> <span class="n">datasets</span><span class="o">.</span><span class="n">load_digits</span><span class="p">()</span>
|
||
|
||
<span class="c1"># define inputs and labels</span>
|
||
<span class="n">inputs</span> <span class="o">=</span> <span class="n">digits</span><span class="o">.</span><span class="n">images</span>
|
||
<span class="n">labels</span> <span class="o">=</span> <span class="n">digits</span><span class="o">.</span><span class="n">target</span>
|
||
|
||
<span class="nb">print</span><span class="p">(</span><span class="s2">"inputs = (n_inputs, pixel_width, pixel_height) = "</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">inputs</span><span class="o">.</span><span class="n">shape</span><span class="p">))</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="s2">"labels = (n_inputs) = "</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">labels</span><span class="o">.</span><span class="n">shape</span><span class="p">))</span>
|
||
|
||
|
||
<span class="c1"># flatten the image</span>
|
||
<span class="c1"># the value -1 means dimension is inferred from the remaining dimensions: 8x8 = 64</span>
|
||
<span class="n">n_inputs</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">inputs</span><span class="p">)</span>
|
||
<span class="n">inputs</span> <span class="o">=</span> <span class="n">inputs</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">n_inputs</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">)</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="s2">"X = (n_inputs, n_features) = "</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">inputs</span><span class="o">.</span><span class="n">shape</span><span class="p">))</span>
|
||
|
||
|
||
<span class="c1"># choose some random images to display</span>
|
||
<span class="n">indices</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">n_inputs</span><span class="p">)</span>
|
||
<span class="n">random_indices</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">choice</span><span class="p">(</span><span class="n">indices</span><span class="p">,</span> <span class="n">size</span><span class="o">=</span><span class="mi">5</span><span class="p">)</span>
|
||
|
||
<span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">image</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">digits</span><span class="o">.</span><span class="n">images</span><span class="p">[</span><span class="n">random_indices</span><span class="p">]):</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="n">i</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">axis</span><span class="p">(</span><span class="s1">'off'</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">imshow</span><span class="p">(</span><span class="n">image</span><span class="p">,</span> <span class="n">cmap</span><span class="o">=</span><span class="n">plt</span><span class="o">.</span><span class="n">cm</span><span class="o">.</span><span class="n">gray_r</span><span class="p">,</span> <span class="n">interpolation</span><span class="o">=</span><span class="s1">'nearest'</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">"Label: </span><span class="si">%d</span><span class="s2">"</span> <span class="o">%</span> <span class="n">digits</span><span class="o">.</span><span class="n">target</span><span class="p">[</span><span class="n">random_indices</span><span class="p">[</span><span class="n">i</span><span class="p">]])</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">tensorflow.keras.layers</span> <span class="kn">import</span> <span class="n">Input</span>
|
||
<span class="kn">from</span> <span class="nn">tensorflow.keras.models</span> <span class="kn">import</span> <span class="n">Sequential</span> <span class="c1">#This allows appending layers to existing models</span>
|
||
<span class="kn">from</span> <span class="nn">tensorflow.keras.layers</span> <span class="kn">import</span> <span class="n">Dense</span> <span class="c1">#This allows defining the characteristics of a particular layer</span>
|
||
<span class="kn">from</span> <span class="nn">tensorflow.keras</span> <span class="kn">import</span> <span class="n">optimizers</span> <span class="c1">#This allows using whichever optimiser we want (sgd,adam,RMSprop)</span>
|
||
<span class="kn">from</span> <span class="nn">tensorflow.keras</span> <span class="kn">import</span> <span class="n">regularizers</span> <span class="c1">#This allows using whichever regularizer we want (l1,l2,l1_l2)</span>
|
||
<span class="kn">from</span> <span class="nn">tensorflow.keras.utils</span> <span class="kn">import</span> <span class="n">to_categorical</span> <span class="c1">#This allows using categorical cross entropy as the cost function</span>
|
||
|
||
<span class="kn">from</span> <span class="nn">sklearn.model_selection</span> <span class="kn">import</span> <span class="n">train_test_split</span>
|
||
|
||
<span class="c1"># one-hot representation of labels</span>
|
||
<span class="n">labels</span> <span class="o">=</span> <span class="n">to_categorical</span><span class="p">(</span><span class="n">labels</span><span class="p">)</span>
|
||
|
||
<span class="c1"># split into train and test data</span>
|
||
<span class="n">train_size</span> <span class="o">=</span> <span class="mf">0.8</span>
|
||
<span class="n">test_size</span> <span class="o">=</span> <span class="mi">1</span> <span class="o">-</span> <span class="n">train_size</span>
|
||
<span class="n">X_train</span><span class="p">,</span> <span class="n">X_test</span><span class="p">,</span> <span class="n">Y_train</span><span class="p">,</span> <span class="n">Y_test</span> <span class="o">=</span> <span class="n">train_test_split</span><span class="p">(</span><span class="n">inputs</span><span class="p">,</span> <span class="n">labels</span><span class="p">,</span> <span class="n">train_size</span><span class="o">=</span><span class="n">train_size</span><span class="p">,</span>
|
||
<span class="n">test_size</span><span class="o">=</span><span class="n">test_size</span><span class="p">)</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">epochs</span> <span class="o">=</span> <span class="mi">100</span>
|
||
<span class="n">batch_size</span> <span class="o">=</span> <span class="mi">100</span>
|
||
<span class="n">n_neurons_layer1</span> <span class="o">=</span> <span class="mi">100</span>
|
||
<span class="n">n_neurons_layer2</span> <span class="o">=</span> <span class="mi">50</span>
|
||
<span class="n">n_categories</span> <span class="o">=</span> <span class="mi">10</span>
|
||
<span class="n">eta_vals</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">logspace</span><span class="p">(</span><span class="o">-</span><span class="mi">5</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">7</span><span class="p">)</span>
|
||
<span class="n">lmbd_vals</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">logspace</span><span class="p">(</span><span class="o">-</span><span class="mi">5</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">7</span><span class="p">)</span>
|
||
<span class="k">def</span> <span class="nf">create_neural_network_keras</span><span class="p">(</span><span class="n">n_neurons_layer1</span><span class="p">,</span> <span class="n">n_neurons_layer2</span><span class="p">,</span> <span class="n">n_categories</span><span class="p">,</span> <span class="n">eta</span><span class="p">,</span> <span class="n">lmbd</span><span class="p">):</span>
|
||
<span class="n">model</span> <span class="o">=</span> <span class="n">Sequential</span><span class="p">()</span>
|
||
<span class="n">model</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">Dense</span><span class="p">(</span><span class="n">n_neurons_layer1</span><span class="p">,</span> <span class="n">activation</span><span class="o">=</span><span class="s1">'sigmoid'</span><span class="p">,</span> <span class="n">kernel_regularizer</span><span class="o">=</span><span class="n">regularizers</span><span class="o">.</span><span class="n">l2</span><span class="p">(</span><span class="n">lmbd</span><span class="p">)))</span>
|
||
<span class="n">model</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">Dense</span><span class="p">(</span><span class="n">n_neurons_layer2</span><span class="p">,</span> <span class="n">activation</span><span class="o">=</span><span class="s1">'sigmoid'</span><span class="p">,</span> <span class="n">kernel_regularizer</span><span class="o">=</span><span class="n">regularizers</span><span class="o">.</span><span class="n">l2</span><span class="p">(</span><span class="n">lmbd</span><span class="p">)))</span>
|
||
<span class="n">model</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">Dense</span><span class="p">(</span><span class="n">n_categories</span><span class="p">,</span> <span class="n">activation</span><span class="o">=</span><span class="s1">'softmax'</span><span class="p">))</span>
|
||
|
||
<span class="n">sgd</span> <span class="o">=</span> <span class="n">optimizers</span><span class="o">.</span><span class="n">SGD</span><span class="p">(</span><span class="n">learning_rate</span><span class="o">=</span><span class="n">eta</span><span class="p">)</span>
|
||
<span class="n">model</span><span class="o">.</span><span class="n">compile</span><span class="p">(</span><span class="n">loss</span><span class="o">=</span><span class="s1">'categorical_crossentropy'</span><span class="p">,</span> <span class="n">optimizer</span><span class="o">=</span><span class="n">sgd</span><span class="p">,</span> <span class="n">metrics</span><span class="o">=</span><span class="p">[</span><span class="s1">'accuracy'</span><span class="p">])</span>
|
||
|
||
<span class="k">return</span> <span class="n">model</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">DNN_keras</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="nb">len</span><span class="p">(</span><span class="n">eta_vals</span><span class="p">),</span> <span class="nb">len</span><span class="p">(</span><span class="n">lmbd_vals</span><span class="p">)),</span> <span class="n">dtype</span><span class="o">=</span><span class="nb">object</span><span class="p">)</span>
|
||
|
||
<span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">eta</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">eta_vals</span><span class="p">):</span>
|
||
<span class="k">for</span> <span class="n">j</span><span class="p">,</span> <span class="n">lmbd</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">lmbd_vals</span><span class="p">):</span>
|
||
<span class="n">DNN</span> <span class="o">=</span> <span class="n">create_neural_network_keras</span><span class="p">(</span><span class="n">n_neurons_layer1</span><span class="p">,</span> <span class="n">n_neurons_layer2</span><span class="p">,</span> <span class="n">n_categories</span><span class="p">,</span>
|
||
<span class="n">eta</span><span class="o">=</span><span class="n">eta</span><span class="p">,</span> <span class="n">lmbd</span><span class="o">=</span><span class="n">lmbd</span><span class="p">)</span>
|
||
<span class="n">DNN</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_train</span><span class="p">,</span> <span class="n">Y_train</span><span class="p">,</span> <span class="n">epochs</span><span class="o">=</span><span class="n">epochs</span><span class="p">,</span> <span class="n">batch_size</span><span class="o">=</span><span class="n">batch_size</span><span class="p">,</span> <span class="n">verbose</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
|
||
<span class="n">scores</span> <span class="o">=</span> <span class="n">DNN</span><span class="o">.</span><span class="n">evaluate</span><span class="p">(</span><span class="n">X_test</span><span class="p">,</span> <span class="n">Y_test</span><span class="p">)</span>
|
||
|
||
<span class="n">DNN_keras</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">DNN</span>
|
||
|
||
<span class="nb">print</span><span class="p">(</span><span class="s2">"Learning rate = "</span><span class="p">,</span> <span class="n">eta</span><span class="p">)</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="s2">"Lambda = "</span><span class="p">,</span> <span class="n">lmbd</span><span class="p">)</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="s2">"Test accuracy: </span><span class="si">%.3f</span><span class="s2">"</span> <span class="o">%</span> <span class="n">scores</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
|
||
<span class="nb">print</span><span class="p">()</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># optional</span>
|
||
<span class="c1"># visual representation of grid search</span>
|
||
<span class="c1"># uses seaborn heatmap, could probably do this in matplotlib</span>
|
||
<span class="kn">import</span> <span class="nn">seaborn</span> <span class="k">as</span> <span class="nn">sns</span>
|
||
|
||
<span class="n">sns</span><span class="o">.</span><span class="n">set</span><span class="p">()</span>
|
||
|
||
<span class="n">train_accuracy</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="nb">len</span><span class="p">(</span><span class="n">eta_vals</span><span class="p">),</span> <span class="nb">len</span><span class="p">(</span><span class="n">lmbd_vals</span><span class="p">)))</span>
|
||
<span class="n">test_accuracy</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="nb">len</span><span class="p">(</span><span class="n">eta_vals</span><span class="p">),</span> <span class="nb">len</span><span class="p">(</span><span class="n">lmbd_vals</span><span class="p">)))</span>
|
||
|
||
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">eta_vals</span><span class="p">)):</span>
|
||
<span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">lmbd_vals</span><span class="p">)):</span>
|
||
<span class="n">DNN</span> <span class="o">=</span> <span class="n">DNN_keras</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span>
|
||
|
||
<span class="n">train_accuracy</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">DNN</span><span class="o">.</span><span class="n">evaluate</span><span class="p">(</span><span class="n">X_train</span><span class="p">,</span> <span class="n">Y_train</span><span class="p">)[</span><span class="mi">1</span><span class="p">]</span>
|
||
<span class="n">test_accuracy</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">DNN</span><span class="o">.</span><span class="n">evaluate</span><span class="p">(</span><span class="n">X_test</span><span class="p">,</span> <span class="n">Y_test</span><span class="p">)[</span><span class="mi">1</span><span class="p">]</span>
|
||
|
||
|
||
<span class="n">fig</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">figsize</span> <span class="o">=</span> <span class="p">(</span><span class="mi">10</span><span class="p">,</span> <span class="mi">10</span><span class="p">))</span>
|
||
<span class="n">sns</span><span class="o">.</span><span class="n">heatmap</span><span class="p">(</span><span class="n">train_accuracy</span><span class="p">,</span> <span class="n">annot</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span> <span class="n">ax</span><span class="o">=</span><span class="n">ax</span><span class="p">,</span> <span class="n">cmap</span><span class="o">=</span><span class="s2">"viridis"</span><span class="p">)</span>
|
||
<span class="n">ax</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s2">"Training Accuracy"</span><span class="p">)</span>
|
||
<span class="n">ax</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s2">"$\eta$"</span><span class="p">)</span>
|
||
<span class="n">ax</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s2">"$\lambda$"</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
|
||
|
||
<span class="n">fig</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">figsize</span> <span class="o">=</span> <span class="p">(</span><span class="mi">10</span><span class="p">,</span> <span class="mi">10</span><span class="p">))</span>
|
||
<span class="n">sns</span><span class="o">.</span><span class="n">heatmap</span><span class="p">(</span><span class="n">test_accuracy</span><span class="p">,</span> <span class="n">annot</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span> <span class="n">ax</span><span class="o">=</span><span class="n">ax</span><span class="p">,</span> <span class="n">cmap</span><span class="o">=</span><span class="s2">"viridis"</span><span class="p">)</span>
|
||
<span class="n">ax</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s2">"Test Accuracy"</span><span class="p">)</span>
|
||
<span class="n">ax</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s2">"$\eta$"</span><span class="p">)</span>
|
||
<span class="n">ax</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s2">"$\lambda$"</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<div class="section" id="the-breast-cancer-data-now-with-keras">
|
||
<h2>The Breast Cancer Data, now with Keras<a class="headerlink" href="#the-breast-cancer-data-now-with-keras" title="Permalink to this headline">¶</a></h2>
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<div class="cell docutils container">
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<div class="cell_input docutils container">
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">tensorflow</span> <span class="k">as</span> <span class="nn">tf</span>
|
||
<span class="kn">from</span> <span class="nn">tensorflow.keras.layers</span> <span class="kn">import</span> <span class="n">Input</span>
|
||
<span class="kn">from</span> <span class="nn">tensorflow.keras.models</span> <span class="kn">import</span> <span class="n">Sequential</span> <span class="c1">#This allows appending layers to existing models</span>
|
||
<span class="kn">from</span> <span class="nn">tensorflow.keras.layers</span> <span class="kn">import</span> <span class="n">Dense</span> <span class="c1">#This allows defining the characteristics of a particular layer</span>
|
||
<span class="kn">from</span> <span class="nn">tensorflow.keras</span> <span class="kn">import</span> <span class="n">optimizers</span> <span class="c1">#This allows using whichever optimiser we want (sgd,adam,RMSprop)</span>
|
||
<span class="kn">from</span> <span class="nn">tensorflow.keras</span> <span class="kn">import</span> <span class="n">regularizers</span> <span class="c1">#This allows using whichever regularizer we want (l1,l2,l1_l2)</span>
|
||
<span class="kn">from</span> <span class="nn">tensorflow.keras.utils</span> <span class="kn">import</span> <span class="n">to_categorical</span> <span class="c1">#This allows using categorical cross entropy as the cost function</span>
|
||
<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
|
||
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
|
||
<span class="kn">import</span> <span class="nn">seaborn</span> <span class="k">as</span> <span class="nn">sns</span>
|
||
<span class="kn">from</span> <span class="nn">sklearn.model_selection</span> <span class="kn">import</span> <span class="n">train_test_split</span> <span class="k">as</span> <span class="n">splitter</span>
|
||
<span class="kn">from</span> <span class="nn">sklearn.datasets</span> <span class="kn">import</span> <span class="n">load_breast_cancer</span>
|
||
<span class="kn">import</span> <span class="nn">pickle</span>
|
||
<span class="kn">import</span> <span class="nn">os</span>
|
||
|
||
|
||
<span class="sd">"""Load breast cancer dataset"""</span>
|
||
|
||
<span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">seed</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span> <span class="c1">#create same seed for random number every time</span>
|
||
|
||
<span class="n">cancer</span><span class="o">=</span><span class="n">load_breast_cancer</span><span class="p">()</span> <span class="c1">#Download breast cancer dataset</span>
|
||
|
||
<span class="n">inputs</span><span class="o">=</span><span class="n">cancer</span><span class="o">.</span><span class="n">data</span> <span class="c1">#Feature matrix of 569 rows (samples) and 30 columns (parameters)</span>
|
||
<span class="n">outputs</span><span class="o">=</span><span class="n">cancer</span><span class="o">.</span><span class="n">target</span> <span class="c1">#Label array of 569 rows (0 for benign and 1 for malignant)</span>
|
||
<span class="n">labels</span><span class="o">=</span><span class="n">cancer</span><span class="o">.</span><span class="n">feature_names</span><span class="p">[</span><span class="mi">0</span><span class="p">:</span><span class="mi">30</span><span class="p">]</span>
|
||
|
||
<span class="nb">print</span><span class="p">(</span><span class="s1">'The content of the breast cancer dataset is:'</span><span class="p">)</span> <span class="c1">#Print information about the datasets</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="n">labels</span><span class="p">)</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="s1">'-------------------------'</span><span class="p">)</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="s2">"inputs = "</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">inputs</span><span class="o">.</span><span class="n">shape</span><span class="p">))</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="s2">"outputs = "</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">outputs</span><span class="o">.</span><span class="n">shape</span><span class="p">))</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="s2">"labels = "</span><span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">labels</span><span class="o">.</span><span class="n">shape</span><span class="p">))</span>
|
||
|
||
<span class="n">x</span><span class="o">=</span><span class="n">inputs</span> <span class="c1">#Reassign the Feature and Label matrices to other variables</span>
|
||
<span class="n">y</span><span class="o">=</span><span class="n">outputs</span>
|
||
|
||
<span class="c1">#%% </span>
|
||
|
||
<span class="c1"># Visualisation of dataset (for correlation analysis)</span>
|
||
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">()</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">x</span><span class="p">[:,</span><span class="mi">0</span><span class="p">],</span><span class="n">x</span><span class="p">[:,</span><span class="mi">2</span><span class="p">],</span><span class="n">s</span><span class="o">=</span><span class="mi">40</span><span class="p">,</span><span class="n">c</span><span class="o">=</span><span class="n">y</span><span class="p">,</span><span class="n">cmap</span><span class="o">=</span><span class="n">plt</span><span class="o">.</span><span class="n">cm</span><span class="o">.</span><span class="n">Spectral</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s1">'Mean radius'</span><span class="p">,</span><span class="n">fontweight</span><span class="o">=</span><span class="s1">'bold'</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s1">'Mean perimeter'</span><span class="p">,</span><span class="n">fontweight</span><span class="o">=</span><span class="s1">'bold'</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
|
||
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">()</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">x</span><span class="p">[:,</span><span class="mi">5</span><span class="p">],</span><span class="n">x</span><span class="p">[:,</span><span class="mi">6</span><span class="p">],</span><span class="n">s</span><span class="o">=</span><span class="mi">40</span><span class="p">,</span><span class="n">c</span><span class="o">=</span><span class="n">y</span><span class="p">,</span> <span class="n">cmap</span><span class="o">=</span><span class="n">plt</span><span class="o">.</span><span class="n">cm</span><span class="o">.</span><span class="n">Spectral</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s1">'Mean compactness'</span><span class="p">,</span><span class="n">fontweight</span><span class="o">=</span><span class="s1">'bold'</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s1">'Mean concavity'</span><span class="p">,</span><span class="n">fontweight</span><span class="o">=</span><span class="s1">'bold'</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
|
||
|
||
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">()</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">x</span><span class="p">[:,</span><span class="mi">0</span><span class="p">],</span><span class="n">x</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],</span><span class="n">s</span><span class="o">=</span><span class="mi">40</span><span class="p">,</span><span class="n">c</span><span class="o">=</span><span class="n">y</span><span class="p">,</span><span class="n">cmap</span><span class="o">=</span><span class="n">plt</span><span class="o">.</span><span class="n">cm</span><span class="o">.</span><span class="n">Spectral</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s1">'Mean radius'</span><span class="p">,</span><span class="n">fontweight</span><span class="o">=</span><span class="s1">'bold'</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s1">'Mean texture'</span><span class="p">,</span><span class="n">fontweight</span><span class="o">=</span><span class="s1">'bold'</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
|
||
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">()</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">x</span><span class="p">[:,</span><span class="mi">2</span><span class="p">],</span><span class="n">x</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],</span><span class="n">s</span><span class="o">=</span><span class="mi">40</span><span class="p">,</span><span class="n">c</span><span class="o">=</span><span class="n">y</span><span class="p">,</span><span class="n">cmap</span><span class="o">=</span><span class="n">plt</span><span class="o">.</span><span class="n">cm</span><span class="o">.</span><span class="n">Spectral</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s1">'Mean perimeter'</span><span class="p">,</span><span class="n">fontweight</span><span class="o">=</span><span class="s1">'bold'</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s1">'Mean compactness'</span><span class="p">,</span><span class="n">fontweight</span><span class="o">=</span><span class="s1">'bold'</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
|
||
|
||
|
||
<span class="c1"># Generate training and testing datasets</span>
|
||
|
||
<span class="c1">#Select features relevant to classification (texture,perimeter,compactness and symmetery) </span>
|
||
<span class="c1">#and add to input matrix</span>
|
||
|
||
<span class="n">temp1</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">x</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],(</span><span class="nb">len</span><span class="p">(</span><span class="n">x</span><span class="p">[:,</span><span class="mi">1</span><span class="p">]),</span><span class="mi">1</span><span class="p">))</span>
|
||
<span class="n">temp2</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">x</span><span class="p">[:,</span><span class="mi">2</span><span class="p">],(</span><span class="nb">len</span><span class="p">(</span><span class="n">x</span><span class="p">[:,</span><span class="mi">2</span><span class="p">]),</span><span class="mi">1</span><span class="p">))</span>
|
||
<span class="n">X</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">hstack</span><span class="p">((</span><span class="n">temp1</span><span class="p">,</span><span class="n">temp2</span><span class="p">))</span>
|
||
<span class="n">temp</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">x</span><span class="p">[:,</span><span class="mi">5</span><span class="p">],(</span><span class="nb">len</span><span class="p">(</span><span class="n">x</span><span class="p">[:,</span><span class="mi">5</span><span class="p">]),</span><span class="mi">1</span><span class="p">))</span>
|
||
<span class="n">X</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">hstack</span><span class="p">((</span><span class="n">X</span><span class="p">,</span><span class="n">temp</span><span class="p">))</span>
|
||
<span class="n">temp</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">x</span><span class="p">[:,</span><span class="mi">8</span><span class="p">],(</span><span class="nb">len</span><span class="p">(</span><span class="n">x</span><span class="p">[:,</span><span class="mi">8</span><span class="p">]),</span><span class="mi">1</span><span class="p">))</span>
|
||
<span class="n">X</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">hstack</span><span class="p">((</span><span class="n">X</span><span class="p">,</span><span class="n">temp</span><span class="p">))</span>
|
||
|
||
<span class="n">X_train</span><span class="p">,</span><span class="n">X_test</span><span class="p">,</span><span class="n">y_train</span><span class="p">,</span><span class="n">y_test</span><span class="o">=</span><span class="n">splitter</span><span class="p">(</span><span class="n">X</span><span class="p">,</span><span class="n">y</span><span class="p">,</span><span class="n">test_size</span><span class="o">=</span><span class="mf">0.1</span><span class="p">)</span> <span class="c1">#Split datasets into training and testing</span>
|
||
|
||
<span class="n">y_train</span><span class="o">=</span><span class="n">to_categorical</span><span class="p">(</span><span class="n">y_train</span><span class="p">)</span> <span class="c1">#Convert labels to categorical when using categorical cross entropy</span>
|
||
<span class="n">y_test</span><span class="o">=</span><span class="n">to_categorical</span><span class="p">(</span><span class="n">y_test</span><span class="p">)</span>
|
||
|
||
<span class="k">del</span> <span class="n">temp1</span><span class="p">,</span><span class="n">temp2</span><span class="p">,</span><span class="n">temp</span>
|
||
|
||
<span class="c1"># %%</span>
|
||
|
||
<span class="c1"># Define tunable parameters"</span>
|
||
|
||
<span class="n">eta</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">logspace</span><span class="p">(</span><span class="o">-</span><span class="mi">3</span><span class="p">,</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="mi">3</span><span class="p">)</span> <span class="c1">#Define vector of learning rates (parameter to SGD optimiser)</span>
|
||
<span class="n">lamda</span><span class="o">=</span><span class="mf">0.01</span> <span class="c1">#Define hyperparameter</span>
|
||
<span class="n">n_layers</span><span class="o">=</span><span class="mi">2</span> <span class="c1">#Define number of hidden layers in the model</span>
|
||
<span class="n">n_neuron</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">logspace</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">,</span><span class="n">dtype</span><span class="o">=</span><span class="nb">int</span><span class="p">)</span> <span class="c1">#Define number of neurons per layer</span>
|
||
<span class="n">epochs</span><span class="o">=</span><span class="mi">100</span> <span class="c1">#Number of reiterations over the input data</span>
|
||
<span class="n">batch_size</span><span class="o">=</span><span class="mi">100</span> <span class="c1">#Number of samples per gradient update</span>
|
||
|
||
<span class="c1"># %%</span>
|
||
|
||
<span class="sd">"""Define function to return Deep Neural Network model"""</span>
|
||
|
||
<span class="k">def</span> <span class="nf">NN_model</span><span class="p">(</span><span class="n">inputsize</span><span class="p">,</span><span class="n">n_layers</span><span class="p">,</span><span class="n">n_neuron</span><span class="p">,</span><span class="n">eta</span><span class="p">,</span><span class="n">lamda</span><span class="p">):</span>
|
||
<span class="n">model</span><span class="o">=</span><span class="n">Sequential</span><span class="p">()</span>
|
||
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n_layers</span><span class="p">):</span> <span class="c1">#Run loop to add hidden layers to the model</span>
|
||
<span class="k">if</span> <span class="p">(</span><span class="n">i</span><span class="o">==</span><span class="mi">0</span><span class="p">):</span> <span class="c1">#First layer requires input dimensions</span>
|
||
<span class="n">model</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">Dense</span><span class="p">(</span><span class="n">n_neuron</span><span class="p">,</span><span class="n">activation</span><span class="o">=</span><span class="s1">'relu'</span><span class="p">,</span><span class="n">kernel_regularizer</span><span class="o">=</span><span class="n">regularizers</span><span class="o">.</span><span class="n">l2</span><span class="p">(</span><span class="n">lamda</span><span class="p">),</span><span class="n">input_dim</span><span class="o">=</span><span class="n">inputsize</span><span class="p">))</span>
|
||
<span class="k">else</span><span class="p">:</span> <span class="c1">#Subsequent layers are capable of automatic shape inferencing</span>
|
||
<span class="n">model</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">Dense</span><span class="p">(</span><span class="n">n_neuron</span><span class="p">,</span><span class="n">activation</span><span class="o">=</span><span class="s1">'relu'</span><span class="p">,</span><span class="n">kernel_regularizer</span><span class="o">=</span><span class="n">regularizers</span><span class="o">.</span><span class="n">l2</span><span class="p">(</span><span class="n">lamda</span><span class="p">)))</span>
|
||
<span class="n">model</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">Dense</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="n">activation</span><span class="o">=</span><span class="s1">'softmax'</span><span class="p">))</span> <span class="c1">#2 outputs - ordered and disordered (softmax for prob)</span>
|
||
<span class="n">sgd</span><span class="o">=</span><span class="n">optimizers</span><span class="o">.</span><span class="n">SGD</span><span class="p">(</span><span class="n">learning_rate</span><span class="o">=</span><span class="n">eta</span><span class="p">)</span>
|
||
<span class="n">model</span><span class="o">.</span><span class="n">compile</span><span class="p">(</span><span class="n">loss</span><span class="o">=</span><span class="s1">'categorical_crossentropy'</span><span class="p">,</span><span class="n">optimizer</span><span class="o">=</span><span class="n">sgd</span><span class="p">,</span><span class="n">metrics</span><span class="o">=</span><span class="p">[</span><span class="s1">'accuracy'</span><span class="p">])</span>
|
||
<span class="k">return</span> <span class="n">model</span>
|
||
|
||
|
||
<span class="n">Train_accuracy</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="nb">len</span><span class="p">(</span><span class="n">n_neuron</span><span class="p">),</span><span class="nb">len</span><span class="p">(</span><span class="n">eta</span><span class="p">)))</span> <span class="c1">#Define matrices to store accuracy scores as a function</span>
|
||
<span class="n">Test_accuracy</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="nb">len</span><span class="p">(</span><span class="n">n_neuron</span><span class="p">),</span><span class="nb">len</span><span class="p">(</span><span class="n">eta</span><span class="p">)))</span> <span class="c1">#of learning rate and number of hidden neurons for </span>
|
||
|
||
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">n_neuron</span><span class="p">)):</span> <span class="c1">#run loops over hidden neurons and learning rates to calculate </span>
|
||
<span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">eta</span><span class="p">)):</span> <span class="c1">#accuracy scores </span>
|
||
<span class="n">DNN_model</span><span class="o">=</span><span class="n">NN_model</span><span class="p">(</span><span class="n">X_train</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span><span class="n">n_layers</span><span class="p">,</span><span class="n">n_neuron</span><span class="p">[</span><span class="n">i</span><span class="p">],</span><span class="n">eta</span><span class="p">[</span><span class="n">j</span><span class="p">],</span><span class="n">lamda</span><span class="p">)</span>
|
||
<span class="n">DNN_model</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_train</span><span class="p">,</span><span class="n">y_train</span><span class="p">,</span><span class="n">epochs</span><span class="o">=</span><span class="n">epochs</span><span class="p">,</span><span class="n">batch_size</span><span class="o">=</span><span class="n">batch_size</span><span class="p">,</span><span class="n">verbose</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
|
||
<span class="n">Train_accuracy</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="n">j</span><span class="p">]</span><span class="o">=</span><span class="n">DNN_model</span><span class="o">.</span><span class="n">evaluate</span><span class="p">(</span><span class="n">X_train</span><span class="p">,</span><span class="n">y_train</span><span class="p">)[</span><span class="mi">1</span><span class="p">]</span>
|
||
<span class="n">Test_accuracy</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="n">j</span><span class="p">]</span><span class="o">=</span><span class="n">DNN_model</span><span class="o">.</span><span class="n">evaluate</span><span class="p">(</span><span class="n">X_test</span><span class="p">,</span><span class="n">y_test</span><span class="p">)[</span><span class="mi">1</span><span class="p">]</span>
|
||
|
||
|
||
<span class="k">def</span> <span class="nf">plot_data</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">,</span><span class="n">data</span><span class="p">,</span><span class="n">title</span><span class="o">=</span><span class="kc">None</span><span class="p">):</span>
|
||
|
||
<span class="c1"># plot results</span>
|
||
<span class="n">fontsize</span><span class="o">=</span><span class="mi">16</span>
|
||
|
||
|
||
<span class="n">fig</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">()</span>
|
||
<span class="n">ax</span> <span class="o">=</span> <span class="n">fig</span><span class="o">.</span><span class="n">add_subplot</span><span class="p">(</span><span class="mi">111</span><span class="p">)</span>
|
||
<span class="n">cax</span> <span class="o">=</span> <span class="n">ax</span><span class="o">.</span><span class="n">matshow</span><span class="p">(</span><span class="n">data</span><span class="p">,</span> <span class="n">interpolation</span><span class="o">=</span><span class="s1">'nearest'</span><span class="p">,</span> <span class="n">vmin</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="n">vmax</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
|
||
|
||
<span class="n">cbar</span><span class="o">=</span><span class="n">fig</span><span class="o">.</span><span class="n">colorbar</span><span class="p">(</span><span class="n">cax</span><span class="p">)</span>
|
||
<span class="n">cbar</span><span class="o">.</span><span class="n">ax</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">'accuracy (%)'</span><span class="p">,</span><span class="n">rotation</span><span class="o">=</span><span class="mi">90</span><span class="p">,</span><span class="n">fontsize</span><span class="o">=</span><span class="n">fontsize</span><span class="p">)</span>
|
||
<span class="n">cbar</span><span class="o">.</span><span class="n">set_ticks</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span><span class="mf">.2</span><span class="p">,</span><span class="mf">.4</span><span class="p">,</span><span class="mf">0.6</span><span class="p">,</span><span class="mf">0.8</span><span class="p">,</span><span class="mf">1.0</span><span class="p">])</span>
|
||
<span class="n">cbar</span><span class="o">.</span><span class="n">set_ticklabels</span><span class="p">([</span><span class="s1">'0%'</span><span class="p">,</span><span class="s1">'20%'</span><span class="p">,</span><span class="s1">'40%'</span><span class="p">,</span><span class="s1">'60%'</span><span class="p">,</span><span class="s1">'80%'</span><span class="p">,</span><span class="s1">'100%'</span><span class="p">])</span>
|
||
|
||
<span class="c1"># put text on matrix elements</span>
|
||
<span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">x_val</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">x</span><span class="p">))):</span>
|
||
<span class="k">for</span> <span class="n">j</span><span class="p">,</span> <span class="n">y_val</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">y</span><span class="p">))):</span>
|
||
<span class="n">c</span> <span class="o">=</span> <span class="s2">"$</span><span class="si">{0:.1f}</span><span class="se">\\</span><span class="s2">%$"</span><span class="o">.</span><span class="n">format</span><span class="p">(</span> <span class="mi">100</span><span class="o">*</span><span class="n">data</span><span class="p">[</span><span class="n">j</span><span class="p">,</span><span class="n">i</span><span class="p">])</span>
|
||
<span class="n">ax</span><span class="o">.</span><span class="n">text</span><span class="p">(</span><span class="n">x_val</span><span class="p">,</span> <span class="n">y_val</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="n">va</span><span class="o">=</span><span class="s1">'center'</span><span class="p">,</span> <span class="n">ha</span><span class="o">=</span><span class="s1">'center'</span><span class="p">)</span>
|
||
|
||
<span class="c1"># convert axis vaues to to string labels</span>
|
||
<span class="n">x</span><span class="o">=</span><span class="p">[</span><span class="nb">str</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">x</span><span class="p">]</span>
|
||
<span class="n">y</span><span class="o">=</span><span class="p">[</span><span class="nb">str</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">y</span><span class="p">]</span>
|
||
|
||
|
||
<span class="n">ax</span><span class="o">.</span><span class="n">set_xticklabels</span><span class="p">([</span><span class="s1">''</span><span class="p">]</span><span class="o">+</span><span class="n">x</span><span class="p">)</span>
|
||
<span class="n">ax</span><span class="o">.</span><span class="n">set_yticklabels</span><span class="p">([</span><span class="s1">''</span><span class="p">]</span><span class="o">+</span><span class="n">y</span><span class="p">)</span>
|
||
|
||
<span class="n">ax</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s1">'$</span><span class="se">\\</span><span class="s1">mathrm{learning</span><span class="se">\\</span><span class="s1"> rate}$'</span><span class="p">,</span><span class="n">fontsize</span><span class="o">=</span><span class="n">fontsize</span><span class="p">)</span>
|
||
<span class="n">ax</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">'$</span><span class="se">\\</span><span class="s1">mathrm{hidden</span><span class="se">\\</span><span class="s1"> neurons}$'</span><span class="p">,</span><span class="n">fontsize</span><span class="o">=</span><span class="n">fontsize</span><span class="p">)</span>
|
||
<span class="k">if</span> <span class="n">title</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span><span class="p">:</span>
|
||
<span class="n">ax</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="n">title</span><span class="p">)</span>
|
||
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">tight_layout</span><span class="p">()</span>
|
||
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
|
||
|
||
<span class="n">plot_data</span><span class="p">(</span><span class="n">eta</span><span class="p">,</span><span class="n">n_neuron</span><span class="p">,</span><span class="n">Train_accuracy</span><span class="p">,</span> <span class="s1">'training'</span><span class="p">)</span>
|
||
<span class="n">plot_data</span><span class="p">(</span><span class="n">eta</span><span class="p">,</span><span class="n">n_neuron</span><span class="p">,</span><span class="n">Test_accuracy</span><span class="p">,</span> <span class="s1">'testing'</span><span class="p">)</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<div class="section" id="building-a-neural-network-code">
|
||
<h2>Building a neural network code<a class="headerlink" href="#building-a-neural-network-code" title="Permalink to this headline">¶</a></h2>
|
||
<p>Here we present a flexible object oriented codebase
|
||
for a feed forward neural network, along with a demonstration of how
|
||
to use it. Before we get into the details of the neural network, we
|
||
will first present some implementations of various schedulers, cost
|
||
functions and activation functions that can be used together with the
|
||
neural network.</p>
|
||
<p>The codes here were developed by Eric Reber and Gregor Kajda during spring 2023.</p>
|
||
<div class="section" id="learning-rate-methods">
|
||
<h3>Learning rate methods<a class="headerlink" href="#learning-rate-methods" title="Permalink to this headline">¶</a></h3>
|
||
<p>The code below shows object oriented implementations of the Constant,
|
||
Momentum, Adagrad, AdagradMomentum, RMS prop and Adam schedulers. All
|
||
of the classes belong to the shared abstract Scheduler class, and
|
||
share the update_change() and reset() methods allowing for any of the
|
||
schedulers to be seamlessly used during the training stage, as will
|
||
later be shown in the fit() method of the neural
|
||
network. Update_change() only has one parameter, the gradient
|
||
(<span class="math notranslate nohighlight">\(δ^l_ja^{l−1}_k\)</span>), and returns the change which will be subtracted
|
||
from the weights. The reset() function takes no parameters, and resets
|
||
the desired variables. For Constant and Momentum, reset does nothing.</p>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">autograd.numpy</span> <span class="k">as</span> <span class="nn">np</span>
|
||
|
||
<span class="k">class</span> <span class="nc">Scheduler</span><span class="p">:</span>
|
||
<span class="w"> </span><span class="sd">"""</span>
|
||
<span class="sd"> Abstract class for Schedulers</span>
|
||
<span class="sd"> """</span>
|
||
|
||
<span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">eta</span><span class="p">):</span>
|
||
<span class="bp">self</span><span class="o">.</span><span class="n">eta</span> <span class="o">=</span> <span class="n">eta</span>
|
||
|
||
<span class="c1"># should be overwritten</span>
|
||
<span class="k">def</span> <span class="nf">update_change</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">gradient</span><span class="p">):</span>
|
||
<span class="k">raise</span> <span class="ne">NotImplementedError</span>
|
||
|
||
<span class="c1"># overwritten if needed</span>
|
||
<span class="k">def</span> <span class="nf">reset</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
|
||
<span class="k">pass</span>
|
||
|
||
|
||
<span class="k">class</span> <span class="nc">Constant</span><span class="p">(</span><span class="n">Scheduler</span><span class="p">):</span>
|
||
<span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">eta</span><span class="p">):</span>
|
||
<span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="fm">__init__</span><span class="p">(</span><span class="n">eta</span><span class="p">)</span>
|
||
|
||
<span class="k">def</span> <span class="nf">update_change</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">gradient</span><span class="p">):</span>
|
||
<span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">eta</span> <span class="o">*</span> <span class="n">gradient</span>
|
||
|
||
<span class="k">def</span> <span class="nf">reset</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
|
||
<span class="k">pass</span>
|
||
|
||
|
||
<span class="k">class</span> <span class="nc">Momentum</span><span class="p">(</span><span class="n">Scheduler</span><span class="p">):</span>
|
||
<span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">eta</span><span class="p">:</span> <span class="nb">float</span><span class="p">,</span> <span class="n">momentum</span><span class="p">:</span> <span class="nb">float</span><span class="p">):</span>
|
||
<span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="fm">__init__</span><span class="p">(</span><span class="n">eta</span><span class="p">)</span>
|
||
<span class="bp">self</span><span class="o">.</span><span class="n">momentum</span> <span class="o">=</span> <span class="n">momentum</span>
|
||
<span class="bp">self</span><span class="o">.</span><span class="n">change</span> <span class="o">=</span> <span class="mi">0</span>
|
||
|
||
<span class="k">def</span> <span class="nf">update_change</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">gradient</span><span class="p">):</span>
|
||
<span class="bp">self</span><span class="o">.</span><span class="n">change</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">momentum</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">change</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">eta</span> <span class="o">*</span> <span class="n">gradient</span>
|
||
<span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">change</span>
|
||
|
||
<span class="k">def</span> <span class="nf">reset</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
|
||
<span class="k">pass</span>
|
||
|
||
|
||
<span class="k">class</span> <span class="nc">Adagrad</span><span class="p">(</span><span class="n">Scheduler</span><span class="p">):</span>
|
||
<span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">eta</span><span class="p">):</span>
|
||
<span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="fm">__init__</span><span class="p">(</span><span class="n">eta</span><span class="p">)</span>
|
||
<span class="bp">self</span><span class="o">.</span><span class="n">G_t</span> <span class="o">=</span> <span class="kc">None</span>
|
||
|
||
<span class="k">def</span> <span class="nf">update_change</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">gradient</span><span class="p">):</span>
|
||
<span class="n">delta</span> <span class="o">=</span> <span class="mf">1e-8</span> <span class="c1"># avoid division ny zero</span>
|
||
|
||
<span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">G_t</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
|
||
<span class="bp">self</span><span class="o">.</span><span class="n">G_t</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">gradient</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">gradient</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">]))</span>
|
||
|
||
<span class="bp">self</span><span class="o">.</span><span class="n">G_t</span> <span class="o">+=</span> <span class="n">gradient</span> <span class="o">@</span> <span class="n">gradient</span><span class="o">.</span><span class="n">T</span>
|
||
|
||
<span class="n">G_t_inverse</span> <span class="o">=</span> <span class="mi">1</span> <span class="o">/</span> <span class="p">(</span>
|
||
<span class="n">delta</span> <span class="o">+</span> <span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">diagonal</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">G_t</span><span class="p">),</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">G_t</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="mi">1</span><span class="p">)))</span>
|
||
<span class="p">)</span>
|
||
<span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">eta</span> <span class="o">*</span> <span class="n">gradient</span> <span class="o">*</span> <span class="n">G_t_inverse</span>
|
||
|
||
<span class="k">def</span> <span class="nf">reset</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
|
||
<span class="bp">self</span><span class="o">.</span><span class="n">G_t</span> <span class="o">=</span> <span class="kc">None</span>
|
||
|
||
|
||
<span class="k">class</span> <span class="nc">AdagradMomentum</span><span class="p">(</span><span class="n">Scheduler</span><span class="p">):</span>
|
||
<span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">eta</span><span class="p">,</span> <span class="n">momentum</span><span class="p">):</span>
|
||
<span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="fm">__init__</span><span class="p">(</span><span class="n">eta</span><span class="p">)</span>
|
||
<span class="bp">self</span><span class="o">.</span><span class="n">G_t</span> <span class="o">=</span> <span class="kc">None</span>
|
||
<span class="bp">self</span><span class="o">.</span><span class="n">momentum</span> <span class="o">=</span> <span class="n">momentum</span>
|
||
<span class="bp">self</span><span class="o">.</span><span class="n">change</span> <span class="o">=</span> <span class="mi">0</span>
|
||
|
||
<span class="k">def</span> <span class="nf">update_change</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">gradient</span><span class="p">):</span>
|
||
<span class="n">delta</span> <span class="o">=</span> <span class="mf">1e-8</span> <span class="c1"># avoid division ny zero</span>
|
||
|
||
<span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">G_t</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
|
||
<span class="bp">self</span><span class="o">.</span><span class="n">G_t</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">gradient</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">gradient</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">]))</span>
|
||
|
||
<span class="bp">self</span><span class="o">.</span><span class="n">G_t</span> <span class="o">+=</span> <span class="n">gradient</span> <span class="o">@</span> <span class="n">gradient</span><span class="o">.</span><span class="n">T</span>
|
||
|
||
<span class="n">G_t_inverse</span> <span class="o">=</span> <span class="mi">1</span> <span class="o">/</span> <span class="p">(</span>
|
||
<span class="n">delta</span> <span class="o">+</span> <span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">diagonal</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">G_t</span><span class="p">),</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">G_t</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="mi">1</span><span class="p">)))</span>
|
||
<span class="p">)</span>
|
||
<span class="bp">self</span><span class="o">.</span><span class="n">change</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">change</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">momentum</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">eta</span> <span class="o">*</span> <span class="n">gradient</span> <span class="o">*</span> <span class="n">G_t_inverse</span>
|
||
<span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">change</span>
|
||
|
||
<span class="k">def</span> <span class="nf">reset</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
|
||
<span class="bp">self</span><span class="o">.</span><span class="n">G_t</span> <span class="o">=</span> <span class="kc">None</span>
|
||
|
||
|
||
<span class="k">class</span> <span class="nc">RMS_prop</span><span class="p">(</span><span class="n">Scheduler</span><span class="p">):</span>
|
||
<span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">eta</span><span class="p">,</span> <span class="n">rho</span><span class="p">):</span>
|
||
<span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="fm">__init__</span><span class="p">(</span><span class="n">eta</span><span class="p">)</span>
|
||
<span class="bp">self</span><span class="o">.</span><span class="n">rho</span> <span class="o">=</span> <span class="n">rho</span>
|
||
<span class="bp">self</span><span class="o">.</span><span class="n">second</span> <span class="o">=</span> <span class="mf">0.0</span>
|
||
|
||
<span class="k">def</span> <span class="nf">update_change</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">gradient</span><span class="p">):</span>
|
||
<span class="n">delta</span> <span class="o">=</span> <span class="mf">1e-8</span> <span class="c1"># avoid division ny zero</span>
|
||
<span class="bp">self</span><span class="o">.</span><span class="n">second</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">rho</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">second</span> <span class="o">+</span> <span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">rho</span><span class="p">)</span> <span class="o">*</span> <span class="n">gradient</span> <span class="o">*</span> <span class="n">gradient</span>
|
||
<span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">eta</span> <span class="o">*</span> <span class="n">gradient</span> <span class="o">/</span> <span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">second</span> <span class="o">+</span> <span class="n">delta</span><span class="p">))</span>
|
||
|
||
<span class="k">def</span> <span class="nf">reset</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
|
||
<span class="bp">self</span><span class="o">.</span><span class="n">second</span> <span class="o">=</span> <span class="mf">0.0</span>
|
||
|
||
|
||
<span class="k">class</span> <span class="nc">Adam</span><span class="p">(</span><span class="n">Scheduler</span><span class="p">):</span>
|
||
<span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">eta</span><span class="p">,</span> <span class="n">rho</span><span class="p">,</span> <span class="n">rho2</span><span class="p">):</span>
|
||
<span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="fm">__init__</span><span class="p">(</span><span class="n">eta</span><span class="p">)</span>
|
||
<span class="bp">self</span><span class="o">.</span><span class="n">rho</span> <span class="o">=</span> <span class="n">rho</span>
|
||
<span class="bp">self</span><span class="o">.</span><span class="n">rho2</span> <span class="o">=</span> <span class="n">rho2</span>
|
||
<span class="bp">self</span><span class="o">.</span><span class="n">moment</span> <span class="o">=</span> <span class="mi">0</span>
|
||
<span class="bp">self</span><span class="o">.</span><span class="n">second</span> <span class="o">=</span> <span class="mi">0</span>
|
||
<span class="bp">self</span><span class="o">.</span><span class="n">n_epochs</span> <span class="o">=</span> <span class="mi">1</span>
|
||
|
||
<span class="k">def</span> <span class="nf">update_change</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">gradient</span><span class="p">):</span>
|
||
<span class="n">delta</span> <span class="o">=</span> <span class="mf">1e-8</span> <span class="c1"># avoid division ny zero</span>
|
||
|
||
<span class="bp">self</span><span class="o">.</span><span class="n">moment</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">rho</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">moment</span> <span class="o">+</span> <span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">rho</span><span class="p">)</span> <span class="o">*</span> <span class="n">gradient</span>
|
||
<span class="bp">self</span><span class="o">.</span><span class="n">second</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">rho2</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">second</span> <span class="o">+</span> <span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">rho2</span><span class="p">)</span> <span class="o">*</span> <span class="n">gradient</span> <span class="o">*</span> <span class="n">gradient</span>
|
||
|
||
<span class="n">moment_corrected</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">moment</span> <span class="o">/</span> <span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">rho</span><span class="o">**</span><span class="bp">self</span><span class="o">.</span><span class="n">n_epochs</span><span class="p">)</span>
|
||
<span class="n">second_corrected</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">second</span> <span class="o">/</span> <span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">rho2</span><span class="o">**</span><span class="bp">self</span><span class="o">.</span><span class="n">n_epochs</span><span class="p">)</span>
|
||
|
||
<span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">eta</span> <span class="o">*</span> <span class="n">moment_corrected</span> <span class="o">/</span> <span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="n">second_corrected</span> <span class="o">+</span> <span class="n">delta</span><span class="p">))</span>
|
||
|
||
<span class="k">def</span> <span class="nf">reset</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
|
||
<span class="bp">self</span><span class="o">.</span><span class="n">n_epochs</span> <span class="o">+=</span> <span class="mi">1</span>
|
||
<span class="bp">self</span><span class="o">.</span><span class="n">moment</span> <span class="o">=</span> <span class="mi">0</span>
|
||
<span class="bp">self</span><span class="o">.</span><span class="n">second</span> <span class="o">=</span> <span class="mi">0</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<div class="section" id="usage-of-the-above-learning-rate-schedulers">
|
||
<h3>Usage of the above learning rate schedulers<a class="headerlink" href="#usage-of-the-above-learning-rate-schedulers" title="Permalink to this headline">¶</a></h3>
|
||
<p>To initalize a scheduler, simply create the object and pass in the
|
||
necessary parameters such as the learning rate and the momentum as
|
||
shown below. As the Scheduler class is an abstract class it should not
|
||
called directly, and will raise an error upon usage.</p>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">momentum_scheduler</span> <span class="o">=</span> <span class="n">Momentum</span><span class="p">(</span><span class="n">eta</span><span class="o">=</span><span class="mf">1e-3</span><span class="p">,</span> <span class="n">momentum</span><span class="o">=</span><span class="mf">0.9</span><span class="p">)</span>
|
||
<span class="n">adam_scheduler</span> <span class="o">=</span> <span class="n">Adam</span><span class="p">(</span><span class="n">eta</span><span class="o">=</span><span class="mf">1e-3</span><span class="p">,</span> <span class="n">rho</span><span class="o">=</span><span class="mf">0.9</span><span class="p">,</span> <span class="n">rho2</span><span class="o">=</span><span class="mf">0.999</span><span class="p">)</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<p>Here is a small example for how a segment of code using schedulers
|
||
could look. Switching out the schedulers is simple.</p>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">weights</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">ones</span><span class="p">((</span><span class="mi">3</span><span class="p">,</span><span class="mi">3</span><span class="p">))</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"Before scheduler:</span><span class="se">\n</span><span class="si">{</span><span class="n">weights</span><span class="si">=}</span><span class="s2">"</span><span class="p">)</span>
|
||
|
||
<span class="n">epochs</span> <span class="o">=</span> <span class="mi">10</span>
|
||
<span class="k">for</span> <span class="n">e</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">epochs</span><span class="p">):</span>
|
||
<span class="n">gradient</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">rand</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
|
||
<span class="n">change</span> <span class="o">=</span> <span class="n">adam_scheduler</span><span class="o">.</span><span class="n">update_change</span><span class="p">(</span><span class="n">gradient</span><span class="p">)</span>
|
||
<span class="n">weights</span> <span class="o">=</span> <span class="n">weights</span> <span class="o">-</span> <span class="n">change</span>
|
||
<span class="n">adam_scheduler</span><span class="o">.</span><span class="n">reset</span><span class="p">()</span>
|
||
|
||
<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"</span><span class="se">\n</span><span class="s2">After scheduler:</span><span class="se">\n</span><span class="si">{</span><span class="n">weights</span><span class="si">=}</span><span class="s2">"</span><span class="p">)</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<div class="section" id="cost-functions">
|
||
<h3>Cost functions<a class="headerlink" href="#cost-functions" title="Permalink to this headline">¶</a></h3>
|
||
<p>Here we discuss cost functions that can be used when creating the
|
||
neural network. Every cost function takes the target vector as its
|
||
parameter, and returns a function valued only at <span class="math notranslate nohighlight">\(x\)</span> such that it may
|
||
easily be differentiated.</p>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">autograd.numpy</span> <span class="k">as</span> <span class="nn">np</span>
|
||
|
||
<span class="k">def</span> <span class="nf">CostOLS</span><span class="p">(</span><span class="n">target</span><span class="p">):</span>
|
||
|
||
<span class="k">def</span> <span class="nf">func</span><span class="p">(</span><span class="n">X</span><span class="p">):</span>
|
||
<span class="k">return</span> <span class="p">(</span><span class="mf">1.0</span> <span class="o">/</span> <span class="n">target</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">((</span><span class="n">target</span> <span class="o">-</span> <span class="n">X</span><span class="p">)</span> <span class="o">**</span> <span class="mi">2</span><span class="p">)</span>
|
||
|
||
<span class="k">return</span> <span class="n">func</span>
|
||
|
||
|
||
<span class="k">def</span> <span class="nf">CostLogReg</span><span class="p">(</span><span class="n">target</span><span class="p">):</span>
|
||
|
||
<span class="k">def</span> <span class="nf">func</span><span class="p">(</span><span class="n">X</span><span class="p">):</span>
|
||
|
||
<span class="k">return</span> <span class="o">-</span><span class="p">(</span><span class="mf">1.0</span> <span class="o">/</span> <span class="n">target</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span>
|
||
<span class="p">(</span><span class="n">target</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">log</span><span class="p">(</span><span class="n">X</span> <span class="o">+</span> <span class="mf">10e-10</span><span class="p">))</span> <span class="o">+</span> <span class="p">((</span><span class="mi">1</span> <span class="o">-</span> <span class="n">target</span><span class="p">)</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">log</span><span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">X</span> <span class="o">+</span> <span class="mf">10e-10</span><span class="p">))</span>
|
||
<span class="p">)</span>
|
||
|
||
<span class="k">return</span> <span class="n">func</span>
|
||
|
||
|
||
<span class="k">def</span> <span class="nf">CostCrossEntropy</span><span class="p">(</span><span class="n">target</span><span class="p">):</span>
|
||
|
||
<span class="k">def</span> <span class="nf">func</span><span class="p">(</span><span class="n">X</span><span class="p">):</span>
|
||
<span class="k">return</span> <span class="o">-</span><span class="p">(</span><span class="mf">1.0</span> <span class="o">/</span> <span class="n">target</span><span class="o">.</span><span class="n">size</span><span class="p">)</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">target</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">log</span><span class="p">(</span><span class="n">X</span> <span class="o">+</span> <span class="mf">10e-10</span><span class="p">))</span>
|
||
|
||
<span class="k">return</span> <span class="n">func</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<p>Below we give a short example of how these cost function may be used
|
||
to obtain results if you wish to test them out on your own using
|
||
AutoGrad’s automatics differentiation.</p>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">autograd</span> <span class="kn">import</span> <span class="n">grad</span>
|
||
|
||
<span class="n">target</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">]])</span><span class="o">.</span><span class="n">T</span>
|
||
<span class="n">a</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span><span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">6</span><span class="p">]])</span><span class="o">.</span><span class="n">T</span>
|
||
|
||
<span class="n">cost_func</span> <span class="o">=</span> <span class="n">CostCrossEntropy</span>
|
||
<span class="n">cost_func_derivative</span> <span class="o">=</span> <span class="n">grad</span><span class="p">(</span><span class="n">cost_func</span><span class="p">(</span><span class="n">target</span><span class="p">))</span>
|
||
|
||
<span class="n">valued_at_a</span> <span class="o">=</span> <span class="n">cost_func_derivative</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"Derivative of cost function </span><span class="si">{</span><span class="n">cost_func</span><span class="o">.</span><span class="vm">__name__</span><span class="si">}</span><span class="s2"> valued at a:</span><span class="se">\n</span><span class="si">{</span><span class="n">valued_at_a</span><span class="si">}</span><span class="s2">"</span><span class="p">)</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<div class="section" id="id3">
|
||
<h3>Activation functions<a class="headerlink" href="#id3" title="Permalink to this headline">¶</a></h3>
|
||
<p>Finally, before we look at the neural network, we will look at the
|
||
activation functions which can be specified between the hidden layers
|
||
and as the output function. Each function can be valued for any given
|
||
vector or matrix X, and can be differentiated via derivate().</p>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">autograd.numpy</span> <span class="k">as</span> <span class="nn">np</span>
|
||
<span class="kn">from</span> <span class="nn">autograd</span> <span class="kn">import</span> <span class="n">elementwise_grad</span>
|
||
|
||
<span class="k">def</span> <span class="nf">identity</span><span class="p">(</span><span class="n">X</span><span class="p">):</span>
|
||
<span class="k">return</span> <span class="n">X</span>
|
||
|
||
|
||
<span class="k">def</span> <span class="nf">sigmoid</span><span class="p">(</span><span class="n">X</span><span class="p">):</span>
|
||
<span class="k">try</span><span class="p">:</span>
|
||
<span class="k">return</span> <span class="mf">1.0</span> <span class="o">/</span> <span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">X</span><span class="p">))</span>
|
||
<span class="k">except</span> <span class="ne">FloatingPointError</span><span class="p">:</span>
|
||
<span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">where</span><span class="p">(</span><span class="n">X</span> <span class="o">></span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">X</span><span class="o">.</span><span class="n">shape</span><span class="p">),</span> <span class="n">np</span><span class="o">.</span><span class="n">ones</span><span class="p">(</span><span class="n">X</span><span class="o">.</span><span class="n">shape</span><span class="p">),</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">X</span><span class="o">.</span><span class="n">shape</span><span class="p">))</span>
|
||
|
||
|
||
<span class="k">def</span> <span class="nf">softmax</span><span class="p">(</span><span class="n">X</span><span class="p">):</span>
|
||
<span class="n">X</span> <span class="o">=</span> <span class="n">X</span> <span class="o">-</span> <span class="n">np</span><span class="o">.</span><span class="n">max</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">axis</span><span class="o">=-</span><span class="mi">1</span><span class="p">,</span> <span class="n">keepdims</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
|
||
<span class="n">delta</span> <span class="o">=</span> <span class="mf">10e-10</span>
|
||
<span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="n">X</span><span class="p">)</span> <span class="o">/</span> <span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="n">X</span><span class="p">),</span> <span class="n">axis</span><span class="o">=-</span><span class="mi">1</span><span class="p">,</span> <span class="n">keepdims</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span> <span class="o">+</span> <span class="n">delta</span><span class="p">)</span>
|
||
|
||
|
||
<span class="k">def</span> <span class="nf">RELU</span><span class="p">(</span><span class="n">X</span><span class="p">):</span>
|
||
<span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">where</span><span class="p">(</span><span class="n">X</span> <span class="o">></span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">X</span><span class="o">.</span><span class="n">shape</span><span class="p">),</span> <span class="n">X</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">X</span><span class="o">.</span><span class="n">shape</span><span class="p">))</span>
|
||
|
||
|
||
<span class="k">def</span> <span class="nf">LRELU</span><span class="p">(</span><span class="n">X</span><span class="p">):</span>
|
||
<span class="n">delta</span> <span class="o">=</span> <span class="mf">10e-4</span>
|
||
<span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">where</span><span class="p">(</span><span class="n">X</span> <span class="o">></span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">X</span><span class="o">.</span><span class="n">shape</span><span class="p">),</span> <span class="n">X</span><span class="p">,</span> <span class="n">delta</span> <span class="o">*</span> <span class="n">X</span><span class="p">)</span>
|
||
|
||
|
||
<span class="k">def</span> <span class="nf">derivate</span><span class="p">(</span><span class="n">func</span><span class="p">):</span>
|
||
<span class="k">if</span> <span class="n">func</span><span class="o">.</span><span class="vm">__name__</span> <span class="o">==</span> <span class="s2">"RELU"</span><span class="p">:</span>
|
||
|
||
<span class="k">def</span> <span class="nf">func</span><span class="p">(</span><span class="n">X</span><span class="p">):</span>
|
||
<span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">where</span><span class="p">(</span><span class="n">X</span> <span class="o">></span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
|
||
|
||
<span class="k">return</span> <span class="n">func</span>
|
||
|
||
<span class="k">elif</span> <span class="n">func</span><span class="o">.</span><span class="vm">__name__</span> <span class="o">==</span> <span class="s2">"LRELU"</span><span class="p">:</span>
|
||
|
||
<span class="k">def</span> <span class="nf">func</span><span class="p">(</span><span class="n">X</span><span class="p">):</span>
|
||
<span class="n">delta</span> <span class="o">=</span> <span class="mf">10e-4</span>
|
||
<span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">where</span><span class="p">(</span><span class="n">X</span> <span class="o">></span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">delta</span><span class="p">)</span>
|
||
|
||
<span class="k">return</span> <span class="n">func</span>
|
||
|
||
<span class="k">else</span><span class="p">:</span>
|
||
<span class="k">return</span> <span class="n">elementwise_grad</span><span class="p">(</span><span class="n">func</span><span class="p">)</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<p>Below follows a short demonstration of how to use an activation
|
||
function. The derivative of the activation function will be important
|
||
when calculating the output delta term during backpropagation. Note
|
||
that derivate() can also be used for cost functions for a more
|
||
generalized approach.</p>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">z</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span><span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">6</span><span class="p">]])</span><span class="o">.</span><span class="n">T</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"Input to activation function:</span><span class="se">\n</span><span class="si">{</span><span class="n">z</span><span class="si">}</span><span class="s2">"</span><span class="p">)</span>
|
||
|
||
<span class="n">act_func</span> <span class="o">=</span> <span class="n">sigmoid</span>
|
||
<span class="n">a</span> <span class="o">=</span> <span class="n">act_func</span><span class="p">(</span><span class="n">z</span><span class="p">)</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"</span><span class="se">\n</span><span class="s2">Output from </span><span class="si">{</span><span class="n">act_func</span><span class="o">.</span><span class="vm">__name__</span><span class="si">}</span><span class="s2"> activation function:</span><span class="se">\n</span><span class="si">{</span><span class="n">a</span><span class="si">}</span><span class="s2">"</span><span class="p">)</span>
|
||
|
||
<span class="n">act_func_derivative</span> <span class="o">=</span> <span class="n">derivate</span><span class="p">(</span><span class="n">act_func</span><span class="p">)</span>
|
||
<span class="n">valued_at_z</span> <span class="o">=</span> <span class="n">act_func_derivative</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"</span><span class="se">\n</span><span class="s2">Derivative of </span><span class="si">{</span><span class="n">act_func</span><span class="o">.</span><span class="vm">__name__</span><span class="si">}</span><span class="s2"> activation function valued at z:</span><span class="se">\n</span><span class="si">{</span><span class="n">valued_at_z</span><span class="si">}</span><span class="s2">"</span><span class="p">)</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<div class="section" id="the-neural-network">
|
||
<h3>The Neural Network<a class="headerlink" href="#the-neural-network" title="Permalink to this headline">¶</a></h3>
|
||
<p>Now that we have gotten a good understanding of the implementation of
|
||
some important components, we can take a look at an object oriented
|
||
implementation of a feed forward neural network. The feed forward
|
||
neural network has been implemented as a class named FFNN, which can
|
||
be initiated as a regressor or classifier dependant on the choice of
|
||
cost function. The FFNN can have any number of input nodes, hidden
|
||
layers with any amount of hidden nodes, and any amount of output nodes
|
||
meaning it can perform multiclass classification as well as binary
|
||
classification and regression problems. Although there is a lot of
|
||
code present, it makes for an easy to use and generalizeable interface
|
||
for creating many types of neural networks as will be demonstrated
|
||
below.</p>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">math</span>
|
||
<span class="kn">import</span> <span class="nn">autograd.numpy</span> <span class="k">as</span> <span class="nn">np</span>
|
||
<span class="kn">import</span> <span class="nn">sys</span>
|
||
<span class="kn">import</span> <span class="nn">warnings</span>
|
||
<span class="kn">from</span> <span class="nn">autograd</span> <span class="kn">import</span> <span class="n">grad</span><span class="p">,</span> <span class="n">elementwise_grad</span>
|
||
<span class="kn">from</span> <span class="nn">random</span> <span class="kn">import</span> <span class="n">random</span><span class="p">,</span> <span class="n">seed</span>
|
||
<span class="kn">from</span> <span class="nn">copy</span> <span class="kn">import</span> <span class="n">deepcopy</span><span class="p">,</span> <span class="n">copy</span>
|
||
<span class="kn">from</span> <span class="nn">typing</span> <span class="kn">import</span> <span class="n">Tuple</span><span class="p">,</span> <span class="n">Callable</span>
|
||
<span class="kn">from</span> <span class="nn">sklearn.utils</span> <span class="kn">import</span> <span class="n">resample</span>
|
||
|
||
<span class="n">warnings</span><span class="o">.</span><span class="n">simplefilter</span><span class="p">(</span><span class="s2">"error"</span><span class="p">)</span>
|
||
|
||
|
||
<span class="k">class</span> <span class="nc">FFNN</span><span class="p">:</span>
|
||
<span class="w"> </span><span class="sd">"""</span>
|
||
<span class="sd"> Description:</span>
|
||
<span class="sd"> ------------</span>
|
||
<span class="sd"> Feed Forward Neural Network with interface enabling flexible design of a</span>
|
||
<span class="sd"> nerual networks architecture and the specification of activation function</span>
|
||
<span class="sd"> in the hidden layers and output layer respectively. This model can be used</span>
|
||
<span class="sd"> for both regression and classification problems, depending on the output function.</span>
|
||
|
||
<span class="sd"> Attributes:</span>
|
||
<span class="sd"> ------------</span>
|
||
<span class="sd"> I dimensions (tuple[int]): A list of positive integers, which specifies the</span>
|
||
<span class="sd"> number of nodes in each of the networks layers. The first integer in the array</span>
|
||
<span class="sd"> defines the number of nodes in the input layer, the second integer defines number</span>
|
||
<span class="sd"> of nodes in the first hidden layer and so on until the last number, which</span>
|
||
<span class="sd"> specifies the number of nodes in the output layer.</span>
|
||
<span class="sd"> II hidden_func (Callable): The activation function for the hidden layers</span>
|
||
<span class="sd"> III output_func (Callable): The activation function for the output layer</span>
|
||
<span class="sd"> IV cost_func (Callable): Our cost function</span>
|
||
<span class="sd"> V seed (int): Sets random seed, makes results reproducible</span>
|
||
<span class="sd"> """</span>
|
||
|
||
<span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span>
|
||
<span class="bp">self</span><span class="p">,</span>
|
||
<span class="n">dimensions</span><span class="p">:</span> <span class="nb">tuple</span><span class="p">[</span><span class="nb">int</span><span class="p">],</span>
|
||
<span class="n">hidden_func</span><span class="p">:</span> <span class="n">Callable</span> <span class="o">=</span> <span class="n">sigmoid</span><span class="p">,</span>
|
||
<span class="n">output_func</span><span class="p">:</span> <span class="n">Callable</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">x</span><span class="p">,</span>
|
||
<span class="n">cost_func</span><span class="p">:</span> <span class="n">Callable</span> <span class="o">=</span> <span class="n">CostOLS</span><span class="p">,</span>
|
||
<span class="n">seed</span><span class="p">:</span> <span class="nb">int</span> <span class="o">=</span> <span class="kc">None</span><span class="p">,</span>
|
||
<span class="p">):</span>
|
||
<span class="bp">self</span><span class="o">.</span><span class="n">dimensions</span> <span class="o">=</span> <span class="n">dimensions</span>
|
||
<span class="bp">self</span><span class="o">.</span><span class="n">hidden_func</span> <span class="o">=</span> <span class="n">hidden_func</span>
|
||
<span class="bp">self</span><span class="o">.</span><span class="n">output_func</span> <span class="o">=</span> <span class="n">output_func</span>
|
||
<span class="bp">self</span><span class="o">.</span><span class="n">cost_func</span> <span class="o">=</span> <span class="n">cost_func</span>
|
||
<span class="bp">self</span><span class="o">.</span><span class="n">seed</span> <span class="o">=</span> <span class="n">seed</span>
|
||
<span class="bp">self</span><span class="o">.</span><span class="n">weights</span> <span class="o">=</span> <span class="nb">list</span><span class="p">()</span>
|
||
<span class="bp">self</span><span class="o">.</span><span class="n">schedulers_weight</span> <span class="o">=</span> <span class="nb">list</span><span class="p">()</span>
|
||
<span class="bp">self</span><span class="o">.</span><span class="n">schedulers_bias</span> <span class="o">=</span> <span class="nb">list</span><span class="p">()</span>
|
||
<span class="bp">self</span><span class="o">.</span><span class="n">a_matrices</span> <span class="o">=</span> <span class="nb">list</span><span class="p">()</span>
|
||
<span class="bp">self</span><span class="o">.</span><span class="n">z_matrices</span> <span class="o">=</span> <span class="nb">list</span><span class="p">()</span>
|
||
<span class="bp">self</span><span class="o">.</span><span class="n">classification</span> <span class="o">=</span> <span class="kc">None</span>
|
||
|
||
<span class="bp">self</span><span class="o">.</span><span class="n">reset_weights</span><span class="p">()</span>
|
||
<span class="bp">self</span><span class="o">.</span><span class="n">_set_classification</span><span class="p">()</span>
|
||
|
||
<span class="k">def</span> <span class="nf">fit</span><span class="p">(</span>
|
||
<span class="bp">self</span><span class="p">,</span>
|
||
<span class="n">X</span><span class="p">:</span> <span class="n">np</span><span class="o">.</span><span class="n">ndarray</span><span class="p">,</span>
|
||
<span class="n">t</span><span class="p">:</span> <span class="n">np</span><span class="o">.</span><span class="n">ndarray</span><span class="p">,</span>
|
||
<span class="n">scheduler</span><span class="p">:</span> <span class="n">Scheduler</span><span class="p">,</span>
|
||
<span class="n">batches</span><span class="p">:</span> <span class="nb">int</span> <span class="o">=</span> <span class="mi">1</span><span class="p">,</span>
|
||
<span class="n">epochs</span><span class="p">:</span> <span class="nb">int</span> <span class="o">=</span> <span class="mi">100</span><span class="p">,</span>
|
||
<span class="n">lam</span><span class="p">:</span> <span class="nb">float</span> <span class="o">=</span> <span class="mi">0</span><span class="p">,</span>
|
||
<span class="n">X_val</span><span class="p">:</span> <span class="n">np</span><span class="o">.</span><span class="n">ndarray</span> <span class="o">=</span> <span class="kc">None</span><span class="p">,</span>
|
||
<span class="n">t_val</span><span class="p">:</span> <span class="n">np</span><span class="o">.</span><span class="n">ndarray</span> <span class="o">=</span> <span class="kc">None</span><span class="p">,</span>
|
||
<span class="p">):</span>
|
||
<span class="w"> </span><span class="sd">"""</span>
|
||
<span class="sd"> Description:</span>
|
||
<span class="sd"> ------------</span>
|
||
<span class="sd"> This function performs the training the neural network by performing the feedforward and backpropagation</span>
|
||
<span class="sd"> algorithm to update the networks weights.</span>
|
||
|
||
<span class="sd"> Parameters:</span>
|
||
<span class="sd"> ------------</span>
|
||
<span class="sd"> I X (np.ndarray) : training data</span>
|
||
<span class="sd"> II t (np.ndarray) : target data</span>
|
||
<span class="sd"> III scheduler (Scheduler) : specified scheduler (algorithm for optimization of gradient descent)</span>
|
||
<span class="sd"> IV scheduler_args (list[int]) : list of all arguments necessary for scheduler</span>
|
||
|
||
<span class="sd"> Optional Parameters:</span>
|
||
<span class="sd"> ------------</span>
|
||
<span class="sd"> V batches (int) : number of batches the datasets are split into, default equal to 1</span>
|
||
<span class="sd"> VI epochs (int) : number of iterations used to train the network, default equal to 100</span>
|
||
<span class="sd"> VII lam (float) : regularization hyperparameter lambda</span>
|
||
<span class="sd"> VIII X_val (np.ndarray) : validation set</span>
|
||
<span class="sd"> IX t_val (np.ndarray) : validation target set</span>
|
||
|
||
<span class="sd"> Returns:</span>
|
||
<span class="sd"> ------------</span>
|
||
<span class="sd"> I scores (dict) : A dictionary containing the performance metrics of the model.</span>
|
||
<span class="sd"> The number of the metrics depends on the parameters passed to the fit-function.</span>
|
||
|
||
<span class="sd"> """</span>
|
||
|
||
<span class="c1"># setup </span>
|
||
<span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">seed</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span><span class="p">:</span>
|
||
<span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">seed</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">seed</span><span class="p">)</span>
|
||
|
||
<span class="n">val_set</span> <span class="o">=</span> <span class="kc">False</span>
|
||
<span class="k">if</span> <span class="n">X_val</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span> <span class="ow">and</span> <span class="n">t_val</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span><span class="p">:</span>
|
||
<span class="n">val_set</span> <span class="o">=</span> <span class="kc">True</span>
|
||
|
||
<span class="c1"># creating arrays for score metrics</span>
|
||
<span class="n">train_errors</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">empty</span><span class="p">(</span><span class="n">epochs</span><span class="p">)</span>
|
||
<span class="n">train_errors</span><span class="o">.</span><span class="n">fill</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">nan</span><span class="p">)</span>
|
||
<span class="n">val_errors</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">empty</span><span class="p">(</span><span class="n">epochs</span><span class="p">)</span>
|
||
<span class="n">val_errors</span><span class="o">.</span><span class="n">fill</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">nan</span><span class="p">)</span>
|
||
|
||
<span class="n">train_accs</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">empty</span><span class="p">(</span><span class="n">epochs</span><span class="p">)</span>
|
||
<span class="n">train_accs</span><span class="o">.</span><span class="n">fill</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">nan</span><span class="p">)</span>
|
||
<span class="n">val_accs</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">empty</span><span class="p">(</span><span class="n">epochs</span><span class="p">)</span>
|
||
<span class="n">val_accs</span><span class="o">.</span><span class="n">fill</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">nan</span><span class="p">)</span>
|
||
|
||
<span class="bp">self</span><span class="o">.</span><span class="n">schedulers_weight</span> <span class="o">=</span> <span class="nb">list</span><span class="p">()</span>
|
||
<span class="bp">self</span><span class="o">.</span><span class="n">schedulers_bias</span> <span class="o">=</span> <span class="nb">list</span><span class="p">()</span>
|
||
|
||
<span class="n">batch_size</span> <span class="o">=</span> <span class="n">X</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">//</span> <span class="n">batches</span>
|
||
|
||
<span class="n">X</span><span class="p">,</span> <span class="n">t</span> <span class="o">=</span> <span class="n">resample</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">t</span><span class="p">)</span>
|
||
|
||
<span class="c1"># this function returns a function valued only at X</span>
|
||
<span class="n">cost_function_train</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">cost_func</span><span class="p">(</span><span class="n">t</span><span class="p">)</span>
|
||
<span class="k">if</span> <span class="n">val_set</span><span class="p">:</span>
|
||
<span class="n">cost_function_val</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">cost_func</span><span class="p">(</span><span class="n">t_val</span><span class="p">)</span>
|
||
|
||
<span class="c1"># create schedulers for each weight matrix</span>
|
||
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">weights</span><span class="p">)):</span>
|
||
<span class="bp">self</span><span class="o">.</span><span class="n">schedulers_weight</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">copy</span><span class="p">(</span><span class="n">scheduler</span><span class="p">))</span>
|
||
<span class="bp">self</span><span class="o">.</span><span class="n">schedulers_bias</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">copy</span><span class="p">(</span><span class="n">scheduler</span><span class="p">))</span>
|
||
|
||
<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"</span><span class="si">{</span><span class="n">scheduler</span><span class="o">.</span><span class="vm">__class__</span><span class="o">.</span><span class="vm">__name__</span><span class="si">}</span><span class="s2">: Eta=</span><span class="si">{</span><span class="n">scheduler</span><span class="o">.</span><span class="n">eta</span><span class="si">}</span><span class="s2">, Lambda=</span><span class="si">{</span><span class="n">lam</span><span class="si">}</span><span class="s2">"</span><span class="p">)</span>
|
||
|
||
<span class="k">try</span><span class="p">:</span>
|
||
<span class="k">for</span> <span class="n">e</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">epochs</span><span class="p">):</span>
|
||
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">batches</span><span class="p">):</span>
|
||
<span class="c1"># allows for minibatch gradient descent</span>
|
||
<span class="k">if</span> <span class="n">i</span> <span class="o">==</span> <span class="n">batches</span> <span class="o">-</span> <span class="mi">1</span><span class="p">:</span>
|
||
<span class="c1"># If the for loop has reached the last batch, take all thats left</span>
|
||
<span class="n">X_batch</span> <span class="o">=</span> <span class="n">X</span><span class="p">[</span><span class="n">i</span> <span class="o">*</span> <span class="n">batch_size</span> <span class="p">:,</span> <span class="p">:]</span>
|
||
<span class="n">t_batch</span> <span class="o">=</span> <span class="n">t</span><span class="p">[</span><span class="n">i</span> <span class="o">*</span> <span class="n">batch_size</span> <span class="p">:,</span> <span class="p">:]</span>
|
||
<span class="k">else</span><span class="p">:</span>
|
||
<span class="n">X_batch</span> <span class="o">=</span> <span class="n">X</span><span class="p">[</span><span class="n">i</span> <span class="o">*</span> <span class="n">batch_size</span> <span class="p">:</span> <span class="p">(</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="o">*</span> <span class="n">batch_size</span><span class="p">,</span> <span class="p">:]</span>
|
||
<span class="n">t_batch</span> <span class="o">=</span> <span class="n">t</span><span class="p">[</span><span class="n">i</span> <span class="o">*</span> <span class="n">batch_size</span> <span class="p">:</span> <span class="p">(</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="o">*</span> <span class="n">batch_size</span><span class="p">,</span> <span class="p">:]</span>
|
||
|
||
<span class="bp">self</span><span class="o">.</span><span class="n">_feedforward</span><span class="p">(</span><span class="n">X_batch</span><span class="p">)</span>
|
||
<span class="bp">self</span><span class="o">.</span><span class="n">_backpropagate</span><span class="p">(</span><span class="n">X_batch</span><span class="p">,</span> <span class="n">t_batch</span><span class="p">,</span> <span class="n">lam</span><span class="p">)</span>
|
||
|
||
<span class="c1"># reset schedulers for each epoch (some schedulers pass in this call)</span>
|
||
<span class="k">for</span> <span class="n">scheduler</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">schedulers_weight</span><span class="p">:</span>
|
||
<span class="n">scheduler</span><span class="o">.</span><span class="n">reset</span><span class="p">()</span>
|
||
|
||
<span class="k">for</span> <span class="n">scheduler</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">schedulers_bias</span><span class="p">:</span>
|
||
<span class="n">scheduler</span><span class="o">.</span><span class="n">reset</span><span class="p">()</span>
|
||
|
||
<span class="c1"># computing performance metrics</span>
|
||
<span class="n">pred_train</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
|
||
<span class="n">train_error</span> <span class="o">=</span> <span class="n">cost_function_train</span><span class="p">(</span><span class="n">pred_train</span><span class="p">)</span>
|
||
|
||
<span class="n">train_errors</span><span class="p">[</span><span class="n">e</span><span class="p">]</span> <span class="o">=</span> <span class="n">train_error</span>
|
||
<span class="k">if</span> <span class="n">val_set</span><span class="p">:</span>
|
||
|
||
<span class="n">pred_val</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X_val</span><span class="p">)</span>
|
||
<span class="n">val_error</span> <span class="o">=</span> <span class="n">cost_function_val</span><span class="p">(</span><span class="n">pred_val</span><span class="p">)</span>
|
||
<span class="n">val_errors</span><span class="p">[</span><span class="n">e</span><span class="p">]</span> <span class="o">=</span> <span class="n">val_error</span>
|
||
|
||
<span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">classification</span><span class="p">:</span>
|
||
<span class="n">train_acc</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_accuracy</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X</span><span class="p">),</span> <span class="n">t</span><span class="p">)</span>
|
||
<span class="n">train_accs</span><span class="p">[</span><span class="n">e</span><span class="p">]</span> <span class="o">=</span> <span class="n">train_acc</span>
|
||
<span class="k">if</span> <span class="n">val_set</span><span class="p">:</span>
|
||
<span class="n">val_acc</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_accuracy</span><span class="p">(</span><span class="n">pred_val</span><span class="p">,</span> <span class="n">t_val</span><span class="p">)</span>
|
||
<span class="n">val_accs</span><span class="p">[</span><span class="n">e</span><span class="p">]</span> <span class="o">=</span> <span class="n">val_acc</span>
|
||
|
||
<span class="c1"># printing progress bar</span>
|
||
<span class="n">progression</span> <span class="o">=</span> <span class="n">e</span> <span class="o">/</span> <span class="n">epochs</span>
|
||
<span class="n">print_length</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_progress_bar</span><span class="p">(</span>
|
||
<span class="n">progression</span><span class="p">,</span>
|
||
<span class="n">train_error</span><span class="o">=</span><span class="n">train_errors</span><span class="p">[</span><span class="n">e</span><span class="p">],</span>
|
||
<span class="n">train_acc</span><span class="o">=</span><span class="n">train_accs</span><span class="p">[</span><span class="n">e</span><span class="p">],</span>
|
||
<span class="n">val_error</span><span class="o">=</span><span class="n">val_errors</span><span class="p">[</span><span class="n">e</span><span class="p">],</span>
|
||
<span class="n">val_acc</span><span class="o">=</span><span class="n">val_accs</span><span class="p">[</span><span class="n">e</span><span class="p">],</span>
|
||
<span class="p">)</span>
|
||
<span class="k">except</span> <span class="ne">KeyboardInterrupt</span><span class="p">:</span>
|
||
<span class="c1"># allows for stopping training at any point and seeing the result</span>
|
||
<span class="k">pass</span>
|
||
|
||
<span class="c1"># visualization of training progression (similiar to tensorflow progression bar)</span>
|
||
<span class="n">sys</span><span class="o">.</span><span class="n">stdout</span><span class="o">.</span><span class="n">write</span><span class="p">(</span><span class="s2">"</span><span class="se">\r</span><span class="s2">"</span> <span class="o">+</span> <span class="s2">" "</span> <span class="o">*</span> <span class="n">print_length</span><span class="p">)</span>
|
||
<span class="n">sys</span><span class="o">.</span><span class="n">stdout</span><span class="o">.</span><span class="n">flush</span><span class="p">()</span>
|
||
<span class="bp">self</span><span class="o">.</span><span class="n">_progress_bar</span><span class="p">(</span>
|
||
<span class="mi">1</span><span class="p">,</span>
|
||
<span class="n">train_error</span><span class="o">=</span><span class="n">train_errors</span><span class="p">[</span><span class="n">e</span><span class="p">],</span>
|
||
<span class="n">train_acc</span><span class="o">=</span><span class="n">train_accs</span><span class="p">[</span><span class="n">e</span><span class="p">],</span>
|
||
<span class="n">val_error</span><span class="o">=</span><span class="n">val_errors</span><span class="p">[</span><span class="n">e</span><span class="p">],</span>
|
||
<span class="n">val_acc</span><span class="o">=</span><span class="n">val_accs</span><span class="p">[</span><span class="n">e</span><span class="p">],</span>
|
||
<span class="p">)</span>
|
||
<span class="n">sys</span><span class="o">.</span><span class="n">stdout</span><span class="o">.</span><span class="n">write</span><span class="p">(</span><span class="s2">""</span><span class="p">)</span>
|
||
|
||
<span class="c1"># return performance metrics for the entire run</span>
|
||
<span class="n">scores</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">()</span>
|
||
|
||
<span class="n">scores</span><span class="p">[</span><span class="s2">"train_errors"</span><span class="p">]</span> <span class="o">=</span> <span class="n">train_errors</span>
|
||
|
||
<span class="k">if</span> <span class="n">val_set</span><span class="p">:</span>
|
||
<span class="n">scores</span><span class="p">[</span><span class="s2">"val_errors"</span><span class="p">]</span> <span class="o">=</span> <span class="n">val_errors</span>
|
||
|
||
<span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">classification</span><span class="p">:</span>
|
||
<span class="n">scores</span><span class="p">[</span><span class="s2">"train_accs"</span><span class="p">]</span> <span class="o">=</span> <span class="n">train_accs</span>
|
||
|
||
<span class="k">if</span> <span class="n">val_set</span><span class="p">:</span>
|
||
<span class="n">scores</span><span class="p">[</span><span class="s2">"val_accs"</span><span class="p">]</span> <span class="o">=</span> <span class="n">val_accs</span>
|
||
|
||
<span class="k">return</span> <span class="n">scores</span>
|
||
|
||
<span class="k">def</span> <span class="nf">predict</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">X</span><span class="p">:</span> <span class="n">np</span><span class="o">.</span><span class="n">ndarray</span><span class="p">,</span> <span class="o">*</span><span class="p">,</span> <span class="n">threshold</span><span class="o">=</span><span class="mf">0.5</span><span class="p">):</span>
|
||
<span class="w"> </span><span class="sd">"""</span>
|
||
<span class="sd"> Description:</span>
|
||
<span class="sd"> ------------</span>
|
||
<span class="sd"> Performs prediction after training of the network has been finished.</span>
|
||
|
||
<span class="sd"> Parameters:</span>
|
||
<span class="sd"> ------------</span>
|
||
<span class="sd"> I X (np.ndarray): The design matrix, with n rows of p features each</span>
|
||
|
||
<span class="sd"> Optional Parameters:</span>
|
||
<span class="sd"> ------------</span>
|
||
<span class="sd"> II threshold (float) : sets minimal value for a prediction to be predicted as the positive class</span>
|
||
<span class="sd"> in classification problems</span>
|
||
|
||
<span class="sd"> Returns:</span>
|
||
<span class="sd"> ------------</span>
|
||
<span class="sd"> I z (np.ndarray): A prediction vector (row) for each row in our design matrix</span>
|
||
<span class="sd"> This vector is thresholded if regression=False, meaning that classification results</span>
|
||
<span class="sd"> in a vector of 1s and 0s, while regressions in an array of decimal numbers</span>
|
||
|
||
<span class="sd"> """</span>
|
||
|
||
<span class="n">predict</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_feedforward</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
|
||
|
||
<span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">classification</span><span class="p">:</span>
|
||
<span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">where</span><span class="p">(</span><span class="n">predict</span> <span class="o">></span> <span class="n">threshold</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
|
||
<span class="k">else</span><span class="p">:</span>
|
||
<span class="k">return</span> <span class="n">predict</span>
|
||
|
||
<span class="k">def</span> <span class="nf">reset_weights</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
|
||
<span class="w"> </span><span class="sd">"""</span>
|
||
<span class="sd"> Description:</span>
|
||
<span class="sd"> ------------</span>
|
||
<span class="sd"> Resets/Reinitializes the weights in order to train the network for a new problem.</span>
|
||
|
||
<span class="sd"> """</span>
|
||
<span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">seed</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span><span class="p">:</span>
|
||
<span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">seed</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">seed</span><span class="p">)</span>
|
||
|
||
<span class="bp">self</span><span class="o">.</span><span class="n">weights</span> <span class="o">=</span> <span class="nb">list</span><span class="p">()</span>
|
||
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">dimensions</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span><span class="p">):</span>
|
||
<span class="n">weight_array</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span>
|
||
<span class="bp">self</span><span class="o">.</span><span class="n">dimensions</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">dimensions</span><span class="p">[</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]</span>
|
||
<span class="p">)</span>
|
||
<span class="n">weight_array</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="p">:]</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">dimensions</span><span class="p">[</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">])</span> <span class="o">*</span> <span class="mf">0.01</span>
|
||
|
||
<span class="bp">self</span><span class="o">.</span><span class="n">weights</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">weight_array</span><span class="p">)</span>
|
||
|
||
<span class="k">def</span> <span class="nf">_feedforward</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">X</span><span class="p">:</span> <span class="n">np</span><span class="o">.</span><span class="n">ndarray</span><span class="p">):</span>
|
||
<span class="w"> </span><span class="sd">"""</span>
|
||
<span class="sd"> Description:</span>
|
||
<span class="sd"> ------------</span>
|
||
<span class="sd"> Calculates the activation of each layer starting at the input and ending at the output.</span>
|
||
<span class="sd"> Each following activation is calculated from a weighted sum of each of the preceeding</span>
|
||
<span class="sd"> activations (except in the case of the input layer).</span>
|
||
|
||
<span class="sd"> Parameters:</span>
|
||
<span class="sd"> ------------</span>
|
||
<span class="sd"> I X (np.ndarray): The design matrix, with n rows of p features each</span>
|
||
|
||
<span class="sd"> Returns:</span>
|
||
<span class="sd"> ------------</span>
|
||
<span class="sd"> I z (np.ndarray): A prediction vector (row) for each row in our design matrix</span>
|
||
<span class="sd"> """</span>
|
||
|
||
<span class="c1"># reset matrices</span>
|
||
<span class="bp">self</span><span class="o">.</span><span class="n">a_matrices</span> <span class="o">=</span> <span class="nb">list</span><span class="p">()</span>
|
||
<span class="bp">self</span><span class="o">.</span><span class="n">z_matrices</span> <span class="o">=</span> <span class="nb">list</span><span class="p">()</span>
|
||
|
||
<span class="c1"># if X is just a vector, make it into a matrix</span>
|
||
<span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">X</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
|
||
<span class="n">X</span> <span class="o">=</span> <span class="n">X</span><span class="o">.</span><span class="n">reshape</span><span class="p">((</span><span class="mi">1</span><span class="p">,</span> <span class="n">X</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">]))</span>
|
||
|
||
<span class="c1"># Add a coloumn of zeros as the first coloumn of the design matrix, in order</span>
|
||
<span class="c1"># to add bias to our data</span>
|
||
<span class="n">bias</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">ones</span><span class="p">((</span><span class="n">X</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="mi">1</span><span class="p">))</span> <span class="o">*</span> <span class="mf">0.01</span>
|
||
<span class="n">X</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">hstack</span><span class="p">([</span><span class="n">bias</span><span class="p">,</span> <span class="n">X</span><span class="p">])</span>
|
||
|
||
<span class="c1"># a^0, the nodes in the input layer (one a^0 for each row in X - where the</span>
|
||
<span class="c1"># exponent indicates layer number).</span>
|
||
<span class="n">a</span> <span class="o">=</span> <span class="n">X</span>
|
||
<span class="bp">self</span><span class="o">.</span><span class="n">a_matrices</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
|
||
<span class="bp">self</span><span class="o">.</span><span class="n">z_matrices</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
|
||
|
||
<span class="c1"># The feed forward algorithm</span>
|
||
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">weights</span><span class="p">)):</span>
|
||
<span class="k">if</span> <span class="n">i</span> <span class="o"><</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">weights</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span><span class="p">:</span>
|
||
<span class="n">z</span> <span class="o">=</span> <span class="n">a</span> <span class="o">@</span> <span class="bp">self</span><span class="o">.</span><span class="n">weights</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
|
||
<span class="bp">self</span><span class="o">.</span><span class="n">z_matrices</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">z</span><span class="p">)</span>
|
||
<span class="n">a</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">hidden_func</span><span class="p">(</span><span class="n">z</span><span class="p">)</span>
|
||
<span class="c1"># bias column again added to the data here</span>
|
||
<span class="n">bias</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">ones</span><span class="p">((</span><span class="n">a</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="mi">1</span><span class="p">))</span> <span class="o">*</span> <span class="mf">0.01</span>
|
||
<span class="n">a</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">hstack</span><span class="p">([</span><span class="n">bias</span><span class="p">,</span> <span class="n">a</span><span class="p">])</span>
|
||
<span class="bp">self</span><span class="o">.</span><span class="n">a_matrices</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
|
||
<span class="k">else</span><span class="p">:</span>
|
||
<span class="k">try</span><span class="p">:</span>
|
||
<span class="c1"># a^L, the nodes in our output layers</span>
|
||
<span class="n">z</span> <span class="o">=</span> <span class="n">a</span> <span class="o">@</span> <span class="bp">self</span><span class="o">.</span><span class="n">weights</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
|
||
<span class="n">a</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">output_func</span><span class="p">(</span><span class="n">z</span><span class="p">)</span>
|
||
<span class="bp">self</span><span class="o">.</span><span class="n">a_matrices</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
|
||
<span class="bp">self</span><span class="o">.</span><span class="n">z_matrices</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">z</span><span class="p">)</span>
|
||
<span class="k">except</span> <span class="ne">Exception</span> <span class="k">as</span> <span class="ne">OverflowError</span><span class="p">:</span>
|
||
<span class="nb">print</span><span class="p">(</span>
|
||
<span class="s2">"OverflowError in fit() in FFNN</span><span class="se">\n</span><span class="s2">HOW TO DEBUG ERROR: Consider lowering your learning rate or scheduler specific parameters such as momentum, or check if your input values need scaling"</span>
|
||
<span class="p">)</span>
|
||
|
||
<span class="c1"># this will be a^L</span>
|
||
<span class="k">return</span> <span class="n">a</span>
|
||
|
||
<span class="k">def</span> <span class="nf">_backpropagate</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">X</span><span class="p">,</span> <span class="n">t</span><span class="p">,</span> <span class="n">lam</span><span class="p">):</span>
|
||
<span class="w"> </span><span class="sd">"""</span>
|
||
<span class="sd"> Description:</span>
|
||
<span class="sd"> ------------</span>
|
||
<span class="sd"> Performs the backpropagation algorithm. In other words, this method</span>
|
||
<span class="sd"> calculates the gradient of all the layers starting at the</span>
|
||
<span class="sd"> output layer, and moving from right to left accumulates the gradient until</span>
|
||
<span class="sd"> the input layer is reached. Each layers respective weights are updated while</span>
|
||
<span class="sd"> the algorithm propagates backwards from the output layer (auto-differentation in reverse mode).</span>
|
||
|
||
<span class="sd"> Parameters:</span>
|
||
<span class="sd"> ------------</span>
|
||
<span class="sd"> I X (np.ndarray): The design matrix, with n rows of p features each.</span>
|
||
<span class="sd"> II t (np.ndarray): The target vector, with n rows of p targets.</span>
|
||
<span class="sd"> III lam (float32): regularization parameter used to punish the weights in case of overfitting</span>
|
||
|
||
<span class="sd"> Returns:</span>
|
||
<span class="sd"> ------------</span>
|
||
<span class="sd"> No return value.</span>
|
||
|
||
<span class="sd"> """</span>
|
||
<span class="n">out_derivative</span> <span class="o">=</span> <span class="n">derivate</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">output_func</span><span class="p">)</span>
|
||
<span class="n">hidden_derivative</span> <span class="o">=</span> <span class="n">derivate</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">hidden_func</span><span class="p">)</span>
|
||
|
||
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">weights</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">):</span>
|
||
<span class="c1"># delta terms for output</span>
|
||
<span class="k">if</span> <span class="n">i</span> <span class="o">==</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">weights</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span><span class="p">:</span>
|
||
<span class="c1"># for multi-class classification</span>
|
||
<span class="k">if</span> <span class="p">(</span>
|
||
<span class="bp">self</span><span class="o">.</span><span class="n">output_func</span><span class="o">.</span><span class="vm">__name__</span> <span class="o">==</span> <span class="s2">"softmax"</span>
|
||
<span class="p">):</span>
|
||
<span class="n">delta_matrix</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">a_matrices</span><span class="p">[</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]</span> <span class="o">-</span> <span class="n">t</span>
|
||
<span class="c1"># for single class classification</span>
|
||
<span class="k">else</span><span class="p">:</span>
|
||
<span class="n">cost_func_derivative</span> <span class="o">=</span> <span class="n">grad</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">cost_func</span><span class="p">(</span><span class="n">t</span><span class="p">))</span>
|
||
<span class="n">delta_matrix</span> <span class="o">=</span> <span class="n">out_derivative</span><span class="p">(</span>
|
||
<span class="bp">self</span><span class="o">.</span><span class="n">z_matrices</span><span class="p">[</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]</span>
|
||
<span class="p">)</span> <span class="o">*</span> <span class="n">cost_func_derivative</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">a_matrices</span><span class="p">[</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">])</span>
|
||
|
||
<span class="c1"># delta terms for hidden layer</span>
|
||
<span class="k">else</span><span class="p">:</span>
|
||
<span class="n">delta_matrix</span> <span class="o">=</span> <span class="p">(</span>
|
||
<span class="bp">self</span><span class="o">.</span><span class="n">weights</span><span class="p">[</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">][</span><span class="mi">1</span><span class="p">:,</span> <span class="p">:]</span> <span class="o">@</span> <span class="n">delta_matrix</span><span class="o">.</span><span class="n">T</span>
|
||
<span class="p">)</span><span class="o">.</span><span class="n">T</span> <span class="o">*</span> <span class="n">hidden_derivative</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">z_matrices</span><span class="p">[</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">])</span>
|
||
|
||
<span class="c1"># calculate gradient</span>
|
||
<span class="n">gradient_weights</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">a_matrices</span><span class="p">[</span><span class="n">i</span><span class="p">][:,</span> <span class="mi">1</span><span class="p">:]</span><span class="o">.</span><span class="n">T</span> <span class="o">@</span> <span class="n">delta_matrix</span>
|
||
<span class="n">gradient_bias</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">delta_matrix</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span>
|
||
<span class="mi">1</span><span class="p">,</span> <span class="n">delta_matrix</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
|
||
<span class="p">)</span>
|
||
|
||
<span class="c1"># regularization term</span>
|
||
<span class="n">gradient_weights</span> <span class="o">+=</span> <span class="bp">self</span><span class="o">.</span><span class="n">weights</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">1</span><span class="p">:,</span> <span class="p">:]</span> <span class="o">*</span> <span class="n">lam</span>
|
||
|
||
<span class="c1"># use scheduler</span>
|
||
<span class="n">update_matrix</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">vstack</span><span class="p">(</span>
|
||
<span class="p">[</span>
|
||
<span class="bp">self</span><span class="o">.</span><span class="n">schedulers_bias</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">update_change</span><span class="p">(</span><span class="n">gradient_bias</span><span class="p">),</span>
|
||
<span class="bp">self</span><span class="o">.</span><span class="n">schedulers_weight</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">update_change</span><span class="p">(</span><span class="n">gradient_weights</span><span class="p">),</span>
|
||
<span class="p">]</span>
|
||
<span class="p">)</span>
|
||
|
||
<span class="c1"># update weights and bias</span>
|
||
<span class="bp">self</span><span class="o">.</span><span class="n">weights</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">-=</span> <span class="n">update_matrix</span>
|
||
|
||
<span class="k">def</span> <span class="nf">_accuracy</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">prediction</span><span class="p">:</span> <span class="n">np</span><span class="o">.</span><span class="n">ndarray</span><span class="p">,</span> <span class="n">target</span><span class="p">:</span> <span class="n">np</span><span class="o">.</span><span class="n">ndarray</span><span class="p">):</span>
|
||
<span class="w"> </span><span class="sd">"""</span>
|
||
<span class="sd"> Description:</span>
|
||
<span class="sd"> ------------</span>
|
||
<span class="sd"> Calculates accuracy of given prediction to target</span>
|
||
|
||
<span class="sd"> Parameters:</span>
|
||
<span class="sd"> ------------</span>
|
||
<span class="sd"> I prediction (np.ndarray): vector of predicitons output network</span>
|
||
<span class="sd"> (1s and 0s in case of classification, and real numbers in case of regression)</span>
|
||
<span class="sd"> II target (np.ndarray): vector of true values (What the network ideally should predict)</span>
|
||
|
||
<span class="sd"> Returns:</span>
|
||
<span class="sd"> ------------</span>
|
||
<span class="sd"> A floating point number representing the percentage of correctly classified instances.</span>
|
||
<span class="sd"> """</span>
|
||
<span class="k">assert</span> <span class="n">prediction</span><span class="o">.</span><span class="n">size</span> <span class="o">==</span> <span class="n">target</span><span class="o">.</span><span class="n">size</span>
|
||
<span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">average</span><span class="p">((</span><span class="n">target</span> <span class="o">==</span> <span class="n">prediction</span><span class="p">))</span>
|
||
<span class="k">def</span> <span class="nf">_set_classification</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
|
||
<span class="w"> </span><span class="sd">"""</span>
|
||
<span class="sd"> Description:</span>
|
||
<span class="sd"> ------------</span>
|
||
<span class="sd"> Decides if FFNN acts as classifier (True) og regressor (False),</span>
|
||
<span class="sd"> sets self.classification during init()</span>
|
||
<span class="sd"> """</span>
|
||
<span class="bp">self</span><span class="o">.</span><span class="n">classification</span> <span class="o">=</span> <span class="kc">False</span>
|
||
<span class="k">if</span> <span class="p">(</span>
|
||
<span class="bp">self</span><span class="o">.</span><span class="n">cost_func</span><span class="o">.</span><span class="vm">__name__</span> <span class="o">==</span> <span class="s2">"CostLogReg"</span>
|
||
<span class="ow">or</span> <span class="bp">self</span><span class="o">.</span><span class="n">cost_func</span><span class="o">.</span><span class="vm">__name__</span> <span class="o">==</span> <span class="s2">"CostCrossEntropy"</span>
|
||
<span class="p">):</span>
|
||
<span class="bp">self</span><span class="o">.</span><span class="n">classification</span> <span class="o">=</span> <span class="kc">True</span>
|
||
|
||
<span class="k">def</span> <span class="nf">_progress_bar</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">progression</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
|
||
<span class="w"> </span><span class="sd">"""</span>
|
||
<span class="sd"> Description:</span>
|
||
<span class="sd"> ------------</span>
|
||
<span class="sd"> Displays progress of training</span>
|
||
<span class="sd"> """</span>
|
||
<span class="n">print_length</span> <span class="o">=</span> <span class="mi">40</span>
|
||
<span class="n">num_equals</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">progression</span> <span class="o">*</span> <span class="n">print_length</span><span class="p">)</span>
|
||
<span class="n">num_not</span> <span class="o">=</span> <span class="n">print_length</span> <span class="o">-</span> <span class="n">num_equals</span>
|
||
<span class="n">arrow</span> <span class="o">=</span> <span class="s2">">"</span> <span class="k">if</span> <span class="n">num_equals</span> <span class="o">></span> <span class="mi">0</span> <span class="k">else</span> <span class="s2">""</span>
|
||
<span class="n">bar</span> <span class="o">=</span> <span class="s2">"["</span> <span class="o">+</span> <span class="s2">"="</span> <span class="o">*</span> <span class="p">(</span><span class="n">num_equals</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span> <span class="o">+</span> <span class="n">arrow</span> <span class="o">+</span> <span class="s2">"-"</span> <span class="o">*</span> <span class="n">num_not</span> <span class="o">+</span> <span class="s2">"]"</span>
|
||
<span class="n">perc_print</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_format</span><span class="p">(</span><span class="n">progression</span> <span class="o">*</span> <span class="mi">100</span><span class="p">,</span> <span class="n">decimals</span><span class="o">=</span><span class="mi">5</span><span class="p">)</span>
|
||
<span class="n">line</span> <span class="o">=</span> <span class="sa">f</span><span class="s2">" </span><span class="si">{</span><span class="n">bar</span><span class="si">}</span><span class="s2"> </span><span class="si">{</span><span class="n">perc_print</span><span class="si">}</span><span class="s2">% "</span>
|
||
|
||
<span class="k">for</span> <span class="n">key</span> <span class="ow">in</span> <span class="n">kwargs</span><span class="p">:</span>
|
||
<span class="k">if</span> <span class="ow">not</span> <span class="n">np</span><span class="o">.</span><span class="n">isnan</span><span class="p">(</span><span class="n">kwargs</span><span class="p">[</span><span class="n">key</span><span class="p">]):</span>
|
||
<span class="n">value</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_format</span><span class="p">(</span><span class="n">kwargs</span><span class="p">[</span><span class="n">key</span><span class="p">],</span> <span class="n">decimals</span><span class="o">=</span><span class="mi">4</span><span class="p">)</span>
|
||
<span class="n">line</span> <span class="o">+=</span> <span class="sa">f</span><span class="s2">"| </span><span class="si">{</span><span class="n">key</span><span class="si">}</span><span class="s2">: </span><span class="si">{</span><span class="n">value</span><span class="si">}</span><span class="s2"> "</span>
|
||
<span class="n">sys</span><span class="o">.</span><span class="n">stdout</span><span class="o">.</span><span class="n">write</span><span class="p">(</span><span class="s2">"</span><span class="se">\r</span><span class="s2">"</span> <span class="o">+</span> <span class="n">line</span><span class="p">)</span>
|
||
<span class="n">sys</span><span class="o">.</span><span class="n">stdout</span><span class="o">.</span><span class="n">flush</span><span class="p">()</span>
|
||
<span class="k">return</span> <span class="nb">len</span><span class="p">(</span><span class="n">line</span><span class="p">)</span>
|
||
|
||
<span class="k">def</span> <span class="nf">_format</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">value</span><span class="p">,</span> <span class="n">decimals</span><span class="o">=</span><span class="mi">4</span><span class="p">):</span>
|
||
<span class="w"> </span><span class="sd">"""</span>
|
||
<span class="sd"> Description:</span>
|
||
<span class="sd"> ------------</span>
|
||
<span class="sd"> Formats decimal numbers for progress bar</span>
|
||
<span class="sd"> """</span>
|
||
<span class="k">if</span> <span class="n">value</span> <span class="o">></span> <span class="mi">0</span><span class="p">:</span>
|
||
<span class="n">v</span> <span class="o">=</span> <span class="n">value</span>
|
||
<span class="k">elif</span> <span class="n">value</span> <span class="o"><</span> <span class="mi">0</span><span class="p">:</span>
|
||
<span class="n">v</span> <span class="o">=</span> <span class="o">-</span><span class="mi">10</span> <span class="o">*</span> <span class="n">value</span>
|
||
<span class="k">else</span><span class="p">:</span>
|
||
<span class="n">v</span> <span class="o">=</span> <span class="mi">1</span>
|
||
<span class="n">n</span> <span class="o">=</span> <span class="mi">1</span> <span class="o">+</span> <span class="n">math</span><span class="o">.</span><span class="n">floor</span><span class="p">(</span><span class="n">math</span><span class="o">.</span><span class="n">log10</span><span class="p">(</span><span class="n">v</span><span class="p">))</span>
|
||
<span class="k">if</span> <span class="n">n</span> <span class="o">>=</span> <span class="n">decimals</span> <span class="o">-</span> <span class="mi">1</span><span class="p">:</span>
|
||
<span class="k">return</span> <span class="nb">str</span><span class="p">(</span><span class="nb">round</span><span class="p">(</span><span class="n">value</span><span class="p">))</span>
|
||
<span class="k">return</span> <span class="sa">f</span><span class="s2">"</span><span class="si">{</span><span class="n">value</span><span class="si">:</span><span class="s2">.</span><span class="si">{</span><span class="n">decimals</span><span class="o">-</span><span class="n">n</span><span class="o">-</span><span class="mi">1</span><span class="si">}</span><span class="s2">f</span><span class="si">}</span><span class="s2">"</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<p>Before we make a model, we will quickly generate a dataset we can use
|
||
for our linear regression problem as shown below</p>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">autograd.numpy</span> <span class="k">as</span> <span class="nn">np</span>
|
||
<span class="kn">from</span> <span class="nn">sklearn.model_selection</span> <span class="kn">import</span> <span class="n">train_test_split</span>
|
||
|
||
<span class="k">def</span> <span class="nf">SkrankeFunction</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">):</span>
|
||
<span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">ravel</span><span class="p">(</span><span class="mi">0</span> <span class="o">+</span> <span class="mi">1</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">y</span> <span class="o">+</span> <span class="mi">3</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">4</span><span class="o">*</span><span class="n">x</span><span class="o">*</span><span class="n">y</span> <span class="o">+</span> <span class="mi">5</span><span class="o">*</span><span class="n">y</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
|
||
|
||
<span class="k">def</span> <span class="nf">create_X</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
|
||
<span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">x</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span> <span class="o">></span> <span class="mi">1</span><span class="p">:</span>
|
||
<span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">ravel</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
|
||
<span class="n">y</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">ravel</span><span class="p">(</span><span class="n">y</span><span class="p">)</span>
|
||
|
||
<span class="n">N</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
|
||
<span class="n">l</span> <span class="o">=</span> <span class="nb">int</span><span class="p">((</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="o">*</span> <span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="mi">2</span><span class="p">)</span> <span class="o">/</span> <span class="mi">2</span><span class="p">)</span> <span class="c1"># Number of elements in beta</span>
|
||
<span class="n">X</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">ones</span><span class="p">((</span><span class="n">N</span><span class="p">,</span> <span class="n">l</span><span class="p">))</span>
|
||
|
||
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
|
||
<span class="n">q</span> <span class="o">=</span> <span class="nb">int</span><span class="p">((</span><span class="n">i</span><span class="p">)</span> <span class="o">*</span> <span class="p">(</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="o">/</span> <span class="mi">2</span><span class="p">)</span>
|
||
<span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
|
||
<span class="n">X</span><span class="p">[:,</span> <span class="n">q</span> <span class="o">+</span> <span class="n">k</span><span class="p">]</span> <span class="o">=</span> <span class="p">(</span><span class="n">x</span> <span class="o">**</span> <span class="p">(</span><span class="n">i</span> <span class="o">-</span> <span class="n">k</span><span class="p">))</span> <span class="o">*</span> <span class="p">(</span><span class="n">y</span><span class="o">**</span><span class="n">k</span><span class="p">)</span>
|
||
|
||
<span class="k">return</span> <span class="n">X</span>
|
||
|
||
<span class="n">step</span><span class="o">=</span><span class="mf">0.5</span>
|
||
<span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">step</span><span class="p">)</span>
|
||
<span class="n">y</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">step</span><span class="p">)</span>
|
||
<span class="n">x</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">meshgrid</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
|
||
<span class="n">target</span> <span class="o">=</span> <span class="n">SkrankeFunction</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
|
||
<span class="n">target</span> <span class="o">=</span> <span class="n">target</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">target</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="mi">1</span><span class="p">)</span>
|
||
|
||
<span class="n">poly_degree</span><span class="o">=</span><span class="mi">3</span>
|
||
<span class="n">X</span> <span class="o">=</span> <span class="n">create_X</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">poly_degree</span><span class="p">)</span>
|
||
|
||
<span class="n">X_train</span><span class="p">,</span> <span class="n">X_test</span><span class="p">,</span> <span class="n">t_train</span><span class="p">,</span> <span class="n">t_test</span> <span class="o">=</span> <span class="n">train_test_split</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">target</span><span class="p">)</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<p>Now that we have our dataset ready for the regression, we can create
|
||
our regressor. Note that with the seed parameter, we can make sure our
|
||
results stay the same every time we run the neural network. For
|
||
inititialization, we simply specify the dimensions (we wish the amount
|
||
of input nodes to be equal to the datapoints, and the output to
|
||
predict one value).</p>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">input_nodes</span> <span class="o">=</span> <span class="n">X_train</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
|
||
<span class="n">output_nodes</span> <span class="o">=</span> <span class="mi">1</span>
|
||
|
||
<span class="n">linear_regression</span> <span class="o">=</span> <span class="n">FFNN</span><span class="p">((</span><span class="n">input_nodes</span><span class="p">,</span> <span class="n">output_nodes</span><span class="p">),</span> <span class="n">output_func</span><span class="o">=</span><span class="n">identity</span><span class="p">,</span> <span class="n">cost_func</span><span class="o">=</span><span class="n">CostOLS</span><span class="p">,</span> <span class="n">seed</span><span class="o">=</span><span class="mi">2023</span><span class="p">)</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<p>We then fit our model with our training data using the scheduler of our choice.</p>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">linear_regression</span><span class="o">.</span><span class="n">reset_weights</span><span class="p">()</span> <span class="c1"># reset weights such that previous runs or reruns don't affect the weights</span>
|
||
|
||
<span class="n">scheduler</span> <span class="o">=</span> <span class="n">Constant</span><span class="p">(</span><span class="n">eta</span><span class="o">=</span><span class="mf">1e-3</span><span class="p">)</span>
|
||
<span class="n">scores</span> <span class="o">=</span> <span class="n">linear_regression</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_train</span><span class="p">,</span> <span class="n">t_train</span><span class="p">,</span> <span class="n">scheduler</span><span class="p">)</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<p>Due to the progress bar we can see the MSE (train_error) throughout
|
||
the FFNN’s training. Note that the fit() function has some optional
|
||
parameters with defualt arguments. For example, the regularization
|
||
hyperparameter can be left ignored if not needed, and equally the FFNN
|
||
will by default run for 100 epochs. These can easily be changed, such
|
||
as for example:</p>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">linear_regression</span><span class="o">.</span><span class="n">reset_weights</span><span class="p">()</span> <span class="c1"># reset weights such that previous runs or reruns don't affect the weights</span>
|
||
|
||
<span class="n">scores</span> <span class="o">=</span> <span class="n">linear_regression</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_train</span><span class="p">,</span> <span class="n">t_train</span><span class="p">,</span> <span class="n">scheduler</span><span class="p">,</span> <span class="n">lam</span><span class="o">=</span><span class="mf">1e-4</span><span class="p">,</span> <span class="n">epochs</span><span class="o">=</span><span class="mi">1000</span><span class="p">)</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<p>We see that given more epochs to train on, the regressor reaches a lower MSE.</p>
|
||
<p>Let us then switch to a binary classification. We use a binary
|
||
classification dataset, and follow a similar setup to the regression
|
||
case.</p>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">sklearn.datasets</span> <span class="kn">import</span> <span class="n">load_breast_cancer</span>
|
||
<span class="kn">from</span> <span class="nn">sklearn.preprocessing</span> <span class="kn">import</span> <span class="n">MinMaxScaler</span>
|
||
|
||
<span class="n">wisconsin</span> <span class="o">=</span> <span class="n">load_breast_cancer</span><span class="p">()</span>
|
||
<span class="n">X</span> <span class="o">=</span> <span class="n">wisconsin</span><span class="o">.</span><span class="n">data</span>
|
||
<span class="n">target</span> <span class="o">=</span> <span class="n">wisconsin</span><span class="o">.</span><span class="n">target</span>
|
||
<span class="n">target</span> <span class="o">=</span> <span class="n">target</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">target</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="mi">1</span><span class="p">)</span>
|
||
|
||
<span class="n">X_train</span><span class="p">,</span> <span class="n">X_val</span><span class="p">,</span> <span class="n">t_train</span><span class="p">,</span> <span class="n">t_val</span> <span class="o">=</span> <span class="n">train_test_split</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">target</span><span class="p">)</span>
|
||
|
||
<span class="n">scaler</span> <span class="o">=</span> <span class="n">MinMaxScaler</span><span class="p">()</span>
|
||
<span class="n">scaler</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_train</span><span class="p">)</span>
|
||
<span class="n">X_train</span> <span class="o">=</span> <span class="n">scaler</span><span class="o">.</span><span class="n">transform</span><span class="p">(</span><span class="n">X_train</span><span class="p">)</span>
|
||
<span class="n">X_val</span> <span class="o">=</span> <span class="n">scaler</span><span class="o">.</span><span class="n">transform</span><span class="p">(</span><span class="n">X_val</span><span class="p">)</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">input_nodes</span> <span class="o">=</span> <span class="n">X_train</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
|
||
<span class="n">output_nodes</span> <span class="o">=</span> <span class="mi">1</span>
|
||
|
||
<span class="n">logistic_regression</span> <span class="o">=</span> <span class="n">FFNN</span><span class="p">((</span><span class="n">input_nodes</span><span class="p">,</span> <span class="n">output_nodes</span><span class="p">),</span> <span class="n">output_func</span><span class="o">=</span><span class="n">sigmoid</span><span class="p">,</span> <span class="n">cost_func</span><span class="o">=</span><span class="n">CostLogReg</span><span class="p">,</span> <span class="n">seed</span><span class="o">=</span><span class="mi">2023</span><span class="p">)</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<p>We will now make use of our validation data by passing it into our fit function as a keyword argument</p>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">logistic_regression</span><span class="o">.</span><span class="n">reset_weights</span><span class="p">()</span> <span class="c1"># reset weights such that previous runs or reruns don't affect the weights</span>
|
||
|
||
<span class="n">scheduler</span> <span class="o">=</span> <span class="n">Adam</span><span class="p">(</span><span class="n">eta</span><span class="o">=</span><span class="mf">1e-3</span><span class="p">,</span> <span class="n">rho</span><span class="o">=</span><span class="mf">0.9</span><span class="p">,</span> <span class="n">rho2</span><span class="o">=</span><span class="mf">0.999</span><span class="p">)</span>
|
||
<span class="n">scores</span> <span class="o">=</span> <span class="n">logistic_regression</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_train</span><span class="p">,</span> <span class="n">t_train</span><span class="p">,</span> <span class="n">scheduler</span><span class="p">,</span> <span class="n">epochs</span><span class="o">=</span><span class="mi">1000</span><span class="p">,</span> <span class="n">X_val</span><span class="o">=</span><span class="n">X_val</span><span class="p">,</span> <span class="n">t_val</span><span class="o">=</span><span class="n">t_val</span><span class="p">)</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<p>Finally, we will create a neural network with 2 hidden layers with activation functions.</p>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">input_nodes</span> <span class="o">=</span> <span class="n">X_train</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
|
||
<span class="n">hidden_nodes1</span> <span class="o">=</span> <span class="mi">100</span>
|
||
<span class="n">hidden_nodes2</span> <span class="o">=</span> <span class="mi">30</span>
|
||
<span class="n">output_nodes</span> <span class="o">=</span> <span class="mi">1</span>
|
||
|
||
<span class="n">dims</span> <span class="o">=</span> <span class="p">(</span><span class="n">input_nodes</span><span class="p">,</span> <span class="n">hidden_nodes1</span><span class="p">,</span> <span class="n">hidden_nodes2</span><span class="p">,</span> <span class="n">output_nodes</span><span class="p">)</span>
|
||
|
||
<span class="n">neural_network</span> <span class="o">=</span> <span class="n">FFNN</span><span class="p">(</span><span class="n">dims</span><span class="p">,</span> <span class="n">hidden_func</span><span class="o">=</span><span class="n">RELU</span><span class="p">,</span> <span class="n">output_func</span><span class="o">=</span><span class="n">sigmoid</span><span class="p">,</span> <span class="n">cost_func</span><span class="o">=</span><span class="n">CostLogReg</span><span class="p">,</span> <span class="n">seed</span><span class="o">=</span><span class="mi">2023</span><span class="p">)</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">neural_network</span><span class="o">.</span><span class="n">reset_weights</span><span class="p">()</span> <span class="c1"># reset weights such that previous runs or reruns don't affect the weights</span>
|
||
|
||
<span class="n">scheduler</span> <span class="o">=</span> <span class="n">Adam</span><span class="p">(</span><span class="n">eta</span><span class="o">=</span><span class="mf">1e-4</span><span class="p">,</span> <span class="n">rho</span><span class="o">=</span><span class="mf">0.9</span><span class="p">,</span> <span class="n">rho2</span><span class="o">=</span><span class="mf">0.999</span><span class="p">)</span>
|
||
<span class="n">scores</span> <span class="o">=</span> <span class="n">neural_network</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_train</span><span class="p">,</span> <span class="n">t_train</span><span class="p">,</span> <span class="n">scheduler</span><span class="p">,</span> <span class="n">epochs</span><span class="o">=</span><span class="mi">1000</span><span class="p">,</span> <span class="n">X_val</span><span class="o">=</span><span class="n">X_val</span><span class="p">,</span> <span class="n">t_val</span><span class="o">=</span><span class="n">t_val</span><span class="p">)</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<div class="section" id="multiclass-classification">
|
||
<h3>Multiclass classification<a class="headerlink" href="#multiclass-classification" title="Permalink to this headline">¶</a></h3>
|
||
<p>Finally, we will demonstrate the use case of multiclass classification
|
||
using our FFNN with the famous MNIST dataset, which contain images of
|
||
digits between the range of 0 to 9.</p>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">sklearn.datasets</span> <span class="kn">import</span> <span class="n">load_digits</span>
|
||
|
||
<span class="k">def</span> <span class="nf">onehot</span><span class="p">(</span><span class="n">target</span><span class="p">:</span> <span class="n">np</span><span class="o">.</span><span class="n">ndarray</span><span class="p">):</span>
|
||
<span class="n">onehot</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">target</span><span class="o">.</span><span class="n">size</span><span class="p">,</span> <span class="n">target</span><span class="o">.</span><span class="n">max</span><span class="p">()</span> <span class="o">+</span> <span class="mi">1</span><span class="p">))</span>
|
||
<span class="n">onehot</span><span class="p">[</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">target</span><span class="o">.</span><span class="n">size</span><span class="p">),</span> <span class="n">target</span><span class="p">]</span> <span class="o">=</span> <span class="mi">1</span>
|
||
<span class="k">return</span> <span class="n">onehot</span>
|
||
|
||
<span class="n">digits</span> <span class="o">=</span> <span class="n">load_digits</span><span class="p">()</span>
|
||
|
||
<span class="n">X</span> <span class="o">=</span> <span class="n">digits</span><span class="o">.</span><span class="n">data</span>
|
||
<span class="n">target</span> <span class="o">=</span> <span class="n">digits</span><span class="o">.</span><span class="n">target</span>
|
||
<span class="n">target</span> <span class="o">=</span> <span class="n">onehot</span><span class="p">(</span><span class="n">target</span><span class="p">)</span>
|
||
|
||
<span class="n">input_nodes</span> <span class="o">=</span> <span class="mi">64</span>
|
||
<span class="n">hidden_nodes1</span> <span class="o">=</span> <span class="mi">100</span>
|
||
<span class="n">hidden_nodes2</span> <span class="o">=</span> <span class="mi">30</span>
|
||
<span class="n">output_nodes</span> <span class="o">=</span> <span class="mi">10</span>
|
||
|
||
<span class="n">dims</span> <span class="o">=</span> <span class="p">(</span><span class="n">input_nodes</span><span class="p">,</span> <span class="n">hidden_nodes1</span><span class="p">,</span> <span class="n">hidden_nodes2</span><span class="p">,</span> <span class="n">output_nodes</span><span class="p">)</span>
|
||
|
||
<span class="n">multiclass</span> <span class="o">=</span> <span class="n">FFNN</span><span class="p">(</span><span class="n">dims</span><span class="p">,</span> <span class="n">hidden_func</span><span class="o">=</span><span class="n">LRELU</span><span class="p">,</span> <span class="n">output_func</span><span class="o">=</span><span class="n">softmax</span><span class="p">,</span> <span class="n">cost_func</span><span class="o">=</span><span class="n">CostCrossEntropy</span><span class="p">)</span>
|
||
|
||
<span class="n">multiclass</span><span class="o">.</span><span class="n">reset_weights</span><span class="p">()</span> <span class="c1"># reset weights such that previous runs or reruns don't affect the weights</span>
|
||
|
||
<span class="n">scheduler</span> <span class="o">=</span> <span class="n">Adam</span><span class="p">(</span><span class="n">eta</span><span class="o">=</span><span class="mf">1e-4</span><span class="p">,</span> <span class="n">rho</span><span class="o">=</span><span class="mf">0.9</span><span class="p">,</span> <span class="n">rho2</span><span class="o">=</span><span class="mf">0.999</span><span class="p">)</span>
|
||
<span class="n">scores</span> <span class="o">=</span> <span class="n">multiclass</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">target</span><span class="p">,</span> <span class="n">scheduler</span><span class="p">,</span> <span class="n">epochs</span><span class="o">=</span><span class="mi">1000</span><span class="p">)</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<div class="section" id="testing-the-xor-gate-and-other-gates">
|
||
<h2>Testing the XOR gate and other gates<a class="headerlink" href="#testing-the-xor-gate-and-other-gates" title="Permalink to this headline">¶</a></h2>
|
||
<p>Let us now use our code to test the XOR gate.</p>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">X</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]],</span><span class="n">dtype</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">float64</span><span class="p">)</span>
|
||
|
||
<span class="c1"># The XOR gate</span>
|
||
<span class="n">yXOR</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span> <span class="p">[[</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="p">,[</span><span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">]])</span>
|
||
|
||
<span class="n">input_nodes</span> <span class="o">=</span> <span class="n">X</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
|
||
<span class="n">output_nodes</span> <span class="o">=</span> <span class="mi">1</span>
|
||
|
||
<span class="n">logistic_regression</span> <span class="o">=</span> <span class="n">FFNN</span><span class="p">((</span><span class="n">input_nodes</span><span class="p">,</span> <span class="n">output_nodes</span><span class="p">),</span> <span class="n">output_func</span><span class="o">=</span><span class="n">sigmoid</span><span class="p">,</span> <span class="n">cost_func</span><span class="o">=</span><span class="n">CostLogReg</span><span class="p">,</span> <span class="n">seed</span><span class="o">=</span><span class="mi">2023</span><span class="p">)</span>
|
||
<span class="n">logistic_regression</span><span class="o">.</span><span class="n">reset_weights</span><span class="p">()</span> <span class="c1"># reset weights such that previous runs or reruns don't affect the weights</span>
|
||
<span class="n">scheduler</span> <span class="o">=</span> <span class="n">Adam</span><span class="p">(</span><span class="n">eta</span><span class="o">=</span><span class="mf">1e-1</span><span class="p">,</span> <span class="n">rho</span><span class="o">=</span><span class="mf">0.9</span><span class="p">,</span> <span class="n">rho2</span><span class="o">=</span><span class="mf">0.999</span><span class="p">)</span>
|
||
<span class="n">scores</span> <span class="o">=</span> <span class="n">logistic_regression</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">yXOR</span><span class="p">,</span> <span class="n">scheduler</span><span class="p">,</span> <span class="n">epochs</span><span class="o">=</span><span class="mi">1000</span><span class="p">)</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<p>Not bad, but the results depend strongly on the learning reate. Try different learning rates.</p>
|
||
</div>
|
||
<div class="section" id="solving-differential-equations-with-deep-learning">
|
||
<h2>Solving differential equations with Deep Learning<a class="headerlink" href="#solving-differential-equations-with-deep-learning" title="Permalink to this headline">¶</a></h2>
|
||
<p>The Universal Approximation Theorem states that a neural network can
|
||
approximate any function at a single hidden layer along with one input
|
||
and output layer to any given precision.</p>
|
||
<p><strong>Book on solving differential equations with ML methods.</strong></p>
|
||
<p><a class="reference external" href="https://www.springer.com/gp/book/9789401798150">An Introduction to Neural Network Methods for Differential Equations</a>, by Yadav and Kumar.</p>
|
||
<p><strong>Master thesis on applying deep learning to problems in mechanics.</strong></p>
|
||
<p><a class="reference external" href="https://www.duo.uio.no/handle/10852/79212">Using Deep Reinforcement Learning for Active Flow Control</a>, by Marius Holm</p>
|
||
<p><strong>Thanks to Kristine Baluka Hein.</strong></p>
|
||
<p>The lectures on differential equations were developed by Kristine Baluka Hein, now PhD student at IFI.
|
||
A great thanks to Kristine.</p>
|
||
</div>
|
||
<div class="section" id="ordinary-differential-equations-first">
|
||
<h2>Ordinary Differential Equations first<a class="headerlink" href="#ordinary-differential-equations-first" title="Permalink to this headline">¶</a></h2>
|
||
<p>An ordinary differential equation (ODE) is an equation involving functions having one variable.</p>
|
||
<p>In general, an ordinary differential equation looks like</p>
|
||
<!-- Equation labels as ordinary links -->
|
||
<div id="ode"></div>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\begin{equation} \label{ode} \tag{1}
|
||
f\left(x, \, g(x), \, g'(x), \, g''(x), \, \dots \, , \, g^{(n)}(x)\right) = 0
|
||
\end{equation}
|
||
\]</div>
|
||
<p>where <span class="math notranslate nohighlight">\(g(x)\)</span> is the function to find, and <span class="math notranslate nohighlight">\(g^{(n)}(x)\)</span> is the <span class="math notranslate nohighlight">\(n\)</span>-th derivative of <span class="math notranslate nohighlight">\(g(x)\)</span>.</p>
|
||
<p>The <span class="math notranslate nohighlight">\(f\left(x, g(x), g'(x), g''(x), \, \dots \, , g^{(n)}(x)\right)\)</span> is just a way to write that there is an expression involving <span class="math notranslate nohighlight">\(x\)</span> and <span class="math notranslate nohighlight">\(g(x), \ g'(x), \ g''(x), \, \dots \, , \text{ and } g^{(n)}(x)\)</span> on the left side of the equality sign in (<a class="reference external" href="#ode">1</a>).
|
||
The highest order of derivative, that is the value of <span class="math notranslate nohighlight">\(n\)</span>, determines to the order of the equation.
|
||
The equation is referred to as a <span class="math notranslate nohighlight">\(n\)</span>-th order ODE.
|
||
Along with (<a class="reference external" href="#ode">1</a>), some additional conditions of the function <span class="math notranslate nohighlight">\(g(x)\)</span> are typically given
|
||
for the solution to be unique.</p>
|
||
</div>
|
||
<div class="section" id="the-trial-solution">
|
||
<h2>The trial solution<a class="headerlink" href="#the-trial-solution" title="Permalink to this headline">¶</a></h2>
|
||
<p>Let the trial solution <span class="math notranslate nohighlight">\(g_t(x)\)</span> be</p>
|
||
<!-- Equation labels as ordinary links -->
|
||
<div id="_auto1"></div>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\begin{equation}
|
||
g_t(x) = h_1(x) + h_2(x,N(x,P))
|
||
\label{_auto1} \tag{2}
|
||
\end{equation}
|
||
\]</div>
|
||
<p>where <span class="math notranslate nohighlight">\(h_1(x)\)</span> is a function that makes <span class="math notranslate nohighlight">\(g_t(x)\)</span> satisfy a given set
|
||
of conditions, <span class="math notranslate nohighlight">\(N(x,P)\)</span> a neural network with weights and biases
|
||
described by <span class="math notranslate nohighlight">\(P\)</span> and <span class="math notranslate nohighlight">\(h_2(x, N(x,P))\)</span> some expression involving the
|
||
neural network. The role of the function <span class="math notranslate nohighlight">\(h_2(x, N(x,P))\)</span>, is to
|
||
ensure that the output from <span class="math notranslate nohighlight">\(N(x,P)\)</span> is zero when <span class="math notranslate nohighlight">\(g_t(x)\)</span> is
|
||
evaluated at the values of <span class="math notranslate nohighlight">\(x\)</span> where the given conditions must be
|
||
satisfied. The function <span class="math notranslate nohighlight">\(h_1(x)\)</span> should alone make <span class="math notranslate nohighlight">\(g_t(x)\)</span> satisfy
|
||
the conditions.</p>
|
||
<p>But what about the network <span class="math notranslate nohighlight">\(N(x,P)\)</span>?</p>
|
||
<p>As described previously, an optimization method could be used to minimize the parameters of a neural network, that being its weights and biases, through backward propagation.</p>
|
||
</div>
|
||
<div class="section" id="minimization-process">
|
||
<h2>Minimization process<a class="headerlink" href="#minimization-process" title="Permalink to this headline">¶</a></h2>
|
||
<p>For the minimization to be defined, we need to have a cost function at hand to minimize.</p>
|
||
<p>It is given that <span class="math notranslate nohighlight">\(f\left(x, \, g(x), \, g'(x), \, g''(x), \, \dots \, , \, g^{(n)}(x)\right)\)</span> should be equal to zero in (<a class="reference external" href="#ode">1</a>).
|
||
We can choose to consider the mean squared error as the cost function for an input <span class="math notranslate nohighlight">\(x\)</span>.
|
||
Since we are looking at one input, the cost function is just <span class="math notranslate nohighlight">\(f\)</span> squared.
|
||
The cost function <span class="math notranslate nohighlight">\(c\left(x, P \right)\)</span> can therefore be expressed as</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
C\left(x, P\right) = \big(f\left(x, \, g(x), \, g'(x), \, g''(x), \, \dots \, , \, g^{(n)}(x)\right)\big)^2
|
||
\]</div>
|
||
<p>If <span class="math notranslate nohighlight">\(N\)</span> inputs are given as a vector <span class="math notranslate nohighlight">\(\boldsymbol{x}\)</span> with elements <span class="math notranslate nohighlight">\(x_i\)</span> for <span class="math notranslate nohighlight">\(i = 1,\dots,N\)</span>,
|
||
the cost function becomes</p>
|
||
<!-- Equation labels as ordinary links -->
|
||
<div id="cost"></div>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\begin{equation} \label{cost} \tag{3}
|
||
C\left(\boldsymbol{x}, P\right) = \frac{1}{N} \sum_{i=1}^N \big(f\left(x_i, \, g(x_i), \, g'(x_i), \, g''(x_i), \, \dots \, , \, g^{(n)}(x_i)\right)\big)^2
|
||
\end{equation}
|
||
\]</div>
|
||
<p>The neural net should then find the parameters <span class="math notranslate nohighlight">\(P\)</span> that minimizes the cost function in
|
||
(<a class="reference external" href="#cost">3</a>) for a set of <span class="math notranslate nohighlight">\(N\)</span> training samples <span class="math notranslate nohighlight">\(x_i\)</span>.</p>
|
||
</div>
|
||
<div class="section" id="minimizing-the-cost-function-using-gradient-descent-and-automatic-differentiation">
|
||
<h2>Minimizing the cost function using gradient descent and automatic differentiation<a class="headerlink" href="#minimizing-the-cost-function-using-gradient-descent-and-automatic-differentiation" title="Permalink to this headline">¶</a></h2>
|
||
<p>To perform the minimization using gradient descent, the gradient of <span class="math notranslate nohighlight">\(C\left(\boldsymbol{x}, P\right)\)</span> is needed.
|
||
It might happen so that finding an analytical expression of the gradient of <span class="math notranslate nohighlight">\(C(\boldsymbol{x}, P)\)</span> from (<a class="reference external" href="#cost">3</a>) gets too messy, depending on which cost function one desires to use.</p>
|
||
<p>Luckily, there exists libraries that makes the job for us through automatic differentiation.
|
||
Automatic differentiation is a method of finding the derivatives numerically with very high precision.</p>
|
||
</div>
|
||
<div class="section" id="example-exponential-decay">
|
||
<h2>Example: Exponential decay<a class="headerlink" href="#example-exponential-decay" title="Permalink to this headline">¶</a></h2>
|
||
<p>An exponential decay of a quantity <span class="math notranslate nohighlight">\(g(x)\)</span> is described by the equation</p>
|
||
<!-- Equation labels as ordinary links -->
|
||
<div id="solve_expdec"></div>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\begin{equation} \label{solve_expdec} \tag{4}
|
||
g'(x) = -\gamma g(x)
|
||
\end{equation}
|
||
\]</div>
|
||
<p>with <span class="math notranslate nohighlight">\(g(0) = g_0\)</span> for some chosen initial value <span class="math notranslate nohighlight">\(g_0\)</span>.</p>
|
||
<p>The analytical solution of (<a class="reference external" href="#solve_expdec">4</a>) is</p>
|
||
<!-- Equation labels as ordinary links -->
|
||
<div id="_auto2"></div>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\begin{equation}
|
||
g(x) = g_0 \exp\left(-\gamma x\right)
|
||
\label{_auto2} \tag{5}
|
||
\end{equation}
|
||
\]</div>
|
||
<p>Having an analytical solution at hand, it is possible to use it to compare how well a neural network finds a solution of (<a class="reference external" href="#solve_expdec">4</a>).</p>
|
||
</div>
|
||
<div class="section" id="the-function-to-solve-for">
|
||
<h2>The function to solve for<a class="headerlink" href="#the-function-to-solve-for" title="Permalink to this headline">¶</a></h2>
|
||
<p>The program will use a neural network to solve</p>
|
||
<!-- Equation labels as ordinary links -->
|
||
<div id="solveode"></div>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\begin{equation} \label{solveode} \tag{6}
|
||
g'(x) = -\gamma g(x)
|
||
\end{equation}
|
||
\]</div>
|
||
<p>where <span class="math notranslate nohighlight">\(g(0) = g_0\)</span> with <span class="math notranslate nohighlight">\(\gamma\)</span> and <span class="math notranslate nohighlight">\(g_0\)</span> being some chosen values.</p>
|
||
<p>In this example, <span class="math notranslate nohighlight">\(\gamma = 2\)</span> and <span class="math notranslate nohighlight">\(g_0 = 10\)</span>.</p>
|
||
</div>
|
||
<div class="section" id="id4">
|
||
<h2>The trial solution<a class="headerlink" href="#id4" title="Permalink to this headline">¶</a></h2>
|
||
<p>To begin with, a trial solution <span class="math notranslate nohighlight">\(g_t(t)\)</span> must be chosen. A general trial solution for ordinary differential equations could be</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
g_t(x, P) = h_1(x) + h_2(x, N(x, P))
|
||
\]</div>
|
||
<p>with <span class="math notranslate nohighlight">\(h_1(x)\)</span> ensuring that <span class="math notranslate nohighlight">\(g_t(x)\)</span> satisfies some conditions and <span class="math notranslate nohighlight">\(h_2(x,N(x, P))\)</span> an expression involving <span class="math notranslate nohighlight">\(x\)</span> and the output from the neural network <span class="math notranslate nohighlight">\(N(x,P)\)</span> with <span class="math notranslate nohighlight">\(P \)</span> being the collection of the weights and biases for each layer. For now, it is assumed that the network consists of one input layer, one hidden layer, and one output layer.</p>
|
||
</div>
|
||
<div class="section" id="setup-of-network">
|
||
<h2>Setup of Network<a class="headerlink" href="#setup-of-network" title="Permalink to this headline">¶</a></h2>
|
||
<p>In this network, there are no weights and bias at the input layer, so <span class="math notranslate nohighlight">\(P = \{ P_{\text{hidden}}, P_{\text{output}} \}\)</span>.
|
||
If there are <span class="math notranslate nohighlight">\(N_{\text{hidden} }\)</span> neurons in the hidden layer, then <span class="math notranslate nohighlight">\(P_{\text{hidden}}\)</span> is a <span class="math notranslate nohighlight">\(N_{\text{hidden} } \times (1 + N_{\text{input}})\)</span> matrix, given that there are <span class="math notranslate nohighlight">\(N_{\text{input}}\)</span> neurons in the input layer.</p>
|
||
<p>The first column in <span class="math notranslate nohighlight">\(P_{\text{hidden} }\)</span> represents the bias for each neuron in the hidden layer and the second column represents the weights for each neuron in the hidden layer from the input layer.
|
||
If there are <span class="math notranslate nohighlight">\(N_{\text{output} }\)</span> neurons in the output layer, then <span class="math notranslate nohighlight">\(P_{\text{output}} \)</span> is a <span class="math notranslate nohighlight">\(N_{\text{output} } \times (1 + N_{\text{hidden} })\)</span> matrix.</p>
|
||
<p>Its first column represents the bias of each neuron and the remaining columns represents the weights to each neuron.</p>
|
||
<p>It is given that <span class="math notranslate nohighlight">\(g(0) = g_0\)</span>. The trial solution must fulfill this condition to be a proper solution of (<a class="reference external" href="#solveode">6</a>). A possible way to ensure that <span class="math notranslate nohighlight">\(g_t(0, P) = g_0\)</span>, is to let <span class="math notranslate nohighlight">\(F(N(x,P)) = x \cdot N(x,P)\)</span> and <span class="math notranslate nohighlight">\(A(x) = g_0\)</span>. This gives the following trial solution:</p>
|
||
<!-- Equation labels as ordinary links -->
|
||
<div id="trial"></div>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\begin{equation} \label{trial} \tag{7}
|
||
g_t(x, P) = g_0 + x \cdot N(x, P)
|
||
\end{equation}
|
||
\]</div>
|
||
</div>
|
||
<div class="section" id="reformulating-the-problem">
|
||
<h2>Reformulating the problem<a class="headerlink" href="#reformulating-the-problem" title="Permalink to this headline">¶</a></h2>
|
||
<p>We wish that our neural network manages to minimize a given cost function.</p>
|
||
<p>A reformulation of out equation, (<a class="reference external" href="#solveode">6</a>), must therefore be done,
|
||
such that it describes the problem a neural network can solve for.</p>
|
||
<p>The neural network must find the set of weights and biases <span class="math notranslate nohighlight">\(P\)</span> such that the trial solution in (<a class="reference external" href="#trial">7</a>) satisfies (<a class="reference external" href="#solveode">6</a>).</p>
|
||
<p>The trial solution</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
g_t(x, P) = g_0 + x \cdot N(x, P)
|
||
\]</div>
|
||
<p>has been chosen such that it already solves the condition <span class="math notranslate nohighlight">\(g(0) = g_0\)</span>. What remains, is to find <span class="math notranslate nohighlight">\(P\)</span> such that</p>
|
||
<!-- Equation labels as ordinary links -->
|
||
<div id="nnmin"></div>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\begin{equation} \label{nnmin} \tag{8}
|
||
g_t'(x, P) = - \gamma g_t(x, P)
|
||
\end{equation}
|
||
\]</div>
|
||
<p>is fulfilled as <em>best as possible</em>.</p>
|
||
</div>
|
||
<div class="section" id="more-technicalities">
|
||
<h2>More technicalities<a class="headerlink" href="#more-technicalities" title="Permalink to this headline">¶</a></h2>
|
||
<p>The left hand side and right hand side of (<a class="reference external" href="#nnmin">8</a>) must be computed separately, and then the neural network must choose weights and biases, contained in <span class="math notranslate nohighlight">\(P\)</span>, such that the sides are equal as best as possible.
|
||
This means that the absolute or squared difference between the sides must be as close to zero, ideally equal to zero.
|
||
In this case, the difference squared shows to be an appropriate measurement of how erroneous the trial solution is with respect to <span class="math notranslate nohighlight">\(P\)</span> of the neural network.</p>
|
||
<p>This gives the following cost function our neural network must solve for:</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\min_{P}\Big\{ \big(g_t'(x, P) - ( -\gamma g_t(x, P) \big)^2 \Big\}
|
||
\]</div>
|
||
<p>(the notation <span class="math notranslate nohighlight">\(\min_{P}\{ f(x, P) \}\)</span> means that we desire to find <span class="math notranslate nohighlight">\(P\)</span> that yields the minimum of <span class="math notranslate nohighlight">\(f(x, P)\)</span>)</p>
|
||
<p>or, in terms of weights and biases for the hidden and output layer in our network:</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\min_{P_{\text{hidden} }, \ P_{\text{output} }}\Big\{ \big(g_t'(x, \{ P_{\text{hidden} }, P_{\text{output} }\}) - ( -\gamma g_t(x, \{ P_{\text{hidden} }, P_{\text{output} }\}) \big)^2 \Big\}
|
||
\]</div>
|
||
<p>for an input value <span class="math notranslate nohighlight">\(x\)</span>.</p>
|
||
</div>
|
||
<div class="section" id="more-details">
|
||
<h2>More details<a class="headerlink" href="#more-details" title="Permalink to this headline">¶</a></h2>
|
||
<p>If the neural network evaluates <span class="math notranslate nohighlight">\(g_t(x, P)\)</span> at more values for <span class="math notranslate nohighlight">\(x\)</span>, say <span class="math notranslate nohighlight">\(N\)</span> values <span class="math notranslate nohighlight">\(x_i\)</span> for <span class="math notranslate nohighlight">\(i = 1, \dots, N\)</span>, then the <em>total</em> error to minimize becomes</p>
|
||
<!-- Equation labels as ordinary links -->
|
||
<div id="min"></div>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\begin{equation} \label{min} \tag{9}
|
||
\min_{P}\Big\{\frac{1}{N} \sum_{i=1}^N \big(g_t'(x_i, P) - ( -\gamma g_t(x_i, P) \big)^2 \Big\}
|
||
\end{equation}
|
||
\]</div>
|
||
<p>Letting <span class="math notranslate nohighlight">\(\boldsymbol{x}\)</span> be a vector with elements <span class="math notranslate nohighlight">\(x_i\)</span> and <span class="math notranslate nohighlight">\(C(\boldsymbol{x}, P) = \frac{1}{N} \sum_i \big(g_t'(x_i, P) - ( -\gamma g_t(x_i, P) \big)^2\)</span> denote the cost function, the minimization problem that our network must solve, becomes</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\min_{P} C(\boldsymbol{x}, P)
|
||
\]</div>
|
||
<p>In terms of <span class="math notranslate nohighlight">\(P_{\text{hidden} }\)</span> and <span class="math notranslate nohighlight">\(P_{\text{output} }\)</span>, this could also be expressed as</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\min_{P_{\text{hidden} }, \ P_{\text{output} }} C(\boldsymbol{x}, \{P_{\text{hidden} }, P_{\text{output} }\})
|
||
\]</div>
|
||
</div>
|
||
<div class="section" id="a-possible-implementation-of-a-neural-network">
|
||
<h2>A possible implementation of a neural network<a class="headerlink" href="#a-possible-implementation-of-a-neural-network" title="Permalink to this headline">¶</a></h2>
|
||
<p>For simplicity, it is assumed that the input is an array <span class="math notranslate nohighlight">\(\boldsymbol{x} = (x_1, \dots, x_N)\)</span> with <span class="math notranslate nohighlight">\(N\)</span> elements. It is at these points the neural network should find <span class="math notranslate nohighlight">\(P\)</span> such that it fulfills (<a class="reference external" href="#min">9</a>).</p>
|
||
<p>First, the neural network must feed forward the inputs.
|
||
This means that <span class="math notranslate nohighlight">\(\boldsymbol{x}s\)</span> must be passed through an input layer, a hidden layer and a output layer. The input layer in this case, does not need to process the data any further.
|
||
The input layer will consist of <span class="math notranslate nohighlight">\(N_{\text{input} }\)</span> neurons, passing its element to each neuron in the hidden layer. The number of neurons in the hidden layer will be <span class="math notranslate nohighlight">\(N_{\text{hidden} }\)</span>.</p>
|
||
</div>
|
||
<div class="section" id="technicalities">
|
||
<h2>Technicalities<a class="headerlink" href="#technicalities" title="Permalink to this headline">¶</a></h2>
|
||
<p>For the <span class="math notranslate nohighlight">\(i\)</span>-th in the hidden layer with weight <span class="math notranslate nohighlight">\(w_i^{\text{hidden} }\)</span> and bias <span class="math notranslate nohighlight">\(b_i^{\text{hidden} }\)</span>, the weighting from the <span class="math notranslate nohighlight">\(j\)</span>-th neuron at the input layer is:</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[\begin{split}
|
||
\begin{aligned}
|
||
z_{i,j}^{\text{hidden}} &= b_i^{\text{hidden}} + w_i^{\text{hidden}}x_j \\
|
||
&=
|
||
\begin{pmatrix}
|
||
b_i^{\text{hidden}} & w_i^{\text{hidden}}
|
||
\end{pmatrix}
|
||
\begin{pmatrix}
|
||
1 \\
|
||
x_j
|
||
\end{pmatrix}
|
||
\end{aligned}
|
||
\end{split}\]</div>
|
||
</div>
|
||
<div class="section" id="final-technicalities-i">
|
||
<h2>Final technicalities I<a class="headerlink" href="#final-technicalities-i" title="Permalink to this headline">¶</a></h2>
|
||
<p>The result after weighting the inputs at the <span class="math notranslate nohighlight">\(i\)</span>-th hidden neuron can be written as a vector:</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[\begin{split}
|
||
\begin{aligned}
|
||
\boldsymbol{z}_{i}^{\text{hidden}} &= \Big( b_i^{\text{hidden}} + w_i^{\text{hidden}}x_1 , \ b_i^{\text{hidden}} + w_i^{\text{hidden}} x_2, \ \dots \, , \ b_i^{\text{hidden}} + w_i^{\text{hidden}} x_N\Big) \\
|
||
&=
|
||
\begin{pmatrix}
|
||
b_i^{\text{hidden}} & w_i^{\text{hidden}}
|
||
\end{pmatrix}
|
||
\begin{pmatrix}
|
||
1 & 1 & \dots & 1 \\
|
||
x_1 & x_2 & \dots & x_N
|
||
\end{pmatrix} \\
|
||
&= \boldsymbol{p}_{i, \text{hidden}}^T X
|
||
\end{aligned}
|
||
\end{split}\]</div>
|
||
</div>
|
||
<div class="section" id="final-technicalities-ii">
|
||
<h2>Final technicalities II<a class="headerlink" href="#final-technicalities-ii" title="Permalink to this headline">¶</a></h2>
|
||
<p>The vector <span class="math notranslate nohighlight">\(\boldsymbol{p}_{i, \text{hidden}}^T\)</span> constitutes each row in <span class="math notranslate nohighlight">\(P_{\text{hidden} }\)</span>, which contains the weights for the neural network to minimize according to (<a class="reference external" href="#min">9</a>).</p>
|
||
<p>After having found <span class="math notranslate nohighlight">\(\boldsymbol{z}_{i}^{\text{hidden}} \)</span> for every <span class="math notranslate nohighlight">\(i\)</span>-th neuron within the hidden layer, the vector will be sent to an activation function <span class="math notranslate nohighlight">\(a_i(\boldsymbol{z})\)</span>.</p>
|
||
<p>In this example, the sigmoid function has been chosen to be the activation function for each hidden neuron:</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
f(z) = \frac{1}{1 + \exp{(-z)}}
|
||
\]</div>
|
||
<p>It is possible to use other activations functions for the hidden layer also.</p>
|
||
<p>The output <span class="math notranslate nohighlight">\(\boldsymbol{x}_i^{\text{hidden}}\)</span> from each <span class="math notranslate nohighlight">\(i\)</span>-th hidden neuron is:</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\boldsymbol{x}_i^{\text{hidden} } = f\big( \boldsymbol{z}_{i}^{\text{hidden}} \big)
|
||
\]</div>
|
||
<p>The outputs <span class="math notranslate nohighlight">\(\boldsymbol{x}_i^{\text{hidden} } \)</span> are then sent to the output layer.</p>
|
||
<p>The output layer consists of one neuron in this case, and combines the
|
||
output from each of the neurons in the hidden layers. The output layer
|
||
combines the results from the hidden layer using some weights <span class="math notranslate nohighlight">\(w_i^{\text{output}}\)</span>
|
||
and biases <span class="math notranslate nohighlight">\(b_i^{\text{output}}\)</span>. In this case,
|
||
it is assumes that the number of neurons in the output layer is one.</p>
|
||
</div>
|
||
<div class="section" id="final-technicalities-iii">
|
||
<h2>Final technicalities III<a class="headerlink" href="#final-technicalities-iii" title="Permalink to this headline">¶</a></h2>
|
||
<p>The procedure of weighting the output neuron <span class="math notranslate nohighlight">\(j\)</span> in the hidden layer to the <span class="math notranslate nohighlight">\(i\)</span>-th neuron in the output layer is similar as for the hidden layer described previously.</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[\begin{split}
|
||
\begin{aligned}
|
||
z_{1,j}^{\text{output}} & =
|
||
\begin{pmatrix}
|
||
b_1^{\text{output}} & \boldsymbol{w}_1^{\text{output}}
|
||
\end{pmatrix}
|
||
\begin{pmatrix}
|
||
1 \\
|
||
\boldsymbol{x}_j^{\text{hidden}}
|
||
\end{pmatrix}
|
||
\end{aligned}
|
||
\end{split}\]</div>
|
||
</div>
|
||
<div class="section" id="final-technicalities-iv">
|
||
<h2>Final technicalities IV<a class="headerlink" href="#final-technicalities-iv" title="Permalink to this headline">¶</a></h2>
|
||
<p>Expressing <span class="math notranslate nohighlight">\(z_{1,j}^{\text{output}}\)</span> as a vector gives the following way of weighting the inputs from the hidden layer:</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[\begin{split}
|
||
\boldsymbol{z}_{1}^{\text{output}} =
|
||
\begin{pmatrix}
|
||
b_1^{\text{output}} & \boldsymbol{w}_1^{\text{output}}
|
||
\end{pmatrix}
|
||
\begin{pmatrix}
|
||
1 & 1 & \dots & 1 \\
|
||
\boldsymbol{x}_1^{\text{hidden}} & \boldsymbol{x}_2^{\text{hidden}} & \dots & \boldsymbol{x}_N^{\text{hidden}}
|
||
\end{pmatrix}
|
||
\end{split}\]</div>
|
||
<p>In this case we seek a continuous range of values since we are approximating a function. This means that after computing <span class="math notranslate nohighlight">\(\boldsymbol{z}_{1}^{\text{output}}\)</span> the neural network has finished its feed forward step, and <span class="math notranslate nohighlight">\(\boldsymbol{z}_{1}^{\text{output}}\)</span> is the final output of the network.</p>
|
||
</div>
|
||
<div class="section" id="back-propagation">
|
||
<h2>Back propagation<a class="headerlink" href="#back-propagation" title="Permalink to this headline">¶</a></h2>
|
||
<p>The next step is to decide how the parameters should be changed such that they minimize the cost function.</p>
|
||
<p>The chosen cost function for this problem is</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
C(\boldsymbol{x}, P) = \frac{1}{N} \sum_i \big(g_t'(x_i, P) - ( -\gamma g_t(x_i, P) \big)^2
|
||
\]</div>
|
||
<p>In order to minimize the cost function, an optimization method must be chosen.</p>
|
||
<p>Here, gradient descent with a constant step size has been chosen.</p>
|
||
</div>
|
||
<div class="section" id="gradient-descent">
|
||
<h2>Gradient descent<a class="headerlink" href="#gradient-descent" title="Permalink to this headline">¶</a></h2>
|
||
<p>The idea of the gradient descent algorithm is to update parameters in
|
||
a direction where the cost function decreases goes to a minimum.</p>
|
||
<p>In general, the update of some parameters <span class="math notranslate nohighlight">\(\boldsymbol{\omega}\)</span> given a cost
|
||
function defined by some weights <span class="math notranslate nohighlight">\(\boldsymbol{\omega}\)</span>, <span class="math notranslate nohighlight">\(C(\boldsymbol{x},
|
||
\boldsymbol{\omega})\)</span>, goes as follows:</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\boldsymbol{\omega}_{\text{new} } = \boldsymbol{\omega} - \lambda \nabla_{\boldsymbol{\omega}} C(\boldsymbol{x}, \boldsymbol{\omega})
|
||
\]</div>
|
||
<p>for a number of iterations or until <span class="math notranslate nohighlight">\( \big|\big| \boldsymbol{\omega}_{\text{new} } - \boldsymbol{\omega} \big|\big|\)</span> becomes smaller than some given tolerance.</p>
|
||
<p>The value of <span class="math notranslate nohighlight">\(\lambda\)</span> decides how large steps the algorithm must take
|
||
in the direction of <span class="math notranslate nohighlight">\( \nabla_{\boldsymbol{\omega}} C(\boldsymbol{x}, \boldsymbol{\omega})\)</span>.
|
||
The notation <span class="math notranslate nohighlight">\(\nabla_{\boldsymbol{\omega}}\)</span> express the gradient with respect
|
||
to the elements in <span class="math notranslate nohighlight">\(\boldsymbol{\omega}\)</span>.</p>
|
||
<p>In our case, we have to minimize the cost function <span class="math notranslate nohighlight">\(C(\boldsymbol{x}, P)\)</span> with
|
||
respect to the two sets of weights and biases, that is for the hidden
|
||
layer <span class="math notranslate nohighlight">\(P_{\text{hidden} }\)</span> and for the output layer <span class="math notranslate nohighlight">\(P_{\text{output}
|
||
}\)</span> .</p>
|
||
<p>This means that <span class="math notranslate nohighlight">\(P_{\text{hidden} }\)</span> and <span class="math notranslate nohighlight">\(P_{\text{output} }\)</span> is updated by</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[\begin{split}
|
||
\begin{aligned}
|
||
P_{\text{hidden},\text{new}} &= P_{\text{hidden}} - \lambda \nabla_{P_{\text{hidden}}} C(\boldsymbol{x}, P) \\
|
||
P_{\text{output},\text{new}} &= P_{\text{output}} - \lambda \nabla_{P_{\text{output}}} C(\boldsymbol{x}, P)
|
||
\end{aligned}
|
||
\end{split}\]</div>
|
||
</div>
|
||
<div class="section" id="the-code-for-solving-the-ode">
|
||
<h2>The code for solving the ODE<a class="headerlink" href="#the-code-for-solving-the-ode" title="Permalink to this headline">¶</a></h2>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">autograd.numpy</span> <span class="k">as</span> <span class="nn">np</span>
|
||
<span class="kn">from</span> <span class="nn">autograd</span> <span class="kn">import</span> <span class="n">grad</span><span class="p">,</span> <span class="n">elementwise_grad</span>
|
||
<span class="kn">import</span> <span class="nn">autograd.numpy.random</span> <span class="k">as</span> <span class="nn">npr</span>
|
||
<span class="kn">from</span> <span class="nn">matplotlib</span> <span class="kn">import</span> <span class="n">pyplot</span> <span class="k">as</span> <span class="n">plt</span>
|
||
|
||
<span class="k">def</span> <span class="nf">sigmoid</span><span class="p">(</span><span class="n">z</span><span class="p">):</span>
|
||
<span class="k">return</span> <span class="mi">1</span><span class="o">/</span><span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">z</span><span class="p">))</span>
|
||
|
||
<span class="c1"># Assuming one input, hidden, and output layer</span>
|
||
<span class="k">def</span> <span class="nf">neural_network</span><span class="p">(</span><span class="n">params</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
|
||
|
||
<span class="c1"># Find the weights (including and biases) for the hidden and output layer.</span>
|
||
<span class="c1"># Assume that params is a list of parameters for each layer.</span>
|
||
<span class="c1"># The biases are the first element for each array in params,</span>
|
||
<span class="c1"># and the weights are the remaning elements in each array in params.</span>
|
||
|
||
<span class="n">w_hidden</span> <span class="o">=</span> <span class="n">params</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
|
||
<span class="n">w_output</span> <span class="o">=</span> <span class="n">params</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
|
||
|
||
<span class="c1"># Assumes input x being an one-dimensional array</span>
|
||
<span class="n">num_values</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">size</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
|
||
<span class="n">x</span> <span class="o">=</span> <span class="n">x</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="n">num_values</span><span class="p">)</span>
|
||
|
||
<span class="c1"># Assume that the input layer does nothing to the input x</span>
|
||
<span class="n">x_input</span> <span class="o">=</span> <span class="n">x</span>
|
||
|
||
<span class="c1">## Hidden layer:</span>
|
||
|
||
<span class="c1"># Add a row of ones to include bias</span>
|
||
<span class="n">x_input</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">concatenate</span><span class="p">((</span><span class="n">np</span><span class="o">.</span><span class="n">ones</span><span class="p">((</span><span class="mi">1</span><span class="p">,</span><span class="n">num_values</span><span class="p">)),</span> <span class="n">x_input</span> <span class="p">),</span> <span class="n">axis</span> <span class="o">=</span> <span class="mi">0</span><span class="p">)</span>
|
||
|
||
<span class="n">z_hidden</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">matmul</span><span class="p">(</span><span class="n">w_hidden</span><span class="p">,</span> <span class="n">x_input</span><span class="p">)</span>
|
||
<span class="n">x_hidden</span> <span class="o">=</span> <span class="n">sigmoid</span><span class="p">(</span><span class="n">z_hidden</span><span class="p">)</span>
|
||
|
||
<span class="c1">## Output layer:</span>
|
||
|
||
<span class="c1"># Include bias:</span>
|
||
<span class="n">x_hidden</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">concatenate</span><span class="p">((</span><span class="n">np</span><span class="o">.</span><span class="n">ones</span><span class="p">((</span><span class="mi">1</span><span class="p">,</span><span class="n">num_values</span><span class="p">)),</span> <span class="n">x_hidden</span> <span class="p">),</span> <span class="n">axis</span> <span class="o">=</span> <span class="mi">0</span><span class="p">)</span>
|
||
|
||
<span class="n">z_output</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">matmul</span><span class="p">(</span><span class="n">w_output</span><span class="p">,</span> <span class="n">x_hidden</span><span class="p">)</span>
|
||
<span class="n">x_output</span> <span class="o">=</span> <span class="n">z_output</span>
|
||
|
||
<span class="k">return</span> <span class="n">x_output</span>
|
||
|
||
<span class="c1"># The trial solution using the deep neural network:</span>
|
||
<span class="k">def</span> <span class="nf">g_trial</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">params</span><span class="p">,</span> <span class="n">g0</span> <span class="o">=</span> <span class="mi">10</span><span class="p">):</span>
|
||
<span class="k">return</span> <span class="n">g0</span> <span class="o">+</span> <span class="n">x</span><span class="o">*</span><span class="n">neural_network</span><span class="p">(</span><span class="n">params</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
|
||
|
||
<span class="c1"># The right side of the ODE:</span>
|
||
<span class="k">def</span> <span class="nf">g</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">g_trial</span><span class="p">,</span> <span class="n">gamma</span> <span class="o">=</span> <span class="mi">2</span><span class="p">):</span>
|
||
<span class="k">return</span> <span class="o">-</span><span class="n">gamma</span><span class="o">*</span><span class="n">g_trial</span>
|
||
|
||
<span class="c1"># The cost function:</span>
|
||
<span class="k">def</span> <span class="nf">cost_function</span><span class="p">(</span><span class="n">P</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
|
||
|
||
<span class="c1"># Evaluate the trial function with the current parameters P</span>
|
||
<span class="n">g_t</span> <span class="o">=</span> <span class="n">g_trial</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">P</span><span class="p">)</span>
|
||
|
||
<span class="c1"># Find the derivative w.r.t x of the neural network</span>
|
||
<span class="n">d_net_out</span> <span class="o">=</span> <span class="n">elementwise_grad</span><span class="p">(</span><span class="n">neural_network</span><span class="p">,</span><span class="mi">1</span><span class="p">)(</span><span class="n">P</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
|
||
|
||
<span class="c1"># Find the derivative w.r.t x of the trial function</span>
|
||
<span class="n">d_g_t</span> <span class="o">=</span> <span class="n">elementwise_grad</span><span class="p">(</span><span class="n">g_trial</span><span class="p">,</span><span class="mi">0</span><span class="p">)(</span><span class="n">x</span><span class="p">,</span><span class="n">P</span><span class="p">)</span>
|
||
|
||
<span class="c1"># The right side of the ODE</span>
|
||
<span class="n">func</span> <span class="o">=</span> <span class="n">g</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">g_t</span><span class="p">)</span>
|
||
|
||
<span class="n">err_sqr</span> <span class="o">=</span> <span class="p">(</span><span class="n">d_g_t</span> <span class="o">-</span> <span class="n">func</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span>
|
||
<span class="n">cost_sum</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">err_sqr</span><span class="p">)</span>
|
||
|
||
<span class="k">return</span> <span class="n">cost_sum</span> <span class="o">/</span> <span class="n">np</span><span class="o">.</span><span class="n">size</span><span class="p">(</span><span class="n">err_sqr</span><span class="p">)</span>
|
||
|
||
<span class="c1"># Solve the exponential decay ODE using neural network with one input, hidden, and output layer</span>
|
||
<span class="k">def</span> <span class="nf">solve_ode_neural_network</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">num_neurons_hidden</span><span class="p">,</span> <span class="n">num_iter</span><span class="p">,</span> <span class="n">lmb</span><span class="p">):</span>
|
||
<span class="c1">## Set up initial weights and biases</span>
|
||
|
||
<span class="c1"># For the hidden layer</span>
|
||
<span class="n">p0</span> <span class="o">=</span> <span class="n">npr</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="n">num_neurons_hidden</span><span class="p">,</span> <span class="mi">2</span> <span class="p">)</span>
|
||
|
||
<span class="c1"># For the output layer</span>
|
||
<span class="n">p1</span> <span class="o">=</span> <span class="n">npr</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">num_neurons_hidden</span> <span class="o">+</span> <span class="mi">1</span> <span class="p">)</span> <span class="c1"># +1 since bias is included</span>
|
||
|
||
<span class="n">P</span> <span class="o">=</span> <span class="p">[</span><span class="n">p0</span><span class="p">,</span> <span class="n">p1</span><span class="p">]</span>
|
||
|
||
<span class="nb">print</span><span class="p">(</span><span class="s1">'Initial cost: </span><span class="si">%g</span><span class="s1">'</span><span class="o">%</span><span class="k">cost_function</span>(P, x))
|
||
|
||
<span class="c1">## Start finding the optimal weights using gradient descent</span>
|
||
|
||
<span class="c1"># Find the Python function that represents the gradient of the cost function</span>
|
||
<span class="c1"># w.r.t the 0-th input argument -- that is the weights and biases in the hidden and output layer</span>
|
||
<span class="n">cost_function_grad</span> <span class="o">=</span> <span class="n">grad</span><span class="p">(</span><span class="n">cost_function</span><span class="p">,</span><span class="mi">0</span><span class="p">)</span>
|
||
|
||
<span class="c1"># Let the update be done num_iter times</span>
|
||
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">num_iter</span><span class="p">):</span>
|
||
<span class="c1"># Evaluate the gradient at the current weights and biases in P.</span>
|
||
<span class="c1"># The cost_grad consist now of two arrays;</span>
|
||
<span class="c1"># one for the gradient w.r.t P_hidden and</span>
|
||
<span class="c1"># one for the gradient w.r.t P_output</span>
|
||
<span class="n">cost_grad</span> <span class="o">=</span> <span class="n">cost_function_grad</span><span class="p">(</span><span class="n">P</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
|
||
|
||
<span class="n">P</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">P</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">-</span> <span class="n">lmb</span> <span class="o">*</span> <span class="n">cost_grad</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
|
||
<span class="n">P</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">P</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">-</span> <span class="n">lmb</span> <span class="o">*</span> <span class="n">cost_grad</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
|
||
|
||
<span class="nb">print</span><span class="p">(</span><span class="s1">'Final cost: </span><span class="si">%g</span><span class="s1">'</span><span class="o">%</span><span class="k">cost_function</span>(P, x))
|
||
|
||
<span class="k">return</span> <span class="n">P</span>
|
||
|
||
<span class="k">def</span> <span class="nf">g_analytic</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">gamma</span> <span class="o">=</span> <span class="mi">2</span><span class="p">,</span> <span class="n">g0</span> <span class="o">=</span> <span class="mi">10</span><span class="p">):</span>
|
||
<span class="k">return</span> <span class="n">g0</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">gamma</span><span class="o">*</span><span class="n">x</span><span class="p">)</span>
|
||
|
||
<span class="c1"># Solve the given problem</span>
|
||
<span class="k">if</span> <span class="vm">__name__</span> <span class="o">==</span> <span class="s1">'__main__'</span><span class="p">:</span>
|
||
<span class="c1"># Set seed such that the weight are initialized</span>
|
||
<span class="c1"># with same weights and biases for every run.</span>
|
||
<span class="n">npr</span><span class="o">.</span><span class="n">seed</span><span class="p">(</span><span class="mi">15</span><span class="p">)</span>
|
||
|
||
<span class="c1">## Decide the vales of arguments to the function to solve</span>
|
||
<span class="n">N</span> <span class="o">=</span> <span class="mi">10</span>
|
||
<span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">N</span><span class="p">)</span>
|
||
|
||
<span class="c1">## Set up the initial parameters</span>
|
||
<span class="n">num_hidden_neurons</span> <span class="o">=</span> <span class="mi">10</span>
|
||
<span class="n">num_iter</span> <span class="o">=</span> <span class="mi">10000</span>
|
||
<span class="n">lmb</span> <span class="o">=</span> <span class="mf">0.001</span>
|
||
|
||
<span class="c1"># Use the network</span>
|
||
<span class="n">P</span> <span class="o">=</span> <span class="n">solve_ode_neural_network</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">num_hidden_neurons</span><span class="p">,</span> <span class="n">num_iter</span><span class="p">,</span> <span class="n">lmb</span><span class="p">)</span>
|
||
|
||
<span class="c1"># Print the deviation from the trial solution and true solution</span>
|
||
<span class="n">res</span> <span class="o">=</span> <span class="n">g_trial</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">P</span><span class="p">)</span>
|
||
<span class="n">res_analytical</span> <span class="o">=</span> <span class="n">g_analytic</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
|
||
|
||
<span class="nb">print</span><span class="p">(</span><span class="s1">'Max absolute difference: </span><span class="si">%g</span><span class="s1">'</span><span class="o">%</span><span class="k">np</span>.max(np.abs(res - res_analytical)))
|
||
|
||
<span class="c1"># Plot the results</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span><span class="mi">10</span><span class="p">))</span>
|
||
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s1">'Performance of neural network solving an ODE compared to the analytical solution'</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">res_analytical</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">res</span><span class="p">[</span><span class="mi">0</span><span class="p">,:])</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">([</span><span class="s1">'analytical'</span><span class="p">,</span><span class="s1">'nn'</span><span class="p">])</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s1">'x'</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s1">'g(x)'</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<div class="section" id="the-network-with-one-input-layer-specified-number-of-hidden-layers-and-one-output-layer">
|
||
<h2>The network with one input layer, specified number of hidden layers, and one output layer<a class="headerlink" href="#the-network-with-one-input-layer-specified-number-of-hidden-layers-and-one-output-layer" title="Permalink to this headline">¶</a></h2>
|
||
<p>It is also possible to extend the construction of our network into a more general one, allowing the network to contain more than one hidden layers.</p>
|
||
<p>The number of neurons within each hidden layer are given as a list of integers in the program below.</p>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">autograd.numpy</span> <span class="k">as</span> <span class="nn">np</span>
|
||
<span class="kn">from</span> <span class="nn">autograd</span> <span class="kn">import</span> <span class="n">grad</span><span class="p">,</span> <span class="n">elementwise_grad</span>
|
||
<span class="kn">import</span> <span class="nn">autograd.numpy.random</span> <span class="k">as</span> <span class="nn">npr</span>
|
||
<span class="kn">from</span> <span class="nn">matplotlib</span> <span class="kn">import</span> <span class="n">pyplot</span> <span class="k">as</span> <span class="n">plt</span>
|
||
|
||
<span class="k">def</span> <span class="nf">sigmoid</span><span class="p">(</span><span class="n">z</span><span class="p">):</span>
|
||
<span class="k">return</span> <span class="mi">1</span><span class="o">/</span><span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">z</span><span class="p">))</span>
|
||
|
||
<span class="c1"># The neural network with one input layer and one output layer,</span>
|
||
<span class="c1"># but with number of hidden layers specified by the user.</span>
|
||
<span class="k">def</span> <span class="nf">deep_neural_network</span><span class="p">(</span><span class="n">deep_params</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
|
||
<span class="c1"># N_hidden is the number of hidden layers</span>
|
||
<span class="c1"># deep_params is a list, len() should be used</span>
|
||
<span class="n">N_hidden</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">deep_params</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span> <span class="c1"># -1 since params consists of</span>
|
||
<span class="c1"># parameters to all the hidden</span>
|
||
<span class="c1"># layers AND the output layer.</span>
|
||
|
||
<span class="c1"># Assumes input x being an one-dimensional array</span>
|
||
<span class="n">num_values</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">size</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
|
||
<span class="n">x</span> <span class="o">=</span> <span class="n">x</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="n">num_values</span><span class="p">)</span>
|
||
|
||
<span class="c1"># Assume that the input layer does nothing to the input x</span>
|
||
<span class="n">x_input</span> <span class="o">=</span> <span class="n">x</span>
|
||
|
||
<span class="c1"># Due to multiple hidden layers, define a variable referencing to the</span>
|
||
<span class="c1"># output of the previous layer:</span>
|
||
<span class="n">x_prev</span> <span class="o">=</span> <span class="n">x_input</span>
|
||
|
||
<span class="c1">## Hidden layers:</span>
|
||
|
||
<span class="k">for</span> <span class="n">l</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">N_hidden</span><span class="p">):</span>
|
||
<span class="c1"># From the list of parameters P; find the correct weigths and bias for this layer</span>
|
||
<span class="n">w_hidden</span> <span class="o">=</span> <span class="n">deep_params</span><span class="p">[</span><span class="n">l</span><span class="p">]</span>
|
||
|
||
<span class="c1"># Add a row of ones to include bias</span>
|
||
<span class="n">x_prev</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">concatenate</span><span class="p">((</span><span class="n">np</span><span class="o">.</span><span class="n">ones</span><span class="p">((</span><span class="mi">1</span><span class="p">,</span><span class="n">num_values</span><span class="p">)),</span> <span class="n">x_prev</span> <span class="p">),</span> <span class="n">axis</span> <span class="o">=</span> <span class="mi">0</span><span class="p">)</span>
|
||
|
||
<span class="n">z_hidden</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">matmul</span><span class="p">(</span><span class="n">w_hidden</span><span class="p">,</span> <span class="n">x_prev</span><span class="p">)</span>
|
||
<span class="n">x_hidden</span> <span class="o">=</span> <span class="n">sigmoid</span><span class="p">(</span><span class="n">z_hidden</span><span class="p">)</span>
|
||
|
||
<span class="c1"># Update x_prev such that next layer can use the output from this layer</span>
|
||
<span class="n">x_prev</span> <span class="o">=</span> <span class="n">x_hidden</span>
|
||
|
||
<span class="c1">## Output layer:</span>
|
||
|
||
<span class="c1"># Get the weights and bias for this layer</span>
|
||
<span class="n">w_output</span> <span class="o">=</span> <span class="n">deep_params</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
|
||
|
||
<span class="c1"># Include bias:</span>
|
||
<span class="n">x_prev</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">concatenate</span><span class="p">((</span><span class="n">np</span><span class="o">.</span><span class="n">ones</span><span class="p">((</span><span class="mi">1</span><span class="p">,</span><span class="n">num_values</span><span class="p">)),</span> <span class="n">x_prev</span><span class="p">),</span> <span class="n">axis</span> <span class="o">=</span> <span class="mi">0</span><span class="p">)</span>
|
||
|
||
<span class="n">z_output</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">matmul</span><span class="p">(</span><span class="n">w_output</span><span class="p">,</span> <span class="n">x_prev</span><span class="p">)</span>
|
||
<span class="n">x_output</span> <span class="o">=</span> <span class="n">z_output</span>
|
||
|
||
<span class="k">return</span> <span class="n">x_output</span>
|
||
|
||
<span class="c1"># The trial solution using the deep neural network:</span>
|
||
<span class="k">def</span> <span class="nf">g_trial_deep</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">params</span><span class="p">,</span> <span class="n">g0</span> <span class="o">=</span> <span class="mi">10</span><span class="p">):</span>
|
||
<span class="k">return</span> <span class="n">g0</span> <span class="o">+</span> <span class="n">x</span><span class="o">*</span><span class="n">deep_neural_network</span><span class="p">(</span><span class="n">params</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
|
||
|
||
<span class="c1"># The right side of the ODE:</span>
|
||
<span class="k">def</span> <span class="nf">g</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">g_trial</span><span class="p">,</span> <span class="n">gamma</span> <span class="o">=</span> <span class="mi">2</span><span class="p">):</span>
|
||
<span class="k">return</span> <span class="o">-</span><span class="n">gamma</span><span class="o">*</span><span class="n">g_trial</span>
|
||
|
||
<span class="c1"># The same cost function as before, but calls deep_neural_network instead.</span>
|
||
<span class="k">def</span> <span class="nf">cost_function_deep</span><span class="p">(</span><span class="n">P</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
|
||
|
||
<span class="c1"># Evaluate the trial function with the current parameters P</span>
|
||
<span class="n">g_t</span> <span class="o">=</span> <span class="n">g_trial_deep</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">P</span><span class="p">)</span>
|
||
|
||
<span class="c1"># Find the derivative w.r.t x of the neural network</span>
|
||
<span class="n">d_net_out</span> <span class="o">=</span> <span class="n">elementwise_grad</span><span class="p">(</span><span class="n">deep_neural_network</span><span class="p">,</span><span class="mi">1</span><span class="p">)(</span><span class="n">P</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
|
||
|
||
<span class="c1"># Find the derivative w.r.t x of the trial function</span>
|
||
<span class="n">d_g_t</span> <span class="o">=</span> <span class="n">elementwise_grad</span><span class="p">(</span><span class="n">g_trial_deep</span><span class="p">,</span><span class="mi">0</span><span class="p">)(</span><span class="n">x</span><span class="p">,</span><span class="n">P</span><span class="p">)</span>
|
||
|
||
<span class="c1"># The right side of the ODE</span>
|
||
<span class="n">func</span> <span class="o">=</span> <span class="n">g</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">g_t</span><span class="p">)</span>
|
||
|
||
<span class="n">err_sqr</span> <span class="o">=</span> <span class="p">(</span><span class="n">d_g_t</span> <span class="o">-</span> <span class="n">func</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span>
|
||
<span class="n">cost_sum</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">err_sqr</span><span class="p">)</span>
|
||
|
||
<span class="k">return</span> <span class="n">cost_sum</span> <span class="o">/</span> <span class="n">np</span><span class="o">.</span><span class="n">size</span><span class="p">(</span><span class="n">err_sqr</span><span class="p">)</span>
|
||
|
||
<span class="c1"># Solve the exponential decay ODE using neural network with one input and one output layer,</span>
|
||
<span class="c1"># but with specified number of hidden layers from the user.</span>
|
||
<span class="k">def</span> <span class="nf">solve_ode_deep_neural_network</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">num_neurons</span><span class="p">,</span> <span class="n">num_iter</span><span class="p">,</span> <span class="n">lmb</span><span class="p">):</span>
|
||
<span class="c1"># num_hidden_neurons is now a list of number of neurons within each hidden layer</span>
|
||
|
||
<span class="c1"># The number of elements in the list num_hidden_neurons thus represents</span>
|
||
<span class="c1"># the number of hidden layers.</span>
|
||
|
||
<span class="c1"># Find the number of hidden layers:</span>
|
||
<span class="n">N_hidden</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">size</span><span class="p">(</span><span class="n">num_neurons</span><span class="p">)</span>
|
||
|
||
<span class="c1">## Set up initial weights and biases</span>
|
||
|
||
<span class="c1"># Initialize the list of parameters:</span>
|
||
<span class="n">P</span> <span class="o">=</span> <span class="p">[</span><span class="kc">None</span><span class="p">]</span><span class="o">*</span><span class="p">(</span><span class="n">N_hidden</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="c1"># + 1 to include the output layer</span>
|
||
|
||
<span class="n">P</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">npr</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="n">num_neurons</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="mi">2</span> <span class="p">)</span>
|
||
<span class="k">for</span> <span class="n">l</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="n">N_hidden</span><span class="p">):</span>
|
||
<span class="n">P</span><span class="p">[</span><span class="n">l</span><span class="p">]</span> <span class="o">=</span> <span class="n">npr</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="n">num_neurons</span><span class="p">[</span><span class="n">l</span><span class="p">],</span> <span class="n">num_neurons</span><span class="p">[</span><span class="n">l</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="c1"># +1 to include bias</span>
|
||
|
||
<span class="c1"># For the output layer</span>
|
||
<span class="n">P</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">npr</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">num_neurons</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">+</span> <span class="mi">1</span> <span class="p">)</span> <span class="c1"># +1 since bias is included</span>
|
||
|
||
<span class="nb">print</span><span class="p">(</span><span class="s1">'Initial cost: </span><span class="si">%g</span><span class="s1">'</span><span class="o">%</span><span class="k">cost_function_deep</span>(P, x))
|
||
|
||
<span class="c1">## Start finding the optimal weights using gradient descent</span>
|
||
|
||
<span class="c1"># Find the Python function that represents the gradient of the cost function</span>
|
||
<span class="c1"># w.r.t the 0-th input argument -- that is the weights and biases in the hidden and output layer</span>
|
||
<span class="n">cost_function_deep_grad</span> <span class="o">=</span> <span class="n">grad</span><span class="p">(</span><span class="n">cost_function_deep</span><span class="p">,</span><span class="mi">0</span><span class="p">)</span>
|
||
|
||
<span class="c1"># Let the update be done num_iter times</span>
|
||
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">num_iter</span><span class="p">):</span>
|
||
<span class="c1"># Evaluate the gradient at the current weights and biases in P.</span>
|
||
<span class="c1"># The cost_grad consist now of N_hidden + 1 arrays; the gradient w.r.t the weights and biases</span>
|
||
<span class="c1"># in the hidden layers and output layers evaluated at x.</span>
|
||
<span class="n">cost_deep_grad</span> <span class="o">=</span> <span class="n">cost_function_deep_grad</span><span class="p">(</span><span class="n">P</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
|
||
|
||
<span class="k">for</span> <span class="n">l</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">N_hidden</span><span class="o">+</span><span class="mi">1</span><span class="p">):</span>
|
||
<span class="n">P</span><span class="p">[</span><span class="n">l</span><span class="p">]</span> <span class="o">=</span> <span class="n">P</span><span class="p">[</span><span class="n">l</span><span class="p">]</span> <span class="o">-</span> <span class="n">lmb</span> <span class="o">*</span> <span class="n">cost_deep_grad</span><span class="p">[</span><span class="n">l</span><span class="p">]</span>
|
||
|
||
<span class="nb">print</span><span class="p">(</span><span class="s1">'Final cost: </span><span class="si">%g</span><span class="s1">'</span><span class="o">%</span><span class="k">cost_function_deep</span>(P, x))
|
||
|
||
<span class="k">return</span> <span class="n">P</span>
|
||
|
||
<span class="k">def</span> <span class="nf">g_analytic</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">gamma</span> <span class="o">=</span> <span class="mi">2</span><span class="p">,</span> <span class="n">g0</span> <span class="o">=</span> <span class="mi">10</span><span class="p">):</span>
|
||
<span class="k">return</span> <span class="n">g0</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">gamma</span><span class="o">*</span><span class="n">x</span><span class="p">)</span>
|
||
|
||
<span class="c1"># Solve the given problem</span>
|
||
<span class="k">if</span> <span class="vm">__name__</span> <span class="o">==</span> <span class="s1">'__main__'</span><span class="p">:</span>
|
||
<span class="n">npr</span><span class="o">.</span><span class="n">seed</span><span class="p">(</span><span class="mi">15</span><span class="p">)</span>
|
||
|
||
<span class="c1">## Decide the vales of arguments to the function to solve</span>
|
||
<span class="n">N</span> <span class="o">=</span> <span class="mi">10</span>
|
||
<span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">N</span><span class="p">)</span>
|
||
|
||
<span class="c1">## Set up the initial parameters</span>
|
||
<span class="n">num_hidden_neurons</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mi">10</span><span class="p">,</span><span class="mi">10</span><span class="p">])</span>
|
||
<span class="n">num_iter</span> <span class="o">=</span> <span class="mi">10000</span>
|
||
<span class="n">lmb</span> <span class="o">=</span> <span class="mf">0.001</span>
|
||
|
||
<span class="n">P</span> <span class="o">=</span> <span class="n">solve_ode_deep_neural_network</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">num_hidden_neurons</span><span class="p">,</span> <span class="n">num_iter</span><span class="p">,</span> <span class="n">lmb</span><span class="p">)</span>
|
||
|
||
<span class="n">res</span> <span class="o">=</span> <span class="n">g_trial_deep</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">P</span><span class="p">)</span>
|
||
<span class="n">res_analytical</span> <span class="o">=</span> <span class="n">g_analytic</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
|
||
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span><span class="mi">10</span><span class="p">))</span>
|
||
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s1">'Performance of a deep neural network solving an ODE compared to the analytical solution'</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">res_analytical</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">res</span><span class="p">[</span><span class="mi">0</span><span class="p">,:])</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">([</span><span class="s1">'analytical'</span><span class="p">,</span><span class="s1">'dnn'</span><span class="p">])</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s1">'g(x)'</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<div class="section" id="example-population-growth">
|
||
<h2>Example: Population growth<a class="headerlink" href="#example-population-growth" title="Permalink to this headline">¶</a></h2>
|
||
<p>A logistic model of population growth assumes that a population converges toward an equilibrium.
|
||
The population growth can be modeled by</p>
|
||
<!-- Equation labels as ordinary links -->
|
||
<div id="log"></div>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\begin{equation} \label{log} \tag{10}
|
||
g'(t) = \alpha g(t)(A - g(t))
|
||
\end{equation}
|
||
\]</div>
|
||
<p>where <span class="math notranslate nohighlight">\(g(t)\)</span> is the population density at time <span class="math notranslate nohighlight">\(t\)</span>, <span class="math notranslate nohighlight">\(\alpha > 0\)</span> the growth rate and <span class="math notranslate nohighlight">\(A > 0\)</span> is the maximum population number in the environment.
|
||
Also, at <span class="math notranslate nohighlight">\(t = 0\)</span> the population has the size <span class="math notranslate nohighlight">\(g(0) = g_0\)</span>, where <span class="math notranslate nohighlight">\(g_0\)</span> is some chosen constant.</p>
|
||
<p>In this example, similar network as for the exponential decay using Autograd has been used to solve the equation. However, as the implementation might suffer from e.g numerical instability
|
||
and high execution time (this might be more apparent in the examples solving PDEs),
|
||
using a library like TensorFlow is recommended.
|
||
Here, we stay with a more simple approach and implement for comparison, the simple forward Euler method.</p>
|
||
</div>
|
||
<div class="section" id="setting-up-the-problem">
|
||
<h2>Setting up the problem<a class="headerlink" href="#setting-up-the-problem" title="Permalink to this headline">¶</a></h2>
|
||
<p>Here, we will model a population <span class="math notranslate nohighlight">\(g(t)\)</span> in an environment having carrying capacity <span class="math notranslate nohighlight">\(A\)</span>.
|
||
The population follows the model</p>
|
||
<!-- Equation labels as ordinary links -->
|
||
<div id="solveode_population"></div>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\begin{equation} \label{solveode_population} \tag{11}
|
||
g'(t) = \alpha g(t)(A - g(t))
|
||
\end{equation}
|
||
\]</div>
|
||
<p>where <span class="math notranslate nohighlight">\(g(0) = g_0\)</span>.</p>
|
||
<p>In this example, we let <span class="math notranslate nohighlight">\(\alpha = 2\)</span>, <span class="math notranslate nohighlight">\(A = 1\)</span>, and <span class="math notranslate nohighlight">\(g_0 = 1.2\)</span>.</p>
|
||
</div>
|
||
<div class="section" id="id5">
|
||
<h2>The trial solution<a class="headerlink" href="#id5" title="Permalink to this headline">¶</a></h2>
|
||
<p>We will get a slightly different trial solution, as the boundary conditions are different
|
||
compared to the case for exponential decay.</p>
|
||
<p>A possible trial solution satisfying the condition <span class="math notranslate nohighlight">\(g(0) = g_0\)</span> could be</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
h_1(t) = g_0 + t \cdot N(t,P)
|
||
\]</div>
|
||
<p>with <span class="math notranslate nohighlight">\(N(t,P)\)</span> being the output from the neural network with weights and biases for each layer collected in the set <span class="math notranslate nohighlight">\(P\)</span>.</p>
|
||
<p>The analytical solution is</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
g(t) = \frac{Ag_0}{g_0 + (A - g_0)\exp(-\alpha A t)}
|
||
\]</div>
|
||
</div>
|
||
<div class="section" id="the-program-using-autograd">
|
||
<h2>The program using Autograd<a class="headerlink" href="#the-program-using-autograd" title="Permalink to this headline">¶</a></h2>
|
||
<p>The network will be the similar as for the exponential decay example, but with some small modifications for our problem.</p>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">autograd.numpy</span> <span class="k">as</span> <span class="nn">np</span>
|
||
<span class="kn">from</span> <span class="nn">autograd</span> <span class="kn">import</span> <span class="n">grad</span><span class="p">,</span> <span class="n">elementwise_grad</span>
|
||
<span class="kn">import</span> <span class="nn">autograd.numpy.random</span> <span class="k">as</span> <span class="nn">npr</span>
|
||
<span class="kn">from</span> <span class="nn">matplotlib</span> <span class="kn">import</span> <span class="n">pyplot</span> <span class="k">as</span> <span class="n">plt</span>
|
||
|
||
<span class="k">def</span> <span class="nf">sigmoid</span><span class="p">(</span><span class="n">z</span><span class="p">):</span>
|
||
<span class="k">return</span> <span class="mi">1</span><span class="o">/</span><span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">z</span><span class="p">))</span>
|
||
|
||
<span class="c1"># Function to get the parameters.</span>
|
||
<span class="c1"># Done such that one can easily change the paramaters after one's liking.</span>
|
||
<span class="k">def</span> <span class="nf">get_parameters</span><span class="p">():</span>
|
||
<span class="n">alpha</span> <span class="o">=</span> <span class="mi">2</span>
|
||
<span class="n">A</span> <span class="o">=</span> <span class="mi">1</span>
|
||
<span class="n">g0</span> <span class="o">=</span> <span class="mf">1.2</span>
|
||
<span class="k">return</span> <span class="n">alpha</span><span class="p">,</span> <span class="n">A</span><span class="p">,</span> <span class="n">g0</span>
|
||
|
||
<span class="k">def</span> <span class="nf">deep_neural_network</span><span class="p">(</span><span class="n">deep_params</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
|
||
<span class="c1"># N_hidden is the number of hidden layers</span>
|
||
<span class="c1"># deep_params is a list, len() should be used</span>
|
||
<span class="n">N_hidden</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">deep_params</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span> <span class="c1"># -1 since params consists of</span>
|
||
<span class="c1"># parameters to all the hidden</span>
|
||
<span class="c1"># layers AND the output layer.</span>
|
||
|
||
<span class="c1"># Assumes input x being an one-dimensional array</span>
|
||
<span class="n">num_values</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">size</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
|
||
<span class="n">x</span> <span class="o">=</span> <span class="n">x</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="n">num_values</span><span class="p">)</span>
|
||
|
||
<span class="c1"># Assume that the input layer does nothing to the input x</span>
|
||
<span class="n">x_input</span> <span class="o">=</span> <span class="n">x</span>
|
||
|
||
<span class="c1"># Due to multiple hidden layers, define a variable referencing to the</span>
|
||
<span class="c1"># output of the previous layer:</span>
|
||
<span class="n">x_prev</span> <span class="o">=</span> <span class="n">x_input</span>
|
||
|
||
<span class="c1">## Hidden layers:</span>
|
||
|
||
<span class="k">for</span> <span class="n">l</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">N_hidden</span><span class="p">):</span>
|
||
<span class="c1"># From the list of parameters P; find the correct weigths and bias for this layer</span>
|
||
<span class="n">w_hidden</span> <span class="o">=</span> <span class="n">deep_params</span><span class="p">[</span><span class="n">l</span><span class="p">]</span>
|
||
|
||
<span class="c1"># Add a row of ones to include bias</span>
|
||
<span class="n">x_prev</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">concatenate</span><span class="p">((</span><span class="n">np</span><span class="o">.</span><span class="n">ones</span><span class="p">((</span><span class="mi">1</span><span class="p">,</span><span class="n">num_values</span><span class="p">)),</span> <span class="n">x_prev</span> <span class="p">),</span> <span class="n">axis</span> <span class="o">=</span> <span class="mi">0</span><span class="p">)</span>
|
||
|
||
<span class="n">z_hidden</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">matmul</span><span class="p">(</span><span class="n">w_hidden</span><span class="p">,</span> <span class="n">x_prev</span><span class="p">)</span>
|
||
<span class="n">x_hidden</span> <span class="o">=</span> <span class="n">sigmoid</span><span class="p">(</span><span class="n">z_hidden</span><span class="p">)</span>
|
||
|
||
<span class="c1"># Update x_prev such that next layer can use the output from this layer</span>
|
||
<span class="n">x_prev</span> <span class="o">=</span> <span class="n">x_hidden</span>
|
||
|
||
<span class="c1">## Output layer:</span>
|
||
|
||
<span class="c1"># Get the weights and bias for this layer</span>
|
||
<span class="n">w_output</span> <span class="o">=</span> <span class="n">deep_params</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
|
||
|
||
<span class="c1"># Include bias:</span>
|
||
<span class="n">x_prev</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">concatenate</span><span class="p">((</span><span class="n">np</span><span class="o">.</span><span class="n">ones</span><span class="p">((</span><span class="mi">1</span><span class="p">,</span><span class="n">num_values</span><span class="p">)),</span> <span class="n">x_prev</span><span class="p">),</span> <span class="n">axis</span> <span class="o">=</span> <span class="mi">0</span><span class="p">)</span>
|
||
|
||
<span class="n">z_output</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">matmul</span><span class="p">(</span><span class="n">w_output</span><span class="p">,</span> <span class="n">x_prev</span><span class="p">)</span>
|
||
<span class="n">x_output</span> <span class="o">=</span> <span class="n">z_output</span>
|
||
|
||
<span class="k">return</span> <span class="n">x_output</span>
|
||
|
||
|
||
|
||
|
||
<span class="k">def</span> <span class="nf">cost_function_deep</span><span class="p">(</span><span class="n">P</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
|
||
|
||
<span class="c1"># Evaluate the trial function with the current parameters P</span>
|
||
<span class="n">g_t</span> <span class="o">=</span> <span class="n">g_trial_deep</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">P</span><span class="p">)</span>
|
||
|
||
<span class="c1"># Find the derivative w.r.t x of the trial function</span>
|
||
<span class="n">d_g_t</span> <span class="o">=</span> <span class="n">elementwise_grad</span><span class="p">(</span><span class="n">g_trial_deep</span><span class="p">,</span><span class="mi">0</span><span class="p">)(</span><span class="n">x</span><span class="p">,</span><span class="n">P</span><span class="p">)</span>
|
||
|
||
<span class="c1"># The right side of the ODE</span>
|
||
<span class="n">func</span> <span class="o">=</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">g_t</span><span class="p">)</span>
|
||
|
||
<span class="n">err_sqr</span> <span class="o">=</span> <span class="p">(</span><span class="n">d_g_t</span> <span class="o">-</span> <span class="n">func</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span>
|
||
<span class="n">cost_sum</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">err_sqr</span><span class="p">)</span>
|
||
|
||
<span class="k">return</span> <span class="n">cost_sum</span> <span class="o">/</span> <span class="n">np</span><span class="o">.</span><span class="n">size</span><span class="p">(</span><span class="n">err_sqr</span><span class="p">)</span>
|
||
|
||
<span class="c1"># The right side of the ODE:</span>
|
||
<span class="k">def</span> <span class="nf">f</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">g_trial</span><span class="p">):</span>
|
||
<span class="n">alpha</span><span class="p">,</span><span class="n">A</span><span class="p">,</span> <span class="n">g0</span> <span class="o">=</span> <span class="n">get_parameters</span><span class="p">()</span>
|
||
<span class="k">return</span> <span class="n">alpha</span><span class="o">*</span><span class="n">g_trial</span><span class="o">*</span><span class="p">(</span><span class="n">A</span> <span class="o">-</span> <span class="n">g_trial</span><span class="p">)</span>
|
||
|
||
<span class="c1"># The trial solution using the deep neural network:</span>
|
||
<span class="k">def</span> <span class="nf">g_trial_deep</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">params</span><span class="p">):</span>
|
||
<span class="n">alpha</span><span class="p">,</span><span class="n">A</span><span class="p">,</span> <span class="n">g0</span> <span class="o">=</span> <span class="n">get_parameters</span><span class="p">()</span>
|
||
<span class="k">return</span> <span class="n">g0</span> <span class="o">+</span> <span class="n">x</span><span class="o">*</span><span class="n">deep_neural_network</span><span class="p">(</span><span class="n">params</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
|
||
|
||
<span class="c1"># The analytical solution:</span>
|
||
<span class="k">def</span> <span class="nf">g_analytic</span><span class="p">(</span><span class="n">t</span><span class="p">):</span>
|
||
<span class="n">alpha</span><span class="p">,</span><span class="n">A</span><span class="p">,</span> <span class="n">g0</span> <span class="o">=</span> <span class="n">get_parameters</span><span class="p">()</span>
|
||
<span class="k">return</span> <span class="n">A</span><span class="o">*</span><span class="n">g0</span><span class="o">/</span><span class="p">(</span><span class="n">g0</span> <span class="o">+</span> <span class="p">(</span><span class="n">A</span> <span class="o">-</span> <span class="n">g0</span><span class="p">)</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">alpha</span><span class="o">*</span><span class="n">A</span><span class="o">*</span><span class="n">t</span><span class="p">))</span>
|
||
|
||
<span class="k">def</span> <span class="nf">solve_ode_deep_neural_network</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">num_neurons</span><span class="p">,</span> <span class="n">num_iter</span><span class="p">,</span> <span class="n">lmb</span><span class="p">):</span>
|
||
<span class="c1"># num_hidden_neurons is now a list of number of neurons within each hidden layer</span>
|
||
|
||
<span class="c1"># Find the number of hidden layers:</span>
|
||
<span class="n">N_hidden</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">size</span><span class="p">(</span><span class="n">num_neurons</span><span class="p">)</span>
|
||
|
||
<span class="c1">## Set up initial weigths and biases</span>
|
||
|
||
<span class="c1"># Initialize the list of parameters:</span>
|
||
<span class="n">P</span> <span class="o">=</span> <span class="p">[</span><span class="kc">None</span><span class="p">]</span><span class="o">*</span><span class="p">(</span><span class="n">N_hidden</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="c1"># + 1 to include the output layer</span>
|
||
|
||
<span class="n">P</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">npr</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="n">num_neurons</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="mi">2</span> <span class="p">)</span>
|
||
<span class="k">for</span> <span class="n">l</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="n">N_hidden</span><span class="p">):</span>
|
||
<span class="n">P</span><span class="p">[</span><span class="n">l</span><span class="p">]</span> <span class="o">=</span> <span class="n">npr</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="n">num_neurons</span><span class="p">[</span><span class="n">l</span><span class="p">],</span> <span class="n">num_neurons</span><span class="p">[</span><span class="n">l</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="c1"># +1 to include bias</span>
|
||
|
||
<span class="c1"># For the output layer</span>
|
||
<span class="n">P</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">npr</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">num_neurons</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">+</span> <span class="mi">1</span> <span class="p">)</span> <span class="c1"># +1 since bias is included</span>
|
||
|
||
<span class="nb">print</span><span class="p">(</span><span class="s1">'Initial cost: </span><span class="si">%g</span><span class="s1">'</span><span class="o">%</span><span class="k">cost_function_deep</span>(P, x))
|
||
|
||
<span class="c1">## Start finding the optimal weigths using gradient descent</span>
|
||
|
||
<span class="c1"># Find the Python function that represents the gradient of the cost function</span>
|
||
<span class="c1"># w.r.t the 0-th input argument -- that is the weights and biases in the hidden and output layer</span>
|
||
<span class="n">cost_function_deep_grad</span> <span class="o">=</span> <span class="n">grad</span><span class="p">(</span><span class="n">cost_function_deep</span><span class="p">,</span><span class="mi">0</span><span class="p">)</span>
|
||
|
||
<span class="c1"># Let the update be done num_iter times</span>
|
||
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">num_iter</span><span class="p">):</span>
|
||
<span class="c1"># Evaluate the gradient at the current weights and biases in P.</span>
|
||
<span class="c1"># The cost_grad consist now of N_hidden + 1 arrays; the gradient w.r.t the weights and biases</span>
|
||
<span class="c1"># in the hidden layers and output layers evaluated at x.</span>
|
||
<span class="n">cost_deep_grad</span> <span class="o">=</span> <span class="n">cost_function_deep_grad</span><span class="p">(</span><span class="n">P</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
|
||
|
||
<span class="k">for</span> <span class="n">l</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">N_hidden</span><span class="o">+</span><span class="mi">1</span><span class="p">):</span>
|
||
<span class="n">P</span><span class="p">[</span><span class="n">l</span><span class="p">]</span> <span class="o">=</span> <span class="n">P</span><span class="p">[</span><span class="n">l</span><span class="p">]</span> <span class="o">-</span> <span class="n">lmb</span> <span class="o">*</span> <span class="n">cost_deep_grad</span><span class="p">[</span><span class="n">l</span><span class="p">]</span>
|
||
|
||
<span class="nb">print</span><span class="p">(</span><span class="s1">'Final cost: </span><span class="si">%g</span><span class="s1">'</span><span class="o">%</span><span class="k">cost_function_deep</span>(P, x))
|
||
|
||
<span class="k">return</span> <span class="n">P</span>
|
||
|
||
<span class="k">if</span> <span class="vm">__name__</span> <span class="o">==</span> <span class="s1">'__main__'</span><span class="p">:</span>
|
||
<span class="n">npr</span><span class="o">.</span><span class="n">seed</span><span class="p">(</span><span class="mi">4155</span><span class="p">)</span>
|
||
|
||
<span class="c1">## Decide the vales of arguments to the function to solve</span>
|
||
<span class="n">Nt</span> <span class="o">=</span> <span class="mi">10</span>
|
||
<span class="n">T</span> <span class="o">=</span> <span class="mi">1</span>
|
||
<span class="n">t</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="n">T</span><span class="p">,</span> <span class="n">Nt</span><span class="p">)</span>
|
||
|
||
<span class="c1">## Set up the initial parameters</span>
|
||
<span class="n">num_hidden_neurons</span> <span class="o">=</span> <span class="p">[</span><span class="mi">100</span><span class="p">,</span> <span class="mi">50</span><span class="p">,</span> <span class="mi">25</span><span class="p">]</span>
|
||
<span class="n">num_iter</span> <span class="o">=</span> <span class="mi">1000</span>
|
||
<span class="n">lmb</span> <span class="o">=</span> <span class="mf">1e-3</span>
|
||
|
||
<span class="n">P</span> <span class="o">=</span> <span class="n">solve_ode_deep_neural_network</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="n">num_hidden_neurons</span><span class="p">,</span> <span class="n">num_iter</span><span class="p">,</span> <span class="n">lmb</span><span class="p">)</span>
|
||
|
||
<span class="n">g_dnn_ag</span> <span class="o">=</span> <span class="n">g_trial_deep</span><span class="p">(</span><span class="n">t</span><span class="p">,</span><span class="n">P</span><span class="p">)</span>
|
||
<span class="n">g_analytical</span> <span class="o">=</span> <span class="n">g_analytic</span><span class="p">(</span><span class="n">t</span><span class="p">)</span>
|
||
|
||
<span class="c1"># Find the maximum absolute difference between the solutons:</span>
|
||
<span class="n">diff_ag</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">max</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">g_dnn_ag</span> <span class="o">-</span> <span class="n">g_analytical</span><span class="p">))</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="s2">"The max absolute difference between the solutions is: </span><span class="si">%g</span><span class="s2">"</span><span class="o">%</span><span class="k">diff_ag</span>)
|
||
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span><span class="mi">10</span><span class="p">))</span>
|
||
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s1">'Performance of neural network solving an ODE compared to the analytical solution'</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="n">g_analytical</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="n">g_dnn_ag</span><span class="p">[</span><span class="mi">0</span><span class="p">,:])</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">([</span><span class="s1">'analytical'</span><span class="p">,</span><span class="s1">'nn'</span><span class="p">])</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s1">'t'</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s1">'g(t)'</span><span class="p">)</span>
|
||
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<div class="section" id="using-forward-euler-to-solve-the-ode">
|
||
<h2>Using forward Euler to solve the ODE<a class="headerlink" href="#using-forward-euler-to-solve-the-ode" title="Permalink to this headline">¶</a></h2>
|
||
<p>A straightforward way of solving an ODE numerically, is to use Euler’s method.</p>
|
||
<p>Euler’s method uses Taylor series to approximate the value at a function <span class="math notranslate nohighlight">\(f\)</span> at a step <span class="math notranslate nohighlight">\(\Delta x\)</span> from <span class="math notranslate nohighlight">\(x\)</span>:</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
f(x + \Delta x) \approx f(x) + \Delta x f'(x)
|
||
\]</div>
|
||
<p>In our case, using Euler’s method to approximate the value of <span class="math notranslate nohighlight">\(g\)</span> at a step <span class="math notranslate nohighlight">\(\Delta t\)</span> from <span class="math notranslate nohighlight">\(t\)</span> yields</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[\begin{split}
|
||
\begin{aligned}
|
||
g(t + \Delta t) &\approx g(t) + \Delta t g'(t) \\
|
||
&= g(t) + \Delta t \big(\alpha g(t)(A - g(t))\big)
|
||
\end{aligned}
|
||
\end{split}\]</div>
|
||
<p>along with the condition that <span class="math notranslate nohighlight">\(g(0) = g_0\)</span>.</p>
|
||
<p>Let <span class="math notranslate nohighlight">\(t_i = i \cdot \Delta t\)</span> where <span class="math notranslate nohighlight">\(\Delta t = \frac{T}{N_t-1}\)</span> where <span class="math notranslate nohighlight">\(T\)</span> is the final time our solver must solve for and <span class="math notranslate nohighlight">\(N_t\)</span> the number of values for <span class="math notranslate nohighlight">\(t \in [0, T]\)</span> for <span class="math notranslate nohighlight">\(i = 0, \dots, N_t-1\)</span>.</p>
|
||
<p>For <span class="math notranslate nohighlight">\(i \geq 1\)</span>, we have that</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[\begin{split}
|
||
\begin{aligned}
|
||
t_i &= i\Delta t \\
|
||
&= (i - 1)\Delta t + \Delta t \\
|
||
&= t_{i-1} + \Delta t
|
||
\end{aligned}
|
||
\end{split}\]</div>
|
||
<p>Now, if <span class="math notranslate nohighlight">\(g_i = g(t_i)\)</span> then</p>
|
||
<!-- Equation labels as ordinary links -->
|
||
<div id="odenum"></div>
|
||
<div class="math notranslate nohighlight">
|
||
\[\begin{split}
|
||
\begin{equation}
|
||
\begin{aligned}
|
||
g_i &= g(t_i) \\
|
||
&= g(t_{i-1} + \Delta t) \\
|
||
&\approx g(t_{i-1}) + \Delta t \big(\alpha g(t_{i-1})(A - g(t_{i-1}))\big) \\
|
||
&= g_{i-1} + \Delta t \big(\alpha g_{i-1}(A - g_{i-1})\big)
|
||
\end{aligned}
|
||
\end{equation} \label{odenum} \tag{12}
|
||
\end{split}\]</div>
|
||
<p>for <span class="math notranslate nohighlight">\(i \geq 1\)</span> and <span class="math notranslate nohighlight">\(g_0 = g(t_0) = g(0) = g_0\)</span>.</p>
|
||
<p>Equation (<a class="reference external" href="#odenum">12</a>) could be implemented in the following way,
|
||
extending the program that uses the network using Autograd:</p>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Assume that all function definitions from the example program using Autograd</span>
|
||
<span class="c1"># are located here.</span>
|
||
|
||
<span class="k">if</span> <span class="vm">__name__</span> <span class="o">==</span> <span class="s1">'__main__'</span><span class="p">:</span>
|
||
<span class="n">npr</span><span class="o">.</span><span class="n">seed</span><span class="p">(</span><span class="mi">4155</span><span class="p">)</span>
|
||
|
||
<span class="c1">## Decide the vales of arguments to the function to solve</span>
|
||
<span class="n">Nt</span> <span class="o">=</span> <span class="mi">10</span>
|
||
<span class="n">T</span> <span class="o">=</span> <span class="mi">1</span>
|
||
<span class="n">t</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="n">T</span><span class="p">,</span> <span class="n">Nt</span><span class="p">)</span>
|
||
|
||
<span class="c1">## Set up the initial parameters</span>
|
||
<span class="n">num_hidden_neurons</span> <span class="o">=</span> <span class="p">[</span><span class="mi">100</span><span class="p">,</span><span class="mi">50</span><span class="p">,</span><span class="mi">25</span><span class="p">]</span>
|
||
<span class="n">num_iter</span> <span class="o">=</span> <span class="mi">1000</span>
|
||
<span class="n">lmb</span> <span class="o">=</span> <span class="mf">1e-3</span>
|
||
|
||
<span class="n">P</span> <span class="o">=</span> <span class="n">solve_ode_deep_neural_network</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="n">num_hidden_neurons</span><span class="p">,</span> <span class="n">num_iter</span><span class="p">,</span> <span class="n">lmb</span><span class="p">)</span>
|
||
|
||
<span class="n">g_dnn_ag</span> <span class="o">=</span> <span class="n">g_trial_deep</span><span class="p">(</span><span class="n">t</span><span class="p">,</span><span class="n">P</span><span class="p">)</span>
|
||
<span class="n">g_analytical</span> <span class="o">=</span> <span class="n">g_analytic</span><span class="p">(</span><span class="n">t</span><span class="p">)</span>
|
||
|
||
<span class="c1"># Find the maximum absolute difference between the solutons:</span>
|
||
<span class="n">diff_ag</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">max</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">g_dnn_ag</span> <span class="o">-</span> <span class="n">g_analytical</span><span class="p">))</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="s2">"The max absolute difference between the solutions is: </span><span class="si">%g</span><span class="s2">"</span><span class="o">%</span><span class="k">diff_ag</span>)
|
||
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span><span class="mi">10</span><span class="p">))</span>
|
||
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s1">'Performance of neural network solving an ODE compared to the analytical solution'</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="n">g_analytical</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="n">g_dnn_ag</span><span class="p">[</span><span class="mi">0</span><span class="p">,:])</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">([</span><span class="s1">'analytical'</span><span class="p">,</span><span class="s1">'nn'</span><span class="p">])</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s1">'t'</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s1">'g(t)'</span><span class="p">)</span>
|
||
|
||
<span class="c1">## Find an approximation to the funtion using forward Euler</span>
|
||
|
||
<span class="n">alpha</span><span class="p">,</span> <span class="n">A</span><span class="p">,</span> <span class="n">g0</span> <span class="o">=</span> <span class="n">get_parameters</span><span class="p">()</span>
|
||
<span class="n">dt</span> <span class="o">=</span> <span class="n">T</span><span class="o">/</span><span class="p">(</span><span class="n">Nt</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>
|
||
|
||
<span class="c1"># Perform forward Euler to solve the ODE</span>
|
||
<span class="n">g_euler</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">Nt</span><span class="p">)</span>
|
||
<span class="n">g_euler</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">g0</span>
|
||
|
||
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="n">Nt</span><span class="p">):</span>
|
||
<span class="n">g_euler</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">g_euler</span><span class="p">[</span><span class="n">i</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">+</span> <span class="n">dt</span><span class="o">*</span><span class="p">(</span><span class="n">alpha</span><span class="o">*</span><span class="n">g_euler</span><span class="p">[</span><span class="n">i</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">*</span><span class="p">(</span><span class="n">A</span> <span class="o">-</span> <span class="n">g_euler</span><span class="p">[</span><span class="n">i</span><span class="o">-</span><span class="mi">1</span><span class="p">]))</span>
|
||
|
||
<span class="c1"># Print the errors done by each method</span>
|
||
<span class="n">diff1</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">max</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">g_euler</span> <span class="o">-</span> <span class="n">g_analytical</span><span class="p">))</span>
|
||
<span class="n">diff2</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">max</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">g_dnn_ag</span><span class="p">[</span><span class="mi">0</span><span class="p">,:]</span> <span class="o">-</span> <span class="n">g_analytical</span><span class="p">))</span>
|
||
|
||
<span class="nb">print</span><span class="p">(</span><span class="s1">'Max absolute difference between Euler method and analytical: </span><span class="si">%g</span><span class="s1">'</span><span class="o">%</span><span class="k">diff1</span>)
|
||
<span class="nb">print</span><span class="p">(</span><span class="s1">'Max absolute difference between deep neural network and analytical: </span><span class="si">%g</span><span class="s1">'</span><span class="o">%</span><span class="k">diff2</span>)
|
||
|
||
<span class="c1"># Plot results</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span><span class="mi">10</span><span class="p">))</span>
|
||
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">t</span><span class="p">,</span><span class="n">g_euler</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">t</span><span class="p">,</span><span class="n">g_analytical</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">t</span><span class="p">,</span><span class="n">g_dnn_ag</span><span class="p">[</span><span class="mi">0</span><span class="p">,:])</span>
|
||
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">([</span><span class="s1">'euler'</span><span class="p">,</span><span class="s1">'analytical'</span><span class="p">,</span><span class="s1">'dnn'</span><span class="p">])</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s1">'Time t'</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s1">'g(t)'</span><span class="p">)</span>
|
||
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<div class="section" id="example-solving-the-one-dimensional-poisson-equation">
|
||
<h2>Example: Solving the one dimensional Poisson equation<a class="headerlink" href="#example-solving-the-one-dimensional-poisson-equation" title="Permalink to this headline">¶</a></h2>
|
||
<p>The Poisson equation for <span class="math notranslate nohighlight">\(g(x)\)</span> in one dimension is</p>
|
||
<!-- Equation labels as ordinary links -->
|
||
<div id="poisson"></div>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\begin{equation} \label{poisson} \tag{13}
|
||
-g''(x) = f(x)
|
||
\end{equation}
|
||
\]</div>
|
||
<p>where <span class="math notranslate nohighlight">\(f(x)\)</span> is a given function for <span class="math notranslate nohighlight">\(x \in (0,1)\)</span>.</p>
|
||
<p>The conditions that <span class="math notranslate nohighlight">\(g(x)\)</span> is chosen to fulfill, are</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[\begin{split}
|
||
\begin{align*}
|
||
g(0) &= 0 \\
|
||
g(1) &= 0
|
||
\end{align*}
|
||
\end{split}\]</div>
|
||
<p>This equation can be solved numerically using programs where e.g Autograd and TensorFlow are used.
|
||
The results from the networks can then be compared to the analytical solution.
|
||
In addition, it could be interesting to see how a typical method for numerically solving second order ODEs compares to the neural networks.</p>
|
||
</div>
|
||
<div class="section" id="the-specific-equation-to-solve-for">
|
||
<h2>The specific equation to solve for<a class="headerlink" href="#the-specific-equation-to-solve-for" title="Permalink to this headline">¶</a></h2>
|
||
<p>Here, the function <span class="math notranslate nohighlight">\(g(x)\)</span> to solve for follows the equation</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
-g''(x) = f(x),\qquad x \in (0,1)
|
||
\]</div>
|
||
<p>where <span class="math notranslate nohighlight">\(f(x)\)</span> is a given function, along with the chosen conditions</p>
|
||
<!-- Equation labels as ordinary links -->
|
||
<div id="cond"></div>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\begin{aligned}
|
||
g(0) = g(1) = 0
|
||
\end{aligned}\label{cond} \tag{14}
|
||
\]</div>
|
||
<p>In this example, we consider the case when <span class="math notranslate nohighlight">\(f(x) = (3x + x^2)\exp(x)\)</span>.</p>
|
||
<p>For this case, a possible trial solution satisfying the conditions could be</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
g_t(x) = x \cdot (1-x) \cdot N(P,x)
|
||
\]</div>
|
||
<p>The analytical solution for this problem is</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
g(x) = x(1 - x)\exp(x)
|
||
\]</div>
|
||
</div>
|
||
<div class="section" id="solving-the-equation-using-autograd">
|
||
<h2>Solving the equation using Autograd<a class="headerlink" href="#solving-the-equation-using-autograd" title="Permalink to this headline">¶</a></h2>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">autograd.numpy</span> <span class="k">as</span> <span class="nn">np</span>
|
||
<span class="kn">from</span> <span class="nn">autograd</span> <span class="kn">import</span> <span class="n">grad</span><span class="p">,</span> <span class="n">elementwise_grad</span>
|
||
<span class="kn">import</span> <span class="nn">autograd.numpy.random</span> <span class="k">as</span> <span class="nn">npr</span>
|
||
<span class="kn">from</span> <span class="nn">matplotlib</span> <span class="kn">import</span> <span class="n">pyplot</span> <span class="k">as</span> <span class="n">plt</span>
|
||
|
||
<span class="k">def</span> <span class="nf">sigmoid</span><span class="p">(</span><span class="n">z</span><span class="p">):</span>
|
||
<span class="k">return</span> <span class="mi">1</span><span class="o">/</span><span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">z</span><span class="p">))</span>
|
||
|
||
<span class="k">def</span> <span class="nf">deep_neural_network</span><span class="p">(</span><span class="n">deep_params</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
|
||
<span class="c1"># N_hidden is the number of hidden layers</span>
|
||
<span class="c1"># deep_params is a list, len() should be used</span>
|
||
<span class="n">N_hidden</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">deep_params</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span> <span class="c1"># -1 since params consists of</span>
|
||
<span class="c1"># parameters to all the hidden</span>
|
||
<span class="c1"># layers AND the output layer.</span>
|
||
|
||
<span class="c1"># Assumes input x being an one-dimensional array</span>
|
||
<span class="n">num_values</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">size</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
|
||
<span class="n">x</span> <span class="o">=</span> <span class="n">x</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="n">num_values</span><span class="p">)</span>
|
||
|
||
<span class="c1"># Assume that the input layer does nothing to the input x</span>
|
||
<span class="n">x_input</span> <span class="o">=</span> <span class="n">x</span>
|
||
|
||
<span class="c1"># Due to multiple hidden layers, define a variable referencing to the</span>
|
||
<span class="c1"># output of the previous layer:</span>
|
||
<span class="n">x_prev</span> <span class="o">=</span> <span class="n">x_input</span>
|
||
|
||
<span class="c1">## Hidden layers:</span>
|
||
|
||
<span class="k">for</span> <span class="n">l</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">N_hidden</span><span class="p">):</span>
|
||
<span class="c1"># From the list of parameters P; find the correct weigths and bias for this layer</span>
|
||
<span class="n">w_hidden</span> <span class="o">=</span> <span class="n">deep_params</span><span class="p">[</span><span class="n">l</span><span class="p">]</span>
|
||
|
||
<span class="c1"># Add a row of ones to include bias</span>
|
||
<span class="n">x_prev</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">concatenate</span><span class="p">((</span><span class="n">np</span><span class="o">.</span><span class="n">ones</span><span class="p">((</span><span class="mi">1</span><span class="p">,</span><span class="n">num_values</span><span class="p">)),</span> <span class="n">x_prev</span> <span class="p">),</span> <span class="n">axis</span> <span class="o">=</span> <span class="mi">0</span><span class="p">)</span>
|
||
|
||
<span class="n">z_hidden</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">matmul</span><span class="p">(</span><span class="n">w_hidden</span><span class="p">,</span> <span class="n">x_prev</span><span class="p">)</span>
|
||
<span class="n">x_hidden</span> <span class="o">=</span> <span class="n">sigmoid</span><span class="p">(</span><span class="n">z_hidden</span><span class="p">)</span>
|
||
|
||
<span class="c1"># Update x_prev such that next layer can use the output from this layer</span>
|
||
<span class="n">x_prev</span> <span class="o">=</span> <span class="n">x_hidden</span>
|
||
|
||
<span class="c1">## Output layer:</span>
|
||
|
||
<span class="c1"># Get the weights and bias for this layer</span>
|
||
<span class="n">w_output</span> <span class="o">=</span> <span class="n">deep_params</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
|
||
|
||
<span class="c1"># Include bias:</span>
|
||
<span class="n">x_prev</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">concatenate</span><span class="p">((</span><span class="n">np</span><span class="o">.</span><span class="n">ones</span><span class="p">((</span><span class="mi">1</span><span class="p">,</span><span class="n">num_values</span><span class="p">)),</span> <span class="n">x_prev</span><span class="p">),</span> <span class="n">axis</span> <span class="o">=</span> <span class="mi">0</span><span class="p">)</span>
|
||
|
||
<span class="n">z_output</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">matmul</span><span class="p">(</span><span class="n">w_output</span><span class="p">,</span> <span class="n">x_prev</span><span class="p">)</span>
|
||
<span class="n">x_output</span> <span class="o">=</span> <span class="n">z_output</span>
|
||
|
||
<span class="k">return</span> <span class="n">x_output</span>
|
||
|
||
|
||
<span class="k">def</span> <span class="nf">solve_ode_deep_neural_network</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">num_neurons</span><span class="p">,</span> <span class="n">num_iter</span><span class="p">,</span> <span class="n">lmb</span><span class="p">):</span>
|
||
<span class="c1"># num_hidden_neurons is now a list of number of neurons within each hidden layer</span>
|
||
|
||
<span class="c1"># Find the number of hidden layers:</span>
|
||
<span class="n">N_hidden</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">size</span><span class="p">(</span><span class="n">num_neurons</span><span class="p">)</span>
|
||
|
||
<span class="c1">## Set up initial weigths and biases</span>
|
||
|
||
<span class="c1"># Initialize the list of parameters:</span>
|
||
<span class="n">P</span> <span class="o">=</span> <span class="p">[</span><span class="kc">None</span><span class="p">]</span><span class="o">*</span><span class="p">(</span><span class="n">N_hidden</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="c1"># + 1 to include the output layer</span>
|
||
|
||
<span class="n">P</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">npr</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="n">num_neurons</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="mi">2</span> <span class="p">)</span>
|
||
<span class="k">for</span> <span class="n">l</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="n">N_hidden</span><span class="p">):</span>
|
||
<span class="n">P</span><span class="p">[</span><span class="n">l</span><span class="p">]</span> <span class="o">=</span> <span class="n">npr</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="n">num_neurons</span><span class="p">[</span><span class="n">l</span><span class="p">],</span> <span class="n">num_neurons</span><span class="p">[</span><span class="n">l</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="c1"># +1 to include bias</span>
|
||
|
||
<span class="c1"># For the output layer</span>
|
||
<span class="n">P</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">npr</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">num_neurons</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">+</span> <span class="mi">1</span> <span class="p">)</span> <span class="c1"># +1 since bias is included</span>
|
||
|
||
<span class="nb">print</span><span class="p">(</span><span class="s1">'Initial cost: </span><span class="si">%g</span><span class="s1">'</span><span class="o">%</span><span class="k">cost_function_deep</span>(P, x))
|
||
|
||
<span class="c1">## Start finding the optimal weigths using gradient descent</span>
|
||
|
||
<span class="c1"># Find the Python function that represents the gradient of the cost function</span>
|
||
<span class="c1"># w.r.t the 0-th input argument -- that is the weights and biases in the hidden and output layer</span>
|
||
<span class="n">cost_function_deep_grad</span> <span class="o">=</span> <span class="n">grad</span><span class="p">(</span><span class="n">cost_function_deep</span><span class="p">,</span><span class="mi">0</span><span class="p">)</span>
|
||
|
||
<span class="c1"># Let the update be done num_iter times</span>
|
||
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">num_iter</span><span class="p">):</span>
|
||
<span class="c1"># Evaluate the gradient at the current weights and biases in P.</span>
|
||
<span class="c1"># The cost_grad consist now of N_hidden + 1 arrays; the gradient w.r.t the weights and biases</span>
|
||
<span class="c1"># in the hidden layers and output layers evaluated at x.</span>
|
||
<span class="n">cost_deep_grad</span> <span class="o">=</span> <span class="n">cost_function_deep_grad</span><span class="p">(</span><span class="n">P</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
|
||
|
||
<span class="k">for</span> <span class="n">l</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">N_hidden</span><span class="o">+</span><span class="mi">1</span><span class="p">):</span>
|
||
<span class="n">P</span><span class="p">[</span><span class="n">l</span><span class="p">]</span> <span class="o">=</span> <span class="n">P</span><span class="p">[</span><span class="n">l</span><span class="p">]</span> <span class="o">-</span> <span class="n">lmb</span> <span class="o">*</span> <span class="n">cost_deep_grad</span><span class="p">[</span><span class="n">l</span><span class="p">]</span>
|
||
|
||
<span class="nb">print</span><span class="p">(</span><span class="s1">'Final cost: </span><span class="si">%g</span><span class="s1">'</span><span class="o">%</span><span class="k">cost_function_deep</span>(P, x))
|
||
|
||
<span class="k">return</span> <span class="n">P</span>
|
||
|
||
<span class="c1">## Set up the cost function specified for this Poisson equation:</span>
|
||
|
||
<span class="c1"># The right side of the ODE</span>
|
||
<span class="k">def</span> <span class="nf">f</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
|
||
<span class="k">return</span> <span class="p">(</span><span class="mi">3</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
|
||
|
||
<span class="k">def</span> <span class="nf">cost_function_deep</span><span class="p">(</span><span class="n">P</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
|
||
|
||
<span class="c1"># Evaluate the trial function with the current parameters P</span>
|
||
<span class="n">g_t</span> <span class="o">=</span> <span class="n">g_trial_deep</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">P</span><span class="p">)</span>
|
||
|
||
<span class="c1"># Find the derivative w.r.t x of the trial function</span>
|
||
<span class="n">d2_g_t</span> <span class="o">=</span> <span class="n">elementwise_grad</span><span class="p">(</span><span class="n">elementwise_grad</span><span class="p">(</span><span class="n">g_trial_deep</span><span class="p">,</span><span class="mi">0</span><span class="p">))(</span><span class="n">x</span><span class="p">,</span><span class="n">P</span><span class="p">)</span>
|
||
|
||
<span class="n">right_side</span> <span class="o">=</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
|
||
|
||
<span class="n">err_sqr</span> <span class="o">=</span> <span class="p">(</span><span class="o">-</span><span class="n">d2_g_t</span> <span class="o">-</span> <span class="n">right_side</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span>
|
||
<span class="n">cost_sum</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">err_sqr</span><span class="p">)</span>
|
||
|
||
<span class="k">return</span> <span class="n">cost_sum</span><span class="o">/</span><span class="n">np</span><span class="o">.</span><span class="n">size</span><span class="p">(</span><span class="n">err_sqr</span><span class="p">)</span>
|
||
|
||
<span class="c1"># The trial solution:</span>
|
||
<span class="k">def</span> <span class="nf">g_trial_deep</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">P</span><span class="p">):</span>
|
||
<span class="k">return</span> <span class="n">x</span><span class="o">*</span><span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">deep_neural_network</span><span class="p">(</span><span class="n">P</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
|
||
|
||
<span class="c1"># The analytic solution;</span>
|
||
<span class="k">def</span> <span class="nf">g_analytic</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
|
||
<span class="k">return</span> <span class="n">x</span><span class="o">*</span><span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
|
||
|
||
<span class="k">if</span> <span class="vm">__name__</span> <span class="o">==</span> <span class="s1">'__main__'</span><span class="p">:</span>
|
||
<span class="n">npr</span><span class="o">.</span><span class="n">seed</span><span class="p">(</span><span class="mi">4155</span><span class="p">)</span>
|
||
|
||
<span class="c1">## Decide the vales of arguments to the function to solve</span>
|
||
<span class="n">Nx</span> <span class="o">=</span> <span class="mi">10</span>
|
||
<span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span> <span class="n">Nx</span><span class="p">)</span>
|
||
|
||
<span class="c1">## Set up the initial parameters</span>
|
||
<span class="n">num_hidden_neurons</span> <span class="o">=</span> <span class="p">[</span><span class="mi">200</span><span class="p">,</span><span class="mi">100</span><span class="p">]</span>
|
||
<span class="n">num_iter</span> <span class="o">=</span> <span class="mi">1000</span>
|
||
<span class="n">lmb</span> <span class="o">=</span> <span class="mf">1e-3</span>
|
||
|
||
<span class="n">P</span> <span class="o">=</span> <span class="n">solve_ode_deep_neural_network</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">num_hidden_neurons</span><span class="p">,</span> <span class="n">num_iter</span><span class="p">,</span> <span class="n">lmb</span><span class="p">)</span>
|
||
|
||
<span class="n">g_dnn_ag</span> <span class="o">=</span> <span class="n">g_trial_deep</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">P</span><span class="p">)</span>
|
||
<span class="n">g_analytical</span> <span class="o">=</span> <span class="n">g_analytic</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
|
||
|
||
<span class="c1"># Find the maximum absolute difference between the solutons:</span>
|
||
<span class="n">max_diff</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">max</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">g_dnn_ag</span> <span class="o">-</span> <span class="n">g_analytical</span><span class="p">))</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="s2">"The max absolute difference between the solutions is: </span><span class="si">%g</span><span class="s2">"</span><span class="o">%</span><span class="k">max_diff</span>)
|
||
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span><span class="mi">10</span><span class="p">))</span>
|
||
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s1">'Performance of neural network solving an ODE compared to the analytical solution'</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">g_analytical</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">g_dnn_ag</span><span class="p">[</span><span class="mi">0</span><span class="p">,:])</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">([</span><span class="s1">'analytical'</span><span class="p">,</span><span class="s1">'nn'</span><span class="p">])</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s1">'x'</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s1">'g(x)'</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<div class="section" id="comparing-with-a-numerical-scheme">
|
||
<h2>Comparing with a numerical scheme<a class="headerlink" href="#comparing-with-a-numerical-scheme" title="Permalink to this headline">¶</a></h2>
|
||
<p>The Poisson equation is possible to solve using Taylor series to approximate the second derivative.</p>
|
||
<p>Using Taylor series, the second derivative can be expressed as</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
g''(x) = \frac{g(x + \Delta x) - 2g(x) + g(x-\Delta x)}{\Delta x^2} + E_{\Delta x}(x)
|
||
\]</div>
|
||
<p>where <span class="math notranslate nohighlight">\(\Delta x\)</span> is a small step size and <span class="math notranslate nohighlight">\(E_{\Delta x}(x)\)</span> being the error term.</p>
|
||
<p>Looking away from the error terms gives an approximation to the second derivative:</p>
|
||
<!-- Equation labels as ordinary links -->
|
||
<div id="approx"></div>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\begin{equation} \label{approx} \tag{15}
|
||
g''(x) \approx \frac{g(x + \Delta x) - 2g(x) + g(x-\Delta x)}{\Delta x^2}
|
||
\end{equation}
|
||
\]</div>
|
||
<p>If <span class="math notranslate nohighlight">\(x_i = i \Delta x = x_{i-1} + \Delta x\)</span> and <span class="math notranslate nohighlight">\(g_i = g(x_i)\)</span> for <span class="math notranslate nohighlight">\(i = 1,\dots N_x - 2\)</span> with <span class="math notranslate nohighlight">\(N_x\)</span> being the number of values for <span class="math notranslate nohighlight">\(x\)</span>, (<a class="reference external" href="#approx">15</a>) becomes</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[\begin{split}
|
||
\begin{aligned}
|
||
g''(x_i) &\approx \frac{g(x_i + \Delta x) - 2g(x_i) + g(x_i -\Delta x)}{\Delta x^2} \\
|
||
&= \frac{g_{i+1} - 2g_i + g_{i-1}}{\Delta x^2}
|
||
\end{aligned}
|
||
\end{split}\]</div>
|
||
<p>Since we know from our problem that</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[\begin{split}
|
||
\begin{aligned}
|
||
-g''(x) &= f(x) \\
|
||
&= (3x + x^2)\exp(x)
|
||
\end{aligned}
|
||
\end{split}\]</div>
|
||
<p>along with the conditions <span class="math notranslate nohighlight">\(g(0) = g(1) = 0\)</span>,
|
||
the following scheme can be used to find an approximate solution for <span class="math notranslate nohighlight">\(g(x)\)</span> numerically:</p>
|
||
<!-- Equation labels as ordinary links -->
|
||
<div id="odesys"></div>
|
||
<div class="math notranslate nohighlight">
|
||
\[\begin{split}
|
||
\begin{equation}
|
||
\begin{aligned}
|
||
-\Big( \frac{g_{i+1} - 2g_i + g_{i-1}}{\Delta x^2} \Big) &= f(x_i) \\
|
||
-g_{i+1} + 2g_i - g_{i-1} &= \Delta x^2 f(x_i)
|
||
\end{aligned}
|
||
\end{equation} \label{odesys} \tag{16}
|
||
\end{split}\]</div>
|
||
<p>for <span class="math notranslate nohighlight">\(i = 1, \dots, N_x - 2\)</span> where <span class="math notranslate nohighlight">\(g_0 = g_{N_x - 1} = 0\)</span> and <span class="math notranslate nohighlight">\(f(x_i) = (3x_i + x_i^2)\exp(x_i)\)</span>, which is given for our specific problem.</p>
|
||
<p>The equation can be rewritten into a matrix equation:</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[\begin{split}
|
||
\begin{aligned}
|
||
\begin{pmatrix}
|
||
2 & -1 & 0 & \dots & 0 \\
|
||
-1 & 2 & -1 & \dots & 0 \\
|
||
\vdots & & \ddots & & \vdots \\
|
||
0 & \dots & -1 & 2 & -1 \\
|
||
0 & \dots & 0 & -1 & 2\\
|
||
\end{pmatrix}
|
||
\begin{pmatrix}
|
||
g_1 \\
|
||
g_2 \\
|
||
\vdots \\
|
||
g_{N_x - 3} \\
|
||
g_{N_x - 2}
|
||
\end{pmatrix}
|
||
&=
|
||
\Delta x^2
|
||
\begin{pmatrix}
|
||
f(x_1) \\
|
||
f(x_2) \\
|
||
\vdots \\
|
||
f(x_{N_x - 3}) \\
|
||
f(x_{N_x - 2})
|
||
\end{pmatrix} \\
|
||
\boldsymbol{A}\boldsymbol{g} &= \boldsymbol{f},
|
||
\end{aligned}
|
||
\end{split}\]</div>
|
||
<p>which makes it possible to solve for the vector <span class="math notranslate nohighlight">\(\boldsymbol{g}\)</span>.</p>
|
||
</div>
|
||
<div class="section" id="setting-up-the-code">
|
||
<h2>Setting up the code<a class="headerlink" href="#setting-up-the-code" title="Permalink to this headline">¶</a></h2>
|
||
<p>We can then compare the result from this numerical scheme with the output from our network using Autograd:</p>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">autograd.numpy</span> <span class="k">as</span> <span class="nn">np</span>
|
||
<span class="kn">from</span> <span class="nn">autograd</span> <span class="kn">import</span> <span class="n">grad</span><span class="p">,</span> <span class="n">elementwise_grad</span>
|
||
<span class="kn">import</span> <span class="nn">autograd.numpy.random</span> <span class="k">as</span> <span class="nn">npr</span>
|
||
<span class="kn">from</span> <span class="nn">matplotlib</span> <span class="kn">import</span> <span class="n">pyplot</span> <span class="k">as</span> <span class="n">plt</span>
|
||
|
||
<span class="k">def</span> <span class="nf">sigmoid</span><span class="p">(</span><span class="n">z</span><span class="p">):</span>
|
||
<span class="k">return</span> <span class="mi">1</span><span class="o">/</span><span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">z</span><span class="p">))</span>
|
||
|
||
<span class="k">def</span> <span class="nf">deep_neural_network</span><span class="p">(</span><span class="n">deep_params</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
|
||
<span class="c1"># N_hidden is the number of hidden layers</span>
|
||
<span class="c1"># deep_params is a list, len() should be used</span>
|
||
<span class="n">N_hidden</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">deep_params</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span> <span class="c1"># -1 since params consists of</span>
|
||
<span class="c1"># parameters to all the hidden</span>
|
||
<span class="c1"># layers AND the output layer.</span>
|
||
|
||
<span class="c1"># Assumes input x being an one-dimensional array</span>
|
||
<span class="n">num_values</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">size</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
|
||
<span class="n">x</span> <span class="o">=</span> <span class="n">x</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="n">num_values</span><span class="p">)</span>
|
||
|
||
<span class="c1"># Assume that the input layer does nothing to the input x</span>
|
||
<span class="n">x_input</span> <span class="o">=</span> <span class="n">x</span>
|
||
|
||
<span class="c1"># Due to multiple hidden layers, define a variable referencing to the</span>
|
||
<span class="c1"># output of the previous layer:</span>
|
||
<span class="n">x_prev</span> <span class="o">=</span> <span class="n">x_input</span>
|
||
|
||
<span class="c1">## Hidden layers:</span>
|
||
|
||
<span class="k">for</span> <span class="n">l</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">N_hidden</span><span class="p">):</span>
|
||
<span class="c1"># From the list of parameters P; find the correct weigths and bias for this layer</span>
|
||
<span class="n">w_hidden</span> <span class="o">=</span> <span class="n">deep_params</span><span class="p">[</span><span class="n">l</span><span class="p">]</span>
|
||
|
||
<span class="c1"># Add a row of ones to include bias</span>
|
||
<span class="n">x_prev</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">concatenate</span><span class="p">((</span><span class="n">np</span><span class="o">.</span><span class="n">ones</span><span class="p">((</span><span class="mi">1</span><span class="p">,</span><span class="n">num_values</span><span class="p">)),</span> <span class="n">x_prev</span> <span class="p">),</span> <span class="n">axis</span> <span class="o">=</span> <span class="mi">0</span><span class="p">)</span>
|
||
|
||
<span class="n">z_hidden</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">matmul</span><span class="p">(</span><span class="n">w_hidden</span><span class="p">,</span> <span class="n">x_prev</span><span class="p">)</span>
|
||
<span class="n">x_hidden</span> <span class="o">=</span> <span class="n">sigmoid</span><span class="p">(</span><span class="n">z_hidden</span><span class="p">)</span>
|
||
|
||
<span class="c1"># Update x_prev such that next layer can use the output from this layer</span>
|
||
<span class="n">x_prev</span> <span class="o">=</span> <span class="n">x_hidden</span>
|
||
|
||
<span class="c1">## Output layer:</span>
|
||
|
||
<span class="c1"># Get the weights and bias for this layer</span>
|
||
<span class="n">w_output</span> <span class="o">=</span> <span class="n">deep_params</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
|
||
|
||
<span class="c1"># Include bias:</span>
|
||
<span class="n">x_prev</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">concatenate</span><span class="p">((</span><span class="n">np</span><span class="o">.</span><span class="n">ones</span><span class="p">((</span><span class="mi">1</span><span class="p">,</span><span class="n">num_values</span><span class="p">)),</span> <span class="n">x_prev</span><span class="p">),</span> <span class="n">axis</span> <span class="o">=</span> <span class="mi">0</span><span class="p">)</span>
|
||
|
||
<span class="n">z_output</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">matmul</span><span class="p">(</span><span class="n">w_output</span><span class="p">,</span> <span class="n">x_prev</span><span class="p">)</span>
|
||
<span class="n">x_output</span> <span class="o">=</span> <span class="n">z_output</span>
|
||
|
||
<span class="k">return</span> <span class="n">x_output</span>
|
||
|
||
|
||
<span class="k">def</span> <span class="nf">solve_ode_deep_neural_network</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">num_neurons</span><span class="p">,</span> <span class="n">num_iter</span><span class="p">,</span> <span class="n">lmb</span><span class="p">):</span>
|
||
<span class="c1"># num_hidden_neurons is now a list of number of neurons within each hidden layer</span>
|
||
|
||
<span class="c1"># Find the number of hidden layers:</span>
|
||
<span class="n">N_hidden</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">size</span><span class="p">(</span><span class="n">num_neurons</span><span class="p">)</span>
|
||
|
||
<span class="c1">## Set up initial weigths and biases</span>
|
||
|
||
<span class="c1"># Initialize the list of parameters:</span>
|
||
<span class="n">P</span> <span class="o">=</span> <span class="p">[</span><span class="kc">None</span><span class="p">]</span><span class="o">*</span><span class="p">(</span><span class="n">N_hidden</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="c1"># + 1 to include the output layer</span>
|
||
|
||
<span class="n">P</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">npr</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="n">num_neurons</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="mi">2</span> <span class="p">)</span>
|
||
<span class="k">for</span> <span class="n">l</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="n">N_hidden</span><span class="p">):</span>
|
||
<span class="n">P</span><span class="p">[</span><span class="n">l</span><span class="p">]</span> <span class="o">=</span> <span class="n">npr</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="n">num_neurons</span><span class="p">[</span><span class="n">l</span><span class="p">],</span> <span class="n">num_neurons</span><span class="p">[</span><span class="n">l</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="c1"># +1 to include bias</span>
|
||
|
||
<span class="c1"># For the output layer</span>
|
||
<span class="n">P</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">npr</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">num_neurons</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">+</span> <span class="mi">1</span> <span class="p">)</span> <span class="c1"># +1 since bias is included</span>
|
||
|
||
<span class="nb">print</span><span class="p">(</span><span class="s1">'Initial cost: </span><span class="si">%g</span><span class="s1">'</span><span class="o">%</span><span class="k">cost_function_deep</span>(P, x))
|
||
|
||
<span class="c1">## Start finding the optimal weigths using gradient descent</span>
|
||
|
||
<span class="c1"># Find the Python function that represents the gradient of the cost function</span>
|
||
<span class="c1"># w.r.t the 0-th input argument -- that is the weights and biases in the hidden and output layer</span>
|
||
<span class="n">cost_function_deep_grad</span> <span class="o">=</span> <span class="n">grad</span><span class="p">(</span><span class="n">cost_function_deep</span><span class="p">,</span><span class="mi">0</span><span class="p">)</span>
|
||
|
||
<span class="c1"># Let the update be done num_iter times</span>
|
||
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">num_iter</span><span class="p">):</span>
|
||
<span class="c1"># Evaluate the gradient at the current weights and biases in P.</span>
|
||
<span class="c1"># The cost_grad consist now of N_hidden + 1 arrays; the gradient w.r.t the weights and biases</span>
|
||
<span class="c1"># in the hidden layers and output layers evaluated at x.</span>
|
||
<span class="n">cost_deep_grad</span> <span class="o">=</span> <span class="n">cost_function_deep_grad</span><span class="p">(</span><span class="n">P</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
|
||
|
||
<span class="k">for</span> <span class="n">l</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">N_hidden</span><span class="o">+</span><span class="mi">1</span><span class="p">):</span>
|
||
<span class="n">P</span><span class="p">[</span><span class="n">l</span><span class="p">]</span> <span class="o">=</span> <span class="n">P</span><span class="p">[</span><span class="n">l</span><span class="p">]</span> <span class="o">-</span> <span class="n">lmb</span> <span class="o">*</span> <span class="n">cost_deep_grad</span><span class="p">[</span><span class="n">l</span><span class="p">]</span>
|
||
|
||
<span class="nb">print</span><span class="p">(</span><span class="s1">'Final cost: </span><span class="si">%g</span><span class="s1">'</span><span class="o">%</span><span class="k">cost_function_deep</span>(P, x))
|
||
|
||
<span class="k">return</span> <span class="n">P</span>
|
||
|
||
<span class="c1">## Set up the cost function specified for this Poisson equation:</span>
|
||
|
||
<span class="c1"># The right side of the ODE</span>
|
||
<span class="k">def</span> <span class="nf">f</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
|
||
<span class="k">return</span> <span class="p">(</span><span class="mi">3</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
|
||
|
||
<span class="k">def</span> <span class="nf">cost_function_deep</span><span class="p">(</span><span class="n">P</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
|
||
|
||
<span class="c1"># Evaluate the trial function with the current parameters P</span>
|
||
<span class="n">g_t</span> <span class="o">=</span> <span class="n">g_trial_deep</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">P</span><span class="p">)</span>
|
||
|
||
<span class="c1"># Find the derivative w.r.t x of the trial function</span>
|
||
<span class="n">d2_g_t</span> <span class="o">=</span> <span class="n">elementwise_grad</span><span class="p">(</span><span class="n">elementwise_grad</span><span class="p">(</span><span class="n">g_trial_deep</span><span class="p">,</span><span class="mi">0</span><span class="p">))(</span><span class="n">x</span><span class="p">,</span><span class="n">P</span><span class="p">)</span>
|
||
|
||
<span class="n">right_side</span> <span class="o">=</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
|
||
|
||
<span class="n">err_sqr</span> <span class="o">=</span> <span class="p">(</span><span class="o">-</span><span class="n">d2_g_t</span> <span class="o">-</span> <span class="n">right_side</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span>
|
||
<span class="n">cost_sum</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">err_sqr</span><span class="p">)</span>
|
||
|
||
<span class="k">return</span> <span class="n">cost_sum</span><span class="o">/</span><span class="n">np</span><span class="o">.</span><span class="n">size</span><span class="p">(</span><span class="n">err_sqr</span><span class="p">)</span>
|
||
|
||
<span class="c1"># The trial solution:</span>
|
||
<span class="k">def</span> <span class="nf">g_trial_deep</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">P</span><span class="p">):</span>
|
||
<span class="k">return</span> <span class="n">x</span><span class="o">*</span><span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">deep_neural_network</span><span class="p">(</span><span class="n">P</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
|
||
|
||
<span class="c1"># The analytic solution;</span>
|
||
<span class="k">def</span> <span class="nf">g_analytic</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
|
||
<span class="k">return</span> <span class="n">x</span><span class="o">*</span><span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
|
||
|
||
<span class="k">if</span> <span class="vm">__name__</span> <span class="o">==</span> <span class="s1">'__main__'</span><span class="p">:</span>
|
||
<span class="n">npr</span><span class="o">.</span><span class="n">seed</span><span class="p">(</span><span class="mi">4155</span><span class="p">)</span>
|
||
|
||
<span class="c1">## Decide the vales of arguments to the function to solve</span>
|
||
<span class="n">Nx</span> <span class="o">=</span> <span class="mi">10</span>
|
||
<span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span> <span class="n">Nx</span><span class="p">)</span>
|
||
|
||
<span class="c1">## Set up the initial parameters</span>
|
||
<span class="n">num_hidden_neurons</span> <span class="o">=</span> <span class="p">[</span><span class="mi">200</span><span class="p">,</span><span class="mi">100</span><span class="p">]</span>
|
||
<span class="n">num_iter</span> <span class="o">=</span> <span class="mi">1000</span>
|
||
<span class="n">lmb</span> <span class="o">=</span> <span class="mf">1e-3</span>
|
||
|
||
<span class="n">P</span> <span class="o">=</span> <span class="n">solve_ode_deep_neural_network</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">num_hidden_neurons</span><span class="p">,</span> <span class="n">num_iter</span><span class="p">,</span> <span class="n">lmb</span><span class="p">)</span>
|
||
|
||
<span class="n">g_dnn_ag</span> <span class="o">=</span> <span class="n">g_trial_deep</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">P</span><span class="p">)</span>
|
||
<span class="n">g_analytical</span> <span class="o">=</span> <span class="n">g_analytic</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
|
||
|
||
<span class="c1"># Find the maximum absolute difference between the solutons:</span>
|
||
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span><span class="mi">10</span><span class="p">))</span>
|
||
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s1">'Performance of neural network solving an ODE compared to the analytical solution'</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">g_analytical</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">g_dnn_ag</span><span class="p">[</span><span class="mi">0</span><span class="p">,:])</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">([</span><span class="s1">'analytical'</span><span class="p">,</span><span class="s1">'nn'</span><span class="p">])</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s1">'x'</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s1">'g(x)'</span><span class="p">)</span>
|
||
|
||
<span class="c1">## Perform the computation using the numerical scheme</span>
|
||
|
||
<span class="n">dx</span> <span class="o">=</span> <span class="mi">1</span><span class="o">/</span><span class="p">(</span><span class="n">Nx</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>
|
||
|
||
<span class="c1"># Set up the matrix A</span>
|
||
<span class="n">A</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">Nx</span><span class="o">-</span><span class="mi">2</span><span class="p">,</span><span class="n">Nx</span><span class="o">-</span><span class="mi">2</span><span class="p">))</span>
|
||
|
||
<span class="n">A</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="mi">2</span>
|
||
<span class="n">A</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="o">-</span><span class="mi">1</span>
|
||
|
||
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="n">Nx</span><span class="o">-</span><span class="mi">3</span><span class="p">):</span>
|
||
<span class="n">A</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="n">i</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="o">-</span><span class="mi">1</span>
|
||
<span class="n">A</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="mi">2</span>
|
||
<span class="n">A</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="n">i</span><span class="o">+</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="o">-</span><span class="mi">1</span>
|
||
|
||
<span class="n">A</span><span class="p">[</span><span class="n">Nx</span> <span class="o">-</span> <span class="mi">3</span><span class="p">,</span> <span class="n">Nx</span> <span class="o">-</span> <span class="mi">4</span><span class="p">]</span> <span class="o">=</span> <span class="o">-</span><span class="mi">1</span>
|
||
<span class="n">A</span><span class="p">[</span><span class="n">Nx</span> <span class="o">-</span> <span class="mi">3</span><span class="p">,</span> <span class="n">Nx</span> <span class="o">-</span> <span class="mi">3</span><span class="p">]</span> <span class="o">=</span> <span class="mi">2</span>
|
||
|
||
<span class="c1"># Set up the vector f</span>
|
||
<span class="n">f_vec</span> <span class="o">=</span> <span class="n">dx</span><span class="o">**</span><span class="mi">2</span> <span class="o">*</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="mi">1</span><span class="p">:</span><span class="o">-</span><span class="mi">1</span><span class="p">])</span>
|
||
|
||
<span class="c1"># Solve the equation</span>
|
||
<span class="n">g_res</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">solve</span><span class="p">(</span><span class="n">A</span><span class="p">,</span><span class="n">f_vec</span><span class="p">)</span>
|
||
|
||
<span class="n">g_vec</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">Nx</span><span class="p">)</span>
|
||
<span class="n">g_vec</span><span class="p">[</span><span class="mi">1</span><span class="p">:</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">g_res</span>
|
||
|
||
<span class="c1"># Print the differences between each method</span>
|
||
<span class="n">max_diff1</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">max</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">g_dnn_ag</span> <span class="o">-</span> <span class="n">g_analytical</span><span class="p">))</span>
|
||
<span class="n">max_diff2</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">max</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">g_vec</span> <span class="o">-</span> <span class="n">g_analytical</span><span class="p">))</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="s2">"The max absolute difference between the analytical solution and DNN Autograd: </span><span class="si">%g</span><span class="s2">"</span><span class="o">%</span><span class="k">max_diff1</span>)
|
||
<span class="nb">print</span><span class="p">(</span><span class="s2">"The max absolute difference between the analytical solution and numerical scheme: </span><span class="si">%g</span><span class="s2">"</span><span class="o">%</span><span class="k">max_diff2</span>)
|
||
|
||
<span class="c1"># Plot the results</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span><span class="mi">10</span><span class="p">))</span>
|
||
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">g_vec</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">g_analytical</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">g_dnn_ag</span><span class="p">[</span><span class="mi">0</span><span class="p">,:])</span>
|
||
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">([</span><span class="s1">'numerical scheme'</span><span class="p">,</span><span class="s1">'analytical'</span><span class="p">,</span><span class="s1">'dnn'</span><span class="p">])</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<div class="section" id="partial-differential-equations">
|
||
<h2>Partial Differential Equations<a class="headerlink" href="#partial-differential-equations" title="Permalink to this headline">¶</a></h2>
|
||
<p>A partial differential equation (PDE) has a solution here the function
|
||
is defined by multiple variables. The equation may involve all kinds
|
||
of combinations of which variables the function is differentiated with
|
||
respect to.</p>
|
||
<p>In general, a partial differential equation for a function <span class="math notranslate nohighlight">\(g(x_1,\dots,x_N)\)</span> with <span class="math notranslate nohighlight">\(N\)</span> variables may be expressed as</p>
|
||
<!-- Equation labels as ordinary links -->
|
||
<div id="PDE"></div>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\begin{equation} \label{PDE} \tag{17}
|
||
f\left(x_1, \, \dots \, , x_N, \frac{\partial g(x_1,\dots,x_N) }{\partial x_1}, \dots , \frac{\partial g(x_1,\dots,x_N) }{\partial x_N}, \frac{\partial g(x_1,\dots,x_N) }{\partial x_1\partial x_2}, \, \dots \, , \frac{\partial^n g(x_1,\dots,x_N) }{\partial x_N^n} \right) = 0
|
||
\end{equation}
|
||
\]</div>
|
||
<p>where <span class="math notranslate nohighlight">\(f\)</span> is an expression involving all kinds of possible mixed derivatives of <span class="math notranslate nohighlight">\(g(x_1,\dots,x_N)\)</span> up to an order <span class="math notranslate nohighlight">\(n\)</span>. In order for the solution to be unique, some additional conditions must also be given.</p>
|
||
</div>
|
||
<div class="section" id="type-of-problem">
|
||
<h2>Type of problem<a class="headerlink" href="#type-of-problem" title="Permalink to this headline">¶</a></h2>
|
||
<p>The problem our network must solve for, is similar to the ODE case.
|
||
We must have a trial solution <span class="math notranslate nohighlight">\(g_t\)</span> at hand.</p>
|
||
<p>For instance, the trial solution could be expressed as</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\begin{align*}
|
||
g_t(x_1,\dots,x_N) = h_1(x_1,\dots,x_N) + h_2(x_1,\dots,x_N,N(x_1,\dots,x_N,P))
|
||
\end{align*}
|
||
\]</div>
|
||
<p>where <span class="math notranslate nohighlight">\(h_1(x_1,\dots,x_N)\)</span> is a function that ensures <span class="math notranslate nohighlight">\(g_t(x_1,\dots,x_N)\)</span> satisfies some given conditions.
|
||
The neural network <span class="math notranslate nohighlight">\(N(x_1,\dots,x_N,P)\)</span> has weights and biases described by <span class="math notranslate nohighlight">\(P\)</span> and <span class="math notranslate nohighlight">\(h_2(x_1,\dots,x_N,N(x_1,\dots,x_N,P))\)</span> is an expression using the output from the neural network in some way.</p>
|
||
<p>The role of the function <span class="math notranslate nohighlight">\(h_2(x_1,\dots,x_N,N(x_1,\dots,x_N,P))\)</span>, is to ensure that the output of <span class="math notranslate nohighlight">\(N(x_1,\dots,x_N,P)\)</span> is zero when <span class="math notranslate nohighlight">\(g_t(x_1,\dots,x_N)\)</span> is evaluated at the values of <span class="math notranslate nohighlight">\(x_1,\dots,x_N\)</span> where the given conditions must be satisfied. The function <span class="math notranslate nohighlight">\(h_1(x_1,\dots,x_N)\)</span> should alone make <span class="math notranslate nohighlight">\(g_t(x_1,\dots,x_N)\)</span> satisfy the conditions.</p>
|
||
</div>
|
||
<div class="section" id="network-requirements">
|
||
<h2>Network requirements<a class="headerlink" href="#network-requirements" title="Permalink to this headline">¶</a></h2>
|
||
<p>The network tries then the minimize the cost function following the
|
||
same ideas as described for the ODE case, but now with more than one
|
||
variables to consider. The concept still remains the same; find a set
|
||
of parameters <span class="math notranslate nohighlight">\(P\)</span> such that the expression <span class="math notranslate nohighlight">\(f\)</span> in (<a class="reference external" href="#PDE">17</a>) is as
|
||
close to zero as possible.</p>
|
||
<p>As for the ODE case, the cost function is the mean squared error that
|
||
the network must try to minimize. The cost function for the network to
|
||
minimize is</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
C\left(x_1, \dots, x_N, P\right) = \left( f\left(x_1, \, \dots \, , x_N, \frac{\partial g(x_1,\dots,x_N) }{\partial x_1}, \dots , \frac{\partial g(x_1,\dots,x_N) }{\partial x_N}, \frac{\partial g(x_1,\dots,x_N) }{\partial x_1\partial x_2}, \, \dots \, , \frac{\partial^n g(x_1,\dots,x_N) }{\partial x_N^n} \right) \right)^2
|
||
\]</div>
|
||
</div>
|
||
<div class="section" id="id6">
|
||
<h2>More details<a class="headerlink" href="#id6" title="Permalink to this headline">¶</a></h2>
|
||
<p>If we let <span class="math notranslate nohighlight">\(\boldsymbol{x} = \big( x_1, \dots, x_N \big)\)</span> be an array containing the values for <span class="math notranslate nohighlight">\(x_1, \dots, x_N\)</span> respectively, the cost function can be reformulated into the following:</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
C\left(\boldsymbol{x}, P\right) = f\left( \left( \boldsymbol{x}, \frac{\partial g(\boldsymbol{x}) }{\partial x_1}, \dots , \frac{\partial g(\boldsymbol{x}) }{\partial x_N}, \frac{\partial g(\boldsymbol{x}) }{\partial x_1\partial x_2}, \, \dots \, , \frac{\partial^n g(\boldsymbol{x}) }{\partial x_N^n} \right) \right)^2
|
||
\]</div>
|
||
<p>If we also have <span class="math notranslate nohighlight">\(M\)</span> different sets of values for <span class="math notranslate nohighlight">\(x_1, \dots, x_N\)</span>, that is <span class="math notranslate nohighlight">\(\boldsymbol{x}_i = \big(x_1^{(i)}, \dots, x_N^{(i)}\big)\)</span> for <span class="math notranslate nohighlight">\(i = 1,\dots,M\)</span> being the rows in matrix <span class="math notranslate nohighlight">\(X\)</span>, the cost function can be generalized into</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
C\left(X, P \right) = \sum_{i=1}^M f\left( \left( \boldsymbol{x}_i, \frac{\partial g(\boldsymbol{x}_i) }{\partial x_1}, \dots , \frac{\partial g(\boldsymbol{x}_i) }{\partial x_N}, \frac{\partial g(\boldsymbol{x}_i) }{\partial x_1\partial x_2}, \, \dots \, , \frac{\partial^n g(\boldsymbol{x}_i) }{\partial x_N^n} \right) \right)^2.
|
||
\]</div>
|
||
</div>
|
||
<div class="section" id="example-the-diffusion-equation">
|
||
<h2>Example: The diffusion equation<a class="headerlink" href="#example-the-diffusion-equation" title="Permalink to this headline">¶</a></h2>
|
||
<p>In one spatial dimension, the equation reads</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\frac{\partial g(x,t)}{\partial t} = \frac{\partial^2 g(x,t)}{\partial x^2}
|
||
\]</div>
|
||
<p>where a possible choice of conditions are</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[\begin{split}
|
||
\begin{align*}
|
||
g(0,t) &= 0 ,\qquad t \geq 0 \\
|
||
g(1,t) &= 0, \qquad t \geq 0 \\
|
||
g(x,0) &= u(x),\qquad x\in [0,1]
|
||
\end{align*}
|
||
\end{split}\]</div>
|
||
<p>with <span class="math notranslate nohighlight">\(u(x)\)</span> being some given function.</p>
|
||
</div>
|
||
<div class="section" id="defining-the-problem">
|
||
<h2>Defining the problem<a class="headerlink" href="#defining-the-problem" title="Permalink to this headline">¶</a></h2>
|
||
<p>For this case, we want to find <span class="math notranslate nohighlight">\(g(x,t)\)</span> such that</p>
|
||
<!-- Equation labels as ordinary links -->
|
||
<div id="diffonedim"></div>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\begin{equation}
|
||
\frac{\partial g(x,t)}{\partial t} = \frac{\partial^2 g(x,t)}{\partial x^2}
|
||
\end{equation} \label{diffonedim} \tag{18}
|
||
\]</div>
|
||
<p>and</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[\begin{split}
|
||
\begin{align*}
|
||
g(0,t) &= 0 ,\qquad t \geq 0 \\
|
||
g(1,t) &= 0, \qquad t \geq 0 \\
|
||
g(x,0) &= u(x),\qquad x\in [0,1]
|
||
\end{align*}
|
||
\end{split}\]</div>
|
||
<p>with <span class="math notranslate nohighlight">\(u(x) = \sin(\pi x)\)</span>.</p>
|
||
<p>First, let us set up the deep neural network.
|
||
The deep neural network will follow the same structure as discussed in the examples solving the ODEs.
|
||
First, we will look into how Autograd could be used in a network tailored to solve for bivariate functions.</p>
|
||
</div>
|
||
<div class="section" id="setting-up-the-network-using-autograd">
|
||
<h2>Setting up the network using Autograd<a class="headerlink" href="#setting-up-the-network-using-autograd" title="Permalink to this headline">¶</a></h2>
|
||
<p>The only change to do here, is to extend our network such that
|
||
functions of multiple parameters are correctly handled. In this case
|
||
we have two variables in our function to solve for, that is time <span class="math notranslate nohighlight">\(t\)</span>
|
||
and position <span class="math notranslate nohighlight">\(x\)</span>. The variables will be represented by a
|
||
one-dimensional array in the program. The program will evaluate the
|
||
network at each possible pair <span class="math notranslate nohighlight">\((x,t)\)</span>, given an array for the desired
|
||
<span class="math notranslate nohighlight">\(x\)</span>-values and <span class="math notranslate nohighlight">\(t\)</span>-values to approximate the solution at.</p>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">sigmoid</span><span class="p">(</span><span class="n">z</span><span class="p">):</span>
|
||
<span class="k">return</span> <span class="mi">1</span><span class="o">/</span><span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">z</span><span class="p">))</span>
|
||
|
||
<span class="k">def</span> <span class="nf">deep_neural_network</span><span class="p">(</span><span class="n">deep_params</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
|
||
<span class="c1"># x is now a point and a 1D numpy array; make it a column vector</span>
|
||
<span class="n">num_coordinates</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">size</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="mi">0</span><span class="p">)</span>
|
||
<span class="n">x</span> <span class="o">=</span> <span class="n">x</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">num_coordinates</span><span class="p">,</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
|
||
|
||
<span class="n">num_points</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">size</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
|
||
|
||
<span class="c1"># N_hidden is the number of hidden layers</span>
|
||
<span class="n">N_hidden</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">deep_params</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span> <span class="c1"># -1 since params consist of parameters to all the hidden layers AND the output layer</span>
|
||
|
||
<span class="c1"># Assume that the input layer does nothing to the input x</span>
|
||
<span class="n">x_input</span> <span class="o">=</span> <span class="n">x</span>
|
||
<span class="n">x_prev</span> <span class="o">=</span> <span class="n">x_input</span>
|
||
|
||
<span class="c1">## Hidden layers:</span>
|
||
|
||
<span class="k">for</span> <span class="n">l</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">N_hidden</span><span class="p">):</span>
|
||
<span class="c1"># From the list of parameters P; find the correct weigths and bias for this layer</span>
|
||
<span class="n">w_hidden</span> <span class="o">=</span> <span class="n">deep_params</span><span class="p">[</span><span class="n">l</span><span class="p">]</span>
|
||
|
||
<span class="c1"># Add a row of ones to include bias</span>
|
||
<span class="n">x_prev</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">concatenate</span><span class="p">((</span><span class="n">np</span><span class="o">.</span><span class="n">ones</span><span class="p">((</span><span class="mi">1</span><span class="p">,</span><span class="n">num_points</span><span class="p">)),</span> <span class="n">x_prev</span> <span class="p">),</span> <span class="n">axis</span> <span class="o">=</span> <span class="mi">0</span><span class="p">)</span>
|
||
|
||
<span class="n">z_hidden</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">matmul</span><span class="p">(</span><span class="n">w_hidden</span><span class="p">,</span> <span class="n">x_prev</span><span class="p">)</span>
|
||
<span class="n">x_hidden</span> <span class="o">=</span> <span class="n">sigmoid</span><span class="p">(</span><span class="n">z_hidden</span><span class="p">)</span>
|
||
|
||
<span class="c1"># Update x_prev such that next layer can use the output from this layer</span>
|
||
<span class="n">x_prev</span> <span class="o">=</span> <span class="n">x_hidden</span>
|
||
|
||
<span class="c1">## Output layer:</span>
|
||
|
||
<span class="c1"># Get the weights and bias for this layer</span>
|
||
<span class="n">w_output</span> <span class="o">=</span> <span class="n">deep_params</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
|
||
|
||
<span class="c1"># Include bias:</span>
|
||
<span class="n">x_prev</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">concatenate</span><span class="p">((</span><span class="n">np</span><span class="o">.</span><span class="n">ones</span><span class="p">((</span><span class="mi">1</span><span class="p">,</span><span class="n">num_points</span><span class="p">)),</span> <span class="n">x_prev</span><span class="p">),</span> <span class="n">axis</span> <span class="o">=</span> <span class="mi">0</span><span class="p">)</span>
|
||
|
||
<span class="n">z_output</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">matmul</span><span class="p">(</span><span class="n">w_output</span><span class="p">,</span> <span class="n">x_prev</span><span class="p">)</span>
|
||
<span class="n">x_output</span> <span class="o">=</span> <span class="n">z_output</span>
|
||
|
||
<span class="k">return</span> <span class="n">x_output</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<div class="section" id="setting-up-the-network-using-autograd-the-trial-solution">
|
||
<h2>Setting up the network using Autograd; The trial solution<a class="headerlink" href="#setting-up-the-network-using-autograd-the-trial-solution" title="Permalink to this headline">¶</a></h2>
|
||
<p>The cost function must then iterate through the given arrays
|
||
containing values for <span class="math notranslate nohighlight">\(x\)</span> and <span class="math notranslate nohighlight">\(t\)</span>, defines a point <span class="math notranslate nohighlight">\((x,t)\)</span> the deep
|
||
neural network and the trial solution is evaluated at, and then finds
|
||
the Jacobian of the trial solution.</p>
|
||
<p>A possible trial solution for this PDE is</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
g_t(x,t) = h_1(x,t) + x(1-x)tN(x,t,P)
|
||
\]</div>
|
||
<p>with <span class="math notranslate nohighlight">\(A(x,t)\)</span> being a function ensuring that <span class="math notranslate nohighlight">\(g_t(x,t)\)</span> satisfies our given conditions, and <span class="math notranslate nohighlight">\(N(x,t,P)\)</span> being the output from the deep neural network using weights and biases for each layer from <span class="math notranslate nohighlight">\(P\)</span>.</p>
|
||
<p>To fulfill the conditions, <span class="math notranslate nohighlight">\(A(x,t)\)</span> could be:</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
h_1(x,t) = (1-t)\Big(u(x) - \big((1-x)u(0) + x u(1)\big)\Big) = (1-t)u(x) = (1-t)\sin(\pi x)
|
||
\]</div>
|
||
<p>since <span class="math notranslate nohighlight">\((0) = u(1) = 0\)</span> and <span class="math notranslate nohighlight">\(u(x) = \sin(\pi x)\)</span>.</p>
|
||
</div>
|
||
<div class="section" id="why-the-jacobian">
|
||
<h2>Why the jacobian?<a class="headerlink" href="#why-the-jacobian" title="Permalink to this headline">¶</a></h2>
|
||
<p>The Jacobian is used because the program must find the derivative of
|
||
the trial solution with respect to <span class="math notranslate nohighlight">\(x\)</span> and <span class="math notranslate nohighlight">\(t\)</span>.</p>
|
||
<p>This gives the necessity of computing the Jacobian matrix, as we want
|
||
to evaluate the gradient with respect to <span class="math notranslate nohighlight">\(x\)</span> and <span class="math notranslate nohighlight">\(t\)</span> (note that the
|
||
Jacobian of a scalar-valued multivariate function is simply its
|
||
gradient).</p>
|
||
<p>In Autograd, the differentiation is by default done with respect to
|
||
the first input argument of your Python function. Since the points is
|
||
an array representing <span class="math notranslate nohighlight">\(x\)</span> and <span class="math notranslate nohighlight">\(t\)</span>, the Jacobian is calculated using
|
||
the values of <span class="math notranslate nohighlight">\(x\)</span> and <span class="math notranslate nohighlight">\(t\)</span>.</p>
|
||
<p>To find the second derivative with respect to <span class="math notranslate nohighlight">\(x\)</span> and <span class="math notranslate nohighlight">\(t\)</span>, the
|
||
Jacobian can be found for the second time. The result is a Hessian
|
||
matrix, which is the matrix containing all the possible second order
|
||
mixed derivatives of <span class="math notranslate nohighlight">\(g(x,t)\)</span>.</p>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Set up the trial function:</span>
|
||
<span class="k">def</span> <span class="nf">u</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
|
||
<span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">pi</span><span class="o">*</span><span class="n">x</span><span class="p">)</span>
|
||
|
||
<span class="k">def</span> <span class="nf">g_trial</span><span class="p">(</span><span class="n">point</span><span class="p">,</span><span class="n">P</span><span class="p">):</span>
|
||
<span class="n">x</span><span class="p">,</span><span class="n">t</span> <span class="o">=</span> <span class="n">point</span>
|
||
<span class="k">return</span> <span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">t</span><span class="p">)</span><span class="o">*</span><span class="n">u</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="o">+</span> <span class="n">x</span><span class="o">*</span><span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">t</span><span class="o">*</span><span class="n">deep_neural_network</span><span class="p">(</span><span class="n">P</span><span class="p">,</span><span class="n">point</span><span class="p">)</span>
|
||
|
||
<span class="c1"># The right side of the ODE:</span>
|
||
<span class="k">def</span> <span class="nf">f</span><span class="p">(</span><span class="n">point</span><span class="p">):</span>
|
||
<span class="k">return</span> <span class="mf">0.</span>
|
||
|
||
<span class="c1"># The cost function:</span>
|
||
<span class="k">def</span> <span class="nf">cost_function</span><span class="p">(</span><span class="n">P</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">t</span><span class="p">):</span>
|
||
<span class="n">cost_sum</span> <span class="o">=</span> <span class="mi">0</span>
|
||
|
||
<span class="n">g_t_jacobian_func</span> <span class="o">=</span> <span class="n">jacobian</span><span class="p">(</span><span class="n">g_trial</span><span class="p">)</span>
|
||
<span class="n">g_t_hessian_func</span> <span class="o">=</span> <span class="n">hessian</span><span class="p">(</span><span class="n">g_trial</span><span class="p">)</span>
|
||
|
||
<span class="k">for</span> <span class="n">x_</span> <span class="ow">in</span> <span class="n">x</span><span class="p">:</span>
|
||
<span class="k">for</span> <span class="n">t_</span> <span class="ow">in</span> <span class="n">t</span><span class="p">:</span>
|
||
<span class="n">point</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="n">x_</span><span class="p">,</span><span class="n">t_</span><span class="p">])</span>
|
||
|
||
<span class="n">g_t</span> <span class="o">=</span> <span class="n">g_trial</span><span class="p">(</span><span class="n">point</span><span class="p">,</span><span class="n">P</span><span class="p">)</span>
|
||
<span class="n">g_t_jacobian</span> <span class="o">=</span> <span class="n">g_t_jacobian_func</span><span class="p">(</span><span class="n">point</span><span class="p">,</span><span class="n">P</span><span class="p">)</span>
|
||
<span class="n">g_t_hessian</span> <span class="o">=</span> <span class="n">g_t_hessian_func</span><span class="p">(</span><span class="n">point</span><span class="p">,</span><span class="n">P</span><span class="p">)</span>
|
||
|
||
<span class="n">g_t_dt</span> <span class="o">=</span> <span class="n">g_t_jacobian</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
|
||
<span class="n">g_t_d2x</span> <span class="o">=</span> <span class="n">g_t_hessian</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span>
|
||
|
||
<span class="n">func</span> <span class="o">=</span> <span class="n">f</span><span class="p">(</span><span class="n">point</span><span class="p">)</span>
|
||
|
||
<span class="n">err_sqr</span> <span class="o">=</span> <span class="p">(</span> <span class="p">(</span><span class="n">g_t_dt</span> <span class="o">-</span> <span class="n">g_t_d2x</span><span class="p">)</span> <span class="o">-</span> <span class="n">func</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span>
|
||
<span class="n">cost_sum</span> <span class="o">+=</span> <span class="n">err_sqr</span>
|
||
|
||
<span class="k">return</span> <span class="n">cost_sum</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<div class="section" id="setting-up-the-network-using-autograd-the-full-program">
|
||
<h2>Setting up the network using Autograd; The full program<a class="headerlink" href="#setting-up-the-network-using-autograd-the-full-program" title="Permalink to this headline">¶</a></h2>
|
||
<p>Having set up the network, along with the trial solution and cost function, we can now see how the deep neural network performs by comparing the results to the analytical solution.</p>
|
||
<p>The analytical solution of our problem is</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
g(x,t) = \exp(-\pi^2 t)\sin(\pi x)
|
||
\]</div>
|
||
<p>A possible way to implement a neural network solving the PDE, is given below.
|
||
Be aware, though, that it is fairly slow for the parameters used.
|
||
A better result is possible, but requires more iterations, and thus longer time to complete.</p>
|
||
<p>Indeed, the program below is not optimal in its implementation, but rather serves as an example on how to implement and use a neural network to solve a PDE.
|
||
Using TensorFlow results in a much better execution time. Try it!</p>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">autograd.numpy</span> <span class="k">as</span> <span class="nn">np</span>
|
||
<span class="kn">from</span> <span class="nn">autograd</span> <span class="kn">import</span> <span class="n">jacobian</span><span class="p">,</span><span class="n">hessian</span><span class="p">,</span><span class="n">grad</span>
|
||
<span class="kn">import</span> <span class="nn">autograd.numpy.random</span> <span class="k">as</span> <span class="nn">npr</span>
|
||
<span class="kn">from</span> <span class="nn">matplotlib</span> <span class="kn">import</span> <span class="n">cm</span>
|
||
<span class="kn">from</span> <span class="nn">matplotlib</span> <span class="kn">import</span> <span class="n">pyplot</span> <span class="k">as</span> <span class="n">plt</span>
|
||
<span class="kn">from</span> <span class="nn">mpl_toolkits.mplot3d</span> <span class="kn">import</span> <span class="n">axes3d</span>
|
||
|
||
<span class="c1">## Set up the network</span>
|
||
|
||
<span class="k">def</span> <span class="nf">sigmoid</span><span class="p">(</span><span class="n">z</span><span class="p">):</span>
|
||
<span class="k">return</span> <span class="mi">1</span><span class="o">/</span><span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">z</span><span class="p">))</span>
|
||
|
||
<span class="k">def</span> <span class="nf">deep_neural_network</span><span class="p">(</span><span class="n">deep_params</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
|
||
<span class="c1"># x is now a point and a 1D numpy array; make it a column vector</span>
|
||
<span class="n">num_coordinates</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">size</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="mi">0</span><span class="p">)</span>
|
||
<span class="n">x</span> <span class="o">=</span> <span class="n">x</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">num_coordinates</span><span class="p">,</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
|
||
|
||
<span class="n">num_points</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">size</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
|
||
|
||
<span class="c1"># N_hidden is the number of hidden layers</span>
|
||
<span class="n">N_hidden</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">deep_params</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span> <span class="c1"># -1 since params consist of parameters to all the hidden layers AND the output layer</span>
|
||
|
||
<span class="c1"># Assume that the input layer does nothing to the input x</span>
|
||
<span class="n">x_input</span> <span class="o">=</span> <span class="n">x</span>
|
||
<span class="n">x_prev</span> <span class="o">=</span> <span class="n">x_input</span>
|
||
|
||
<span class="c1">## Hidden layers:</span>
|
||
|
||
<span class="k">for</span> <span class="n">l</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">N_hidden</span><span class="p">):</span>
|
||
<span class="c1"># From the list of parameters P; find the correct weigths and bias for this layer</span>
|
||
<span class="n">w_hidden</span> <span class="o">=</span> <span class="n">deep_params</span><span class="p">[</span><span class="n">l</span><span class="p">]</span>
|
||
|
||
<span class="c1"># Add a row of ones to include bias</span>
|
||
<span class="n">x_prev</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">concatenate</span><span class="p">((</span><span class="n">np</span><span class="o">.</span><span class="n">ones</span><span class="p">((</span><span class="mi">1</span><span class="p">,</span><span class="n">num_points</span><span class="p">)),</span> <span class="n">x_prev</span> <span class="p">),</span> <span class="n">axis</span> <span class="o">=</span> <span class="mi">0</span><span class="p">)</span>
|
||
|
||
<span class="n">z_hidden</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">matmul</span><span class="p">(</span><span class="n">w_hidden</span><span class="p">,</span> <span class="n">x_prev</span><span class="p">)</span>
|
||
<span class="n">x_hidden</span> <span class="o">=</span> <span class="n">sigmoid</span><span class="p">(</span><span class="n">z_hidden</span><span class="p">)</span>
|
||
|
||
<span class="c1"># Update x_prev such that next layer can use the output from this layer</span>
|
||
<span class="n">x_prev</span> <span class="o">=</span> <span class="n">x_hidden</span>
|
||
|
||
<span class="c1">## Output layer:</span>
|
||
|
||
<span class="c1"># Get the weights and bias for this layer</span>
|
||
<span class="n">w_output</span> <span class="o">=</span> <span class="n">deep_params</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
|
||
|
||
<span class="c1"># Include bias:</span>
|
||
<span class="n">x_prev</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">concatenate</span><span class="p">((</span><span class="n">np</span><span class="o">.</span><span class="n">ones</span><span class="p">((</span><span class="mi">1</span><span class="p">,</span><span class="n">num_points</span><span class="p">)),</span> <span class="n">x_prev</span><span class="p">),</span> <span class="n">axis</span> <span class="o">=</span> <span class="mi">0</span><span class="p">)</span>
|
||
|
||
<span class="n">z_output</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">matmul</span><span class="p">(</span><span class="n">w_output</span><span class="p">,</span> <span class="n">x_prev</span><span class="p">)</span>
|
||
<span class="n">x_output</span> <span class="o">=</span> <span class="n">z_output</span>
|
||
|
||
<span class="k">return</span> <span class="n">x_output</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span>
|
||
|
||
<span class="c1">## Define the trial solution and cost function</span>
|
||
<span class="k">def</span> <span class="nf">u</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
|
||
<span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">pi</span><span class="o">*</span><span class="n">x</span><span class="p">)</span>
|
||
|
||
<span class="k">def</span> <span class="nf">g_trial</span><span class="p">(</span><span class="n">point</span><span class="p">,</span><span class="n">P</span><span class="p">):</span>
|
||
<span class="n">x</span><span class="p">,</span><span class="n">t</span> <span class="o">=</span> <span class="n">point</span>
|
||
<span class="k">return</span> <span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">t</span><span class="p">)</span><span class="o">*</span><span class="n">u</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="o">+</span> <span class="n">x</span><span class="o">*</span><span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">t</span><span class="o">*</span><span class="n">deep_neural_network</span><span class="p">(</span><span class="n">P</span><span class="p">,</span><span class="n">point</span><span class="p">)</span>
|
||
|
||
<span class="c1"># The right side of the ODE:</span>
|
||
<span class="k">def</span> <span class="nf">f</span><span class="p">(</span><span class="n">point</span><span class="p">):</span>
|
||
<span class="k">return</span> <span class="mf">0.</span>
|
||
|
||
<span class="c1"># The cost function:</span>
|
||
<span class="k">def</span> <span class="nf">cost_function</span><span class="p">(</span><span class="n">P</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">t</span><span class="p">):</span>
|
||
<span class="n">cost_sum</span> <span class="o">=</span> <span class="mi">0</span>
|
||
|
||
<span class="n">g_t_jacobian_func</span> <span class="o">=</span> <span class="n">jacobian</span><span class="p">(</span><span class="n">g_trial</span><span class="p">)</span>
|
||
<span class="n">g_t_hessian_func</span> <span class="o">=</span> <span class="n">hessian</span><span class="p">(</span><span class="n">g_trial</span><span class="p">)</span>
|
||
|
||
<span class="k">for</span> <span class="n">x_</span> <span class="ow">in</span> <span class="n">x</span><span class="p">:</span>
|
||
<span class="k">for</span> <span class="n">t_</span> <span class="ow">in</span> <span class="n">t</span><span class="p">:</span>
|
||
<span class="n">point</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="n">x_</span><span class="p">,</span><span class="n">t_</span><span class="p">])</span>
|
||
|
||
<span class="n">g_t</span> <span class="o">=</span> <span class="n">g_trial</span><span class="p">(</span><span class="n">point</span><span class="p">,</span><span class="n">P</span><span class="p">)</span>
|
||
<span class="n">g_t_jacobian</span> <span class="o">=</span> <span class="n">g_t_jacobian_func</span><span class="p">(</span><span class="n">point</span><span class="p">,</span><span class="n">P</span><span class="p">)</span>
|
||
<span class="n">g_t_hessian</span> <span class="o">=</span> <span class="n">g_t_hessian_func</span><span class="p">(</span><span class="n">point</span><span class="p">,</span><span class="n">P</span><span class="p">)</span>
|
||
|
||
<span class="n">g_t_dt</span> <span class="o">=</span> <span class="n">g_t_jacobian</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
|
||
<span class="n">g_t_d2x</span> <span class="o">=</span> <span class="n">g_t_hessian</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span>
|
||
|
||
<span class="n">func</span> <span class="o">=</span> <span class="n">f</span><span class="p">(</span><span class="n">point</span><span class="p">)</span>
|
||
|
||
<span class="n">err_sqr</span> <span class="o">=</span> <span class="p">(</span> <span class="p">(</span><span class="n">g_t_dt</span> <span class="o">-</span> <span class="n">g_t_d2x</span><span class="p">)</span> <span class="o">-</span> <span class="n">func</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span>
|
||
<span class="n">cost_sum</span> <span class="o">+=</span> <span class="n">err_sqr</span>
|
||
|
||
<span class="k">return</span> <span class="n">cost_sum</span> <span class="o">/</span><span class="p">(</span> <span class="n">np</span><span class="o">.</span><span class="n">size</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">size</span><span class="p">(</span><span class="n">t</span><span class="p">)</span> <span class="p">)</span>
|
||
|
||
<span class="c1">## For comparison, define the analytical solution</span>
|
||
<span class="k">def</span> <span class="nf">g_analytic</span><span class="p">(</span><span class="n">point</span><span class="p">):</span>
|
||
<span class="n">x</span><span class="p">,</span><span class="n">t</span> <span class="o">=</span> <span class="n">point</span>
|
||
<span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">np</span><span class="o">.</span><span class="n">pi</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="n">t</span><span class="p">)</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">pi</span><span class="o">*</span><span class="n">x</span><span class="p">)</span>
|
||
|
||
<span class="c1">## Set up a function for training the network to solve for the equation</span>
|
||
<span class="k">def</span> <span class="nf">solve_pde_deep_neural_network</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">t</span><span class="p">,</span> <span class="n">num_neurons</span><span class="p">,</span> <span class="n">num_iter</span><span class="p">,</span> <span class="n">lmb</span><span class="p">):</span>
|
||
<span class="c1">## Set up initial weigths and biases</span>
|
||
<span class="n">N_hidden</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">size</span><span class="p">(</span><span class="n">num_neurons</span><span class="p">)</span>
|
||
|
||
<span class="c1">## Set up initial weigths and biases</span>
|
||
|
||
<span class="c1"># Initialize the list of parameters:</span>
|
||
<span class="n">P</span> <span class="o">=</span> <span class="p">[</span><span class="kc">None</span><span class="p">]</span><span class="o">*</span><span class="p">(</span><span class="n">N_hidden</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="c1"># + 1 to include the output layer</span>
|
||
|
||
<span class="n">P</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">npr</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="n">num_neurons</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="mi">2</span> <span class="o">+</span> <span class="mi">1</span> <span class="p">)</span> <span class="c1"># 2 since we have two points, +1 to include bias</span>
|
||
<span class="k">for</span> <span class="n">l</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="n">N_hidden</span><span class="p">):</span>
|
||
<span class="n">P</span><span class="p">[</span><span class="n">l</span><span class="p">]</span> <span class="o">=</span> <span class="n">npr</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="n">num_neurons</span><span class="p">[</span><span class="n">l</span><span class="p">],</span> <span class="n">num_neurons</span><span class="p">[</span><span class="n">l</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="c1"># +1 to include bias</span>
|
||
|
||
<span class="c1"># For the output layer</span>
|
||
<span class="n">P</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">npr</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">num_neurons</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">+</span> <span class="mi">1</span> <span class="p">)</span> <span class="c1"># +1 since bias is included</span>
|
||
|
||
<span class="nb">print</span><span class="p">(</span><span class="s1">'Initial cost: '</span><span class="p">,</span><span class="n">cost_function</span><span class="p">(</span><span class="n">P</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">t</span><span class="p">))</span>
|
||
|
||
<span class="n">cost_function_grad</span> <span class="o">=</span> <span class="n">grad</span><span class="p">(</span><span class="n">cost_function</span><span class="p">,</span><span class="mi">0</span><span class="p">)</span>
|
||
|
||
<span class="c1"># Let the update be done num_iter times</span>
|
||
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">num_iter</span><span class="p">):</span>
|
||
<span class="n">cost_grad</span> <span class="o">=</span> <span class="n">cost_function_grad</span><span class="p">(</span><span class="n">P</span><span class="p">,</span> <span class="n">x</span> <span class="p">,</span> <span class="n">t</span><span class="p">)</span>
|
||
|
||
<span class="k">for</span> <span class="n">l</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">N_hidden</span><span class="o">+</span><span class="mi">1</span><span class="p">):</span>
|
||
<span class="n">P</span><span class="p">[</span><span class="n">l</span><span class="p">]</span> <span class="o">=</span> <span class="n">P</span><span class="p">[</span><span class="n">l</span><span class="p">]</span> <span class="o">-</span> <span class="n">lmb</span> <span class="o">*</span> <span class="n">cost_grad</span><span class="p">[</span><span class="n">l</span><span class="p">]</span>
|
||
|
||
<span class="nb">print</span><span class="p">(</span><span class="s1">'Final cost: '</span><span class="p">,</span><span class="n">cost_function</span><span class="p">(</span><span class="n">P</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">t</span><span class="p">))</span>
|
||
|
||
<span class="k">return</span> <span class="n">P</span>
|
||
|
||
<span class="k">if</span> <span class="vm">__name__</span> <span class="o">==</span> <span class="s1">'__main__'</span><span class="p">:</span>
|
||
<span class="c1">### Use the neural network:</span>
|
||
<span class="n">npr</span><span class="o">.</span><span class="n">seed</span><span class="p">(</span><span class="mi">15</span><span class="p">)</span>
|
||
|
||
<span class="c1">## Decide the vales of arguments to the function to solve</span>
|
||
<span class="n">Nx</span> <span class="o">=</span> <span class="mi">10</span><span class="p">;</span> <span class="n">Nt</span> <span class="o">=</span> <span class="mi">10</span>
|
||
<span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">Nx</span><span class="p">)</span>
|
||
<span class="n">t</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="n">Nt</span><span class="p">)</span>
|
||
|
||
<span class="c1">## Set up the parameters for the network</span>
|
||
<span class="n">num_hidden_neurons</span> <span class="o">=</span> <span class="p">[</span><span class="mi">100</span><span class="p">,</span> <span class="mi">25</span><span class="p">]</span>
|
||
<span class="n">num_iter</span> <span class="o">=</span> <span class="mi">250</span>
|
||
<span class="n">lmb</span> <span class="o">=</span> <span class="mf">0.01</span>
|
||
|
||
<span class="n">P</span> <span class="o">=</span> <span class="n">solve_pde_deep_neural_network</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">t</span><span class="p">,</span> <span class="n">num_hidden_neurons</span><span class="p">,</span> <span class="n">num_iter</span><span class="p">,</span> <span class="n">lmb</span><span class="p">)</span>
|
||
|
||
<span class="c1">## Store the results</span>
|
||
<span class="n">g_dnn_ag</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">Nx</span><span class="p">,</span> <span class="n">Nt</span><span class="p">))</span>
|
||
<span class="n">G_analytical</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">Nx</span><span class="p">,</span> <span class="n">Nt</span><span class="p">))</span>
|
||
<span class="k">for</span> <span class="n">i</span><span class="p">,</span><span class="n">x_</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
|
||
<span class="k">for</span> <span class="n">j</span><span class="p">,</span> <span class="n">t_</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">t</span><span class="p">):</span>
|
||
<span class="n">point</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="n">x_</span><span class="p">,</span> <span class="n">t_</span><span class="p">])</span>
|
||
<span class="n">g_dnn_ag</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">g_trial</span><span class="p">(</span><span class="n">point</span><span class="p">,</span><span class="n">P</span><span class="p">)</span>
|
||
|
||
<span class="n">G_analytical</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">g_analytic</span><span class="p">(</span><span class="n">point</span><span class="p">)</span>
|
||
|
||
<span class="c1"># Find the map difference between the analytical and the computed solution</span>
|
||
<span class="n">diff_ag</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">g_dnn_ag</span> <span class="o">-</span> <span class="n">G_analytical</span><span class="p">)</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="s1">'Max absolute difference between the analytical solution and the network: </span><span class="si">%g</span><span class="s1">'</span><span class="o">%</span><span class="k">np</span>.max(diff_ag))
|
||
|
||
<span class="c1">## Plot the solutions in two dimensions, that being in position and time</span>
|
||
|
||
<span class="n">T</span><span class="p">,</span><span class="n">X</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">meshgrid</span><span class="p">(</span><span class="n">t</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
|
||
|
||
<span class="n">fig</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span><span class="mi">10</span><span class="p">))</span>
|
||
<span class="n">ax</span> <span class="o">=</span> <span class="n">fig</span><span class="o">.</span><span class="n">add_suplot</span><span class="p">(</span><span class="n">projection</span><span class="o">=</span><span class="s1">'3d'</span><span class="p">)</span>
|
||
<span class="n">ax</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">'Solution from the deep neural network w/ </span><span class="si">%d</span><span class="s1"> layer'</span><span class="o">%</span><span class="k">len</span>(num_hidden_neurons))
|
||
<span class="n">s</span> <span class="o">=</span> <span class="n">ax</span><span class="o">.</span><span class="n">plot_surface</span><span class="p">(</span><span class="n">T</span><span class="p">,</span><span class="n">X</span><span class="p">,</span><span class="n">g_dnn_ag</span><span class="p">,</span><span class="n">linewidth</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span><span class="n">antialiased</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span><span class="n">cmap</span><span class="o">=</span><span class="n">cm</span><span class="o">.</span><span class="n">viridis</span><span class="p">)</span>
|
||
<span class="n">ax</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s1">'Time $t$'</span><span class="p">)</span>
|
||
<span class="n">ax</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">'Position $x$'</span><span class="p">);</span>
|
||
|
||
|
||
<span class="n">fig</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span><span class="mi">10</span><span class="p">))</span>
|
||
<span class="n">ax</span> <span class="o">=</span> <span class="n">fig</span><span class="o">.</span><span class="n">add_suplot</span><span class="p">(</span><span class="n">projection</span><span class="o">=</span><span class="s1">'3d'</span><span class="p">)</span>
|
||
<span class="n">ax</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">'Analytical solution'</span><span class="p">)</span>
|
||
<span class="n">s</span> <span class="o">=</span> <span class="n">ax</span><span class="o">.</span><span class="n">plot_surface</span><span class="p">(</span><span class="n">T</span><span class="p">,</span><span class="n">X</span><span class="p">,</span><span class="n">G_analytical</span><span class="p">,</span><span class="n">linewidth</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span><span class="n">antialiased</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span><span class="n">cmap</span><span class="o">=</span><span class="n">cm</span><span class="o">.</span><span class="n">viridis</span><span class="p">)</span>
|
||
<span class="n">ax</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s1">'Time $t$'</span><span class="p">)</span>
|
||
<span class="n">ax</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">'Position $x$'</span><span class="p">);</span>
|
||
|
||
<span class="n">fig</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span><span class="mi">10</span><span class="p">))</span>
|
||
<span class="n">ax</span> <span class="o">=</span> <span class="n">fig</span><span class="o">.</span><span class="n">add_suplot</span><span class="p">(</span><span class="n">projection</span><span class="o">=</span><span class="s1">'3d'</span><span class="p">)</span>
|
||
<span class="n">ax</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">'Difference'</span><span class="p">)</span>
|
||
<span class="n">s</span> <span class="o">=</span> <span class="n">ax</span><span class="o">.</span><span class="n">plot_surface</span><span class="p">(</span><span class="n">T</span><span class="p">,</span><span class="n">X</span><span class="p">,</span><span class="n">diff_ag</span><span class="p">,</span><span class="n">linewidth</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span><span class="n">antialiased</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span><span class="n">cmap</span><span class="o">=</span><span class="n">cm</span><span class="o">.</span><span class="n">viridis</span><span class="p">)</span>
|
||
<span class="n">ax</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s1">'Time $t$'</span><span class="p">)</span>
|
||
<span class="n">ax</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">'Position $x$'</span><span class="p">);</span>
|
||
|
||
<span class="c1">## Take some slices of the 3D plots just to see the solutions at particular times</span>
|
||
<span class="n">indx1</span> <span class="o">=</span> <span class="mi">0</span>
|
||
<span class="n">indx2</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">Nt</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span>
|
||
<span class="n">indx3</span> <span class="o">=</span> <span class="n">Nt</span><span class="o">-</span><span class="mi">1</span>
|
||
|
||
<span class="n">t1</span> <span class="o">=</span> <span class="n">t</span><span class="p">[</span><span class="n">indx1</span><span class="p">]</span>
|
||
<span class="n">t2</span> <span class="o">=</span> <span class="n">t</span><span class="p">[</span><span class="n">indx2</span><span class="p">]</span>
|
||
<span class="n">t3</span> <span class="o">=</span> <span class="n">t</span><span class="p">[</span><span class="n">indx3</span><span class="p">]</span>
|
||
|
||
<span class="c1"># Slice the results from the DNN</span>
|
||
<span class="n">res1</span> <span class="o">=</span> <span class="n">g_dnn_ag</span><span class="p">[:,</span><span class="n">indx1</span><span class="p">]</span>
|
||
<span class="n">res2</span> <span class="o">=</span> <span class="n">g_dnn_ag</span><span class="p">[:,</span><span class="n">indx2</span><span class="p">]</span>
|
||
<span class="n">res3</span> <span class="o">=</span> <span class="n">g_dnn_ag</span><span class="p">[:,</span><span class="n">indx3</span><span class="p">]</span>
|
||
|
||
<span class="c1"># Slice the analytical results</span>
|
||
<span class="n">res_analytical1</span> <span class="o">=</span> <span class="n">G_analytical</span><span class="p">[:,</span><span class="n">indx1</span><span class="p">]</span>
|
||
<span class="n">res_analytical2</span> <span class="o">=</span> <span class="n">G_analytical</span><span class="p">[:,</span><span class="n">indx2</span><span class="p">]</span>
|
||
<span class="n">res_analytical3</span> <span class="o">=</span> <span class="n">G_analytical</span><span class="p">[:,</span><span class="n">indx3</span><span class="p">]</span>
|
||
|
||
<span class="c1"># Plot the slices</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span><span class="mi">10</span><span class="p">))</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">"Computed solutions at time = </span><span class="si">%g</span><span class="s2">"</span><span class="o">%</span><span class="k">t1</span>)
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">res1</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">res_analytical1</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">([</span><span class="s1">'dnn'</span><span class="p">,</span><span class="s1">'analytical'</span><span class="p">])</span>
|
||
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span><span class="mi">10</span><span class="p">))</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">"Computed solutions at time = </span><span class="si">%g</span><span class="s2">"</span><span class="o">%</span><span class="k">t2</span>)
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">res2</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">res_analytical2</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">([</span><span class="s1">'dnn'</span><span class="p">,</span><span class="s1">'analytical'</span><span class="p">])</span>
|
||
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span><span class="mi">10</span><span class="p">))</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">"Computed solutions at time = </span><span class="si">%g</span><span class="s2">"</span><span class="o">%</span><span class="k">t3</span>)
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">res3</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">res_analytical3</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">([</span><span class="s1">'dnn'</span><span class="p">,</span><span class="s1">'analytical'</span><span class="p">])</span>
|
||
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<div class="section" id="example-solving-the-wave-equation-with-neural-networks">
|
||
<h2>Example: Solving the wave equation with Neural Networks<a class="headerlink" href="#example-solving-the-wave-equation-with-neural-networks" title="Permalink to this headline">¶</a></h2>
|
||
<p>The wave equation is</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\frac{\partial^2 g(x,t)}{\partial t^2} = c^2\frac{\partial^2 g(x,t)}{\partial x^2}
|
||
\]</div>
|
||
<p>with <span class="math notranslate nohighlight">\(c\)</span> being the specified wave speed.</p>
|
||
<p>Here, the chosen conditions are</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[\begin{split}
|
||
\begin{align*}
|
||
g(0,t) &= 0 \\
|
||
g(1,t) &= 0 \\
|
||
g(x,0) &= u(x) \\
|
||
\frac{\partial g(x,t)}{\partial t} \Big |_{t = 0} &= v(x)
|
||
\end{align*}
|
||
\end{split}\]</div>
|
||
<p>where <span class="math notranslate nohighlight">\(\frac{\partial g(x,t)}{\partial t} \Big |_{t = 0}\)</span> means the derivative of <span class="math notranslate nohighlight">\(g(x,t)\)</span> with respect to <span class="math notranslate nohighlight">\(t\)</span> is evaluated at <span class="math notranslate nohighlight">\(t = 0\)</span>, and <span class="math notranslate nohighlight">\(u(x)\)</span> and <span class="math notranslate nohighlight">\(v(x)\)</span> being given functions.</p>
|
||
</div>
|
||
<div class="section" id="the-problem-to-solve-for">
|
||
<h2>The problem to solve for<a class="headerlink" href="#the-problem-to-solve-for" title="Permalink to this headline">¶</a></h2>
|
||
<p>The wave equation to solve for, is</p>
|
||
<!-- Equation labels as ordinary links -->
|
||
<div id="wave"></div>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\begin{equation} \label{wave} \tag{19}
|
||
\frac{\partial^2 g(x,t)}{\partial t^2} = c^2 \frac{\partial^2 g(x,t)}{\partial x^2}
|
||
\end{equation}
|
||
\]</div>
|
||
<p>where <span class="math notranslate nohighlight">\(c\)</span> is the given wave speed.
|
||
The chosen conditions for this equation are</p>
|
||
<!-- Equation labels as ordinary links -->
|
||
<div id="condwave"></div>
|
||
<div class="math notranslate nohighlight">
|
||
\[\begin{split}
|
||
\begin{aligned}
|
||
g(0,t) &= 0, &t \geq 0 \\
|
||
g(1,t) &= 0, &t \geq 0 \\
|
||
g(x,0) &= u(x), &x\in[0,1] \\
|
||
\frac{\partial g(x,t)}{\partial t}\Big |_{t = 0} &= v(x), &x \in [0,1]
|
||
\end{aligned} \label{condwave} \tag{20}
|
||
\end{split}\]</div>
|
||
<p>In this example, let <span class="math notranslate nohighlight">\(c = 1\)</span> and <span class="math notranslate nohighlight">\(u(x) = \sin(\pi x)\)</span> and <span class="math notranslate nohighlight">\(v(x) = -\pi\sin(\pi x)\)</span>.</p>
|
||
</div>
|
||
<div class="section" id="id7">
|
||
<h2>The trial solution<a class="headerlink" href="#id7" title="Permalink to this headline">¶</a></h2>
|
||
<p>Setting up the network is done in similar matter as for the example of solving the diffusion equation.
|
||
The only things we have to change, is the trial solution such that it satisfies the conditions from (<a class="reference external" href="#condwave">20</a>) and the cost function.</p>
|
||
<p>The trial solution becomes slightly different since we have other conditions than in the example of solving the diffusion equation. Here, a possible trial solution <span class="math notranslate nohighlight">\(g_t(x,t)\)</span> is</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
g_t(x,t) = h_1(x,t) + x(1-x)t^2N(x,t,P)
|
||
\]</div>
|
||
<p>where</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
h_1(x,t) = (1-t^2)u(x) + tv(x)
|
||
\]</div>
|
||
<p>Note that this trial solution satisfies the conditions only if <span class="math notranslate nohighlight">\(u(0) = v(0) = u(1) = v(1) = 0\)</span>, which is the case in this example.</p>
|
||
</div>
|
||
<div class="section" id="the-analytical-solution">
|
||
<h2>The analytical solution<a class="headerlink" href="#the-analytical-solution" title="Permalink to this headline">¶</a></h2>
|
||
<p>The analytical solution for our specific problem, is</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
g(x,t) = \sin(\pi x)\cos(\pi t) - \sin(\pi x)\sin(\pi t)
|
||
\]</div>
|
||
</div>
|
||
<div class="section" id="solving-the-wave-equation-the-full-program-using-autograd">
|
||
<h2>Solving the wave equation - the full program using Autograd<a class="headerlink" href="#solving-the-wave-equation-the-full-program-using-autograd" title="Permalink to this headline">¶</a></h2>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">autograd.numpy</span> <span class="k">as</span> <span class="nn">np</span>
|
||
<span class="kn">from</span> <span class="nn">autograd</span> <span class="kn">import</span> <span class="n">hessian</span><span class="p">,</span><span class="n">grad</span>
|
||
<span class="kn">import</span> <span class="nn">autograd.numpy.random</span> <span class="k">as</span> <span class="nn">npr</span>
|
||
<span class="kn">from</span> <span class="nn">matplotlib</span> <span class="kn">import</span> <span class="n">cm</span>
|
||
<span class="kn">from</span> <span class="nn">matplotlib</span> <span class="kn">import</span> <span class="n">pyplot</span> <span class="k">as</span> <span class="n">plt</span>
|
||
<span class="kn">from</span> <span class="nn">mpl_toolkits.mplot3d</span> <span class="kn">import</span> <span class="n">axes3d</span>
|
||
|
||
<span class="c1">## Set up the trial function:</span>
|
||
<span class="k">def</span> <span class="nf">u</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
|
||
<span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">pi</span><span class="o">*</span><span class="n">x</span><span class="p">)</span>
|
||
|
||
<span class="k">def</span> <span class="nf">v</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
|
||
<span class="k">return</span> <span class="o">-</span><span class="n">np</span><span class="o">.</span><span class="n">pi</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">pi</span><span class="o">*</span><span class="n">x</span><span class="p">)</span>
|
||
|
||
<span class="k">def</span> <span class="nf">h1</span><span class="p">(</span><span class="n">point</span><span class="p">):</span>
|
||
<span class="n">x</span><span class="p">,</span><span class="n">t</span> <span class="o">=</span> <span class="n">point</span>
|
||
<span class="k">return</span> <span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">t</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">u</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="o">+</span> <span class="n">t</span><span class="o">*</span><span class="n">v</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
|
||
|
||
<span class="k">def</span> <span class="nf">g_trial</span><span class="p">(</span><span class="n">point</span><span class="p">,</span><span class="n">P</span><span class="p">):</span>
|
||
<span class="n">x</span><span class="p">,</span><span class="n">t</span> <span class="o">=</span> <span class="n">point</span>
|
||
<span class="k">return</span> <span class="n">h1</span><span class="p">(</span><span class="n">point</span><span class="p">)</span> <span class="o">+</span> <span class="n">x</span><span class="o">*</span><span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">t</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="n">deep_neural_network</span><span class="p">(</span><span class="n">P</span><span class="p">,</span><span class="n">point</span><span class="p">)</span>
|
||
|
||
<span class="c1">## Define the cost function</span>
|
||
<span class="k">def</span> <span class="nf">cost_function</span><span class="p">(</span><span class="n">P</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">t</span><span class="p">):</span>
|
||
<span class="n">cost_sum</span> <span class="o">=</span> <span class="mi">0</span>
|
||
|
||
<span class="n">g_t_hessian_func</span> <span class="o">=</span> <span class="n">hessian</span><span class="p">(</span><span class="n">g_trial</span><span class="p">)</span>
|
||
|
||
<span class="k">for</span> <span class="n">x_</span> <span class="ow">in</span> <span class="n">x</span><span class="p">:</span>
|
||
<span class="k">for</span> <span class="n">t_</span> <span class="ow">in</span> <span class="n">t</span><span class="p">:</span>
|
||
<span class="n">point</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="n">x_</span><span class="p">,</span><span class="n">t_</span><span class="p">])</span>
|
||
|
||
<span class="n">g_t_hessian</span> <span class="o">=</span> <span class="n">g_t_hessian_func</span><span class="p">(</span><span class="n">point</span><span class="p">,</span><span class="n">P</span><span class="p">)</span>
|
||
|
||
<span class="n">g_t_d2x</span> <span class="o">=</span> <span class="n">g_t_hessian</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span>
|
||
<span class="n">g_t_d2t</span> <span class="o">=</span> <span class="n">g_t_hessian</span><span class="p">[</span><span class="mi">1</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span>
|
||
|
||
<span class="n">err_sqr</span> <span class="o">=</span> <span class="p">(</span> <span class="p">(</span><span class="n">g_t_d2t</span> <span class="o">-</span> <span class="n">g_t_d2x</span><span class="p">)</span> <span class="p">)</span><span class="o">**</span><span class="mi">2</span>
|
||
<span class="n">cost_sum</span> <span class="o">+=</span> <span class="n">err_sqr</span>
|
||
|
||
<span class="k">return</span> <span class="n">cost_sum</span> <span class="o">/</span> <span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">size</span><span class="p">(</span><span class="n">t</span><span class="p">)</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">size</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
|
||
|
||
<span class="c1">## The neural network</span>
|
||
<span class="k">def</span> <span class="nf">sigmoid</span><span class="p">(</span><span class="n">z</span><span class="p">):</span>
|
||
<span class="k">return</span> <span class="mi">1</span><span class="o">/</span><span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">z</span><span class="p">))</span>
|
||
|
||
<span class="k">def</span> <span class="nf">deep_neural_network</span><span class="p">(</span><span class="n">deep_params</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
|
||
<span class="c1"># x is now a point and a 1D numpy array; make it a column vector</span>
|
||
<span class="n">num_coordinates</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">size</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="mi">0</span><span class="p">)</span>
|
||
<span class="n">x</span> <span class="o">=</span> <span class="n">x</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">num_coordinates</span><span class="p">,</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
|
||
|
||
<span class="n">num_points</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">size</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
|
||
|
||
<span class="c1"># N_hidden is the number of hidden layers</span>
|
||
<span class="n">N_hidden</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">deep_params</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span> <span class="c1"># -1 since params consist of parameters to all the hidden layers AND the output layer</span>
|
||
|
||
<span class="c1"># Assume that the input layer does nothing to the input x</span>
|
||
<span class="n">x_input</span> <span class="o">=</span> <span class="n">x</span>
|
||
<span class="n">x_prev</span> <span class="o">=</span> <span class="n">x_input</span>
|
||
|
||
<span class="c1">## Hidden layers:</span>
|
||
|
||
<span class="k">for</span> <span class="n">l</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">N_hidden</span><span class="p">):</span>
|
||
<span class="c1"># From the list of parameters P; find the correct weigths and bias for this layer</span>
|
||
<span class="n">w_hidden</span> <span class="o">=</span> <span class="n">deep_params</span><span class="p">[</span><span class="n">l</span><span class="p">]</span>
|
||
|
||
<span class="c1"># Add a row of ones to include bias</span>
|
||
<span class="n">x_prev</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">concatenate</span><span class="p">((</span><span class="n">np</span><span class="o">.</span><span class="n">ones</span><span class="p">((</span><span class="mi">1</span><span class="p">,</span><span class="n">num_points</span><span class="p">)),</span> <span class="n">x_prev</span> <span class="p">),</span> <span class="n">axis</span> <span class="o">=</span> <span class="mi">0</span><span class="p">)</span>
|
||
|
||
<span class="n">z_hidden</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">matmul</span><span class="p">(</span><span class="n">w_hidden</span><span class="p">,</span> <span class="n">x_prev</span><span class="p">)</span>
|
||
<span class="n">x_hidden</span> <span class="o">=</span> <span class="n">sigmoid</span><span class="p">(</span><span class="n">z_hidden</span><span class="p">)</span>
|
||
|
||
<span class="c1"># Update x_prev such that next layer can use the output from this layer</span>
|
||
<span class="n">x_prev</span> <span class="o">=</span> <span class="n">x_hidden</span>
|
||
|
||
<span class="c1">## Output layer:</span>
|
||
|
||
<span class="c1"># Get the weights and bias for this layer</span>
|
||
<span class="n">w_output</span> <span class="o">=</span> <span class="n">deep_params</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
|
||
|
||
<span class="c1"># Include bias:</span>
|
||
<span class="n">x_prev</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">concatenate</span><span class="p">((</span><span class="n">np</span><span class="o">.</span><span class="n">ones</span><span class="p">((</span><span class="mi">1</span><span class="p">,</span><span class="n">num_points</span><span class="p">)),</span> <span class="n">x_prev</span><span class="p">),</span> <span class="n">axis</span> <span class="o">=</span> <span class="mi">0</span><span class="p">)</span>
|
||
|
||
<span class="n">z_output</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">matmul</span><span class="p">(</span><span class="n">w_output</span><span class="p">,</span> <span class="n">x_prev</span><span class="p">)</span>
|
||
<span class="n">x_output</span> <span class="o">=</span> <span class="n">z_output</span>
|
||
|
||
<span class="k">return</span> <span class="n">x_output</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span>
|
||
|
||
<span class="c1">## The analytical solution</span>
|
||
<span class="k">def</span> <span class="nf">g_analytic</span><span class="p">(</span><span class="n">point</span><span class="p">):</span>
|
||
<span class="n">x</span><span class="p">,</span><span class="n">t</span> <span class="o">=</span> <span class="n">point</span>
|
||
<span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">pi</span><span class="o">*</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">pi</span><span class="o">*</span><span class="n">t</span><span class="p">)</span> <span class="o">-</span> <span class="n">np</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">pi</span><span class="o">*</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">pi</span><span class="o">*</span><span class="n">t</span><span class="p">)</span>
|
||
|
||
<span class="k">def</span> <span class="nf">solve_pde_deep_neural_network</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">t</span><span class="p">,</span> <span class="n">num_neurons</span><span class="p">,</span> <span class="n">num_iter</span><span class="p">,</span> <span class="n">lmb</span><span class="p">):</span>
|
||
<span class="c1">## Set up initial weigths and biases</span>
|
||
<span class="n">N_hidden</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">size</span><span class="p">(</span><span class="n">num_neurons</span><span class="p">)</span>
|
||
|
||
<span class="c1">## Set up initial weigths and biases</span>
|
||
|
||
<span class="c1"># Initialize the list of parameters:</span>
|
||
<span class="n">P</span> <span class="o">=</span> <span class="p">[</span><span class="kc">None</span><span class="p">]</span><span class="o">*</span><span class="p">(</span><span class="n">N_hidden</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="c1"># + 1 to include the output layer</span>
|
||
|
||
<span class="n">P</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">npr</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="n">num_neurons</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="mi">2</span> <span class="o">+</span> <span class="mi">1</span> <span class="p">)</span> <span class="c1"># 2 since we have two points, +1 to include bias</span>
|
||
<span class="k">for</span> <span class="n">l</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="n">N_hidden</span><span class="p">):</span>
|
||
<span class="n">P</span><span class="p">[</span><span class="n">l</span><span class="p">]</span> <span class="o">=</span> <span class="n">npr</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="n">num_neurons</span><span class="p">[</span><span class="n">l</span><span class="p">],</span> <span class="n">num_neurons</span><span class="p">[</span><span class="n">l</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="c1"># +1 to include bias</span>
|
||
|
||
<span class="c1"># For the output layer</span>
|
||
<span class="n">P</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">npr</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">num_neurons</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">+</span> <span class="mi">1</span> <span class="p">)</span> <span class="c1"># +1 since bias is included</span>
|
||
|
||
<span class="nb">print</span><span class="p">(</span><span class="s1">'Initial cost: '</span><span class="p">,</span><span class="n">cost_function</span><span class="p">(</span><span class="n">P</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">t</span><span class="p">))</span>
|
||
|
||
<span class="n">cost_function_grad</span> <span class="o">=</span> <span class="n">grad</span><span class="p">(</span><span class="n">cost_function</span><span class="p">,</span><span class="mi">0</span><span class="p">)</span>
|
||
|
||
<span class="c1"># Let the update be done num_iter times</span>
|
||
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">num_iter</span><span class="p">):</span>
|
||
<span class="n">cost_grad</span> <span class="o">=</span> <span class="n">cost_function_grad</span><span class="p">(</span><span class="n">P</span><span class="p">,</span> <span class="n">x</span> <span class="p">,</span> <span class="n">t</span><span class="p">)</span>
|
||
|
||
<span class="k">for</span> <span class="n">l</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">N_hidden</span><span class="o">+</span><span class="mi">1</span><span class="p">):</span>
|
||
<span class="n">P</span><span class="p">[</span><span class="n">l</span><span class="p">]</span> <span class="o">=</span> <span class="n">P</span><span class="p">[</span><span class="n">l</span><span class="p">]</span> <span class="o">-</span> <span class="n">lmb</span> <span class="o">*</span> <span class="n">cost_grad</span><span class="p">[</span><span class="n">l</span><span class="p">]</span>
|
||
|
||
|
||
<span class="nb">print</span><span class="p">(</span><span class="s1">'Final cost: '</span><span class="p">,</span><span class="n">cost_function</span><span class="p">(</span><span class="n">P</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">t</span><span class="p">))</span>
|
||
|
||
<span class="k">return</span> <span class="n">P</span>
|
||
|
||
<span class="k">if</span> <span class="vm">__name__</span> <span class="o">==</span> <span class="s1">'__main__'</span><span class="p">:</span>
|
||
<span class="c1">### Use the neural network:</span>
|
||
<span class="n">npr</span><span class="o">.</span><span class="n">seed</span><span class="p">(</span><span class="mi">15</span><span class="p">)</span>
|
||
|
||
<span class="c1">## Decide the vales of arguments to the function to solve</span>
|
||
<span class="n">Nx</span> <span class="o">=</span> <span class="mi">10</span><span class="p">;</span> <span class="n">Nt</span> <span class="o">=</span> <span class="mi">10</span>
|
||
<span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">Nx</span><span class="p">)</span>
|
||
<span class="n">t</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="n">Nt</span><span class="p">)</span>
|
||
|
||
<span class="c1">## Set up the parameters for the network</span>
|
||
<span class="n">num_hidden_neurons</span> <span class="o">=</span> <span class="p">[</span><span class="mi">50</span><span class="p">,</span><span class="mi">20</span><span class="p">]</span>
|
||
<span class="n">num_iter</span> <span class="o">=</span> <span class="mi">1000</span>
|
||
<span class="n">lmb</span> <span class="o">=</span> <span class="mf">0.01</span>
|
||
|
||
<span class="n">P</span> <span class="o">=</span> <span class="n">solve_pde_deep_neural_network</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">t</span><span class="p">,</span> <span class="n">num_hidden_neurons</span><span class="p">,</span> <span class="n">num_iter</span><span class="p">,</span> <span class="n">lmb</span><span class="p">)</span>
|
||
|
||
<span class="c1">## Store the results</span>
|
||
<span class="n">res</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">Nx</span><span class="p">,</span> <span class="n">Nt</span><span class="p">))</span>
|
||
<span class="n">res_analytical</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">Nx</span><span class="p">,</span> <span class="n">Nt</span><span class="p">))</span>
|
||
<span class="k">for</span> <span class="n">i</span><span class="p">,</span><span class="n">x_</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
|
||
<span class="k">for</span> <span class="n">j</span><span class="p">,</span> <span class="n">t_</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">t</span><span class="p">):</span>
|
||
<span class="n">point</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="n">x_</span><span class="p">,</span> <span class="n">t_</span><span class="p">])</span>
|
||
<span class="n">res</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">g_trial</span><span class="p">(</span><span class="n">point</span><span class="p">,</span><span class="n">P</span><span class="p">)</span>
|
||
|
||
<span class="n">res_analytical</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">g_analytic</span><span class="p">(</span><span class="n">point</span><span class="p">)</span>
|
||
|
||
<span class="n">diff</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">res</span> <span class="o">-</span> <span class="n">res_analytical</span><span class="p">)</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="s2">"Max difference between analytical and solution from nn: </span><span class="si">%g</span><span class="s2">"</span><span class="o">%</span><span class="k">np</span>.max(diff))
|
||
|
||
<span class="c1">## Plot the solutions in two dimensions, that being in position and time</span>
|
||
|
||
<span class="n">T</span><span class="p">,</span><span class="n">X</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">meshgrid</span><span class="p">(</span><span class="n">t</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
|
||
|
||
<span class="n">fig</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span><span class="mi">10</span><span class="p">))</span>
|
||
<span class="n">ax</span> <span class="o">=</span> <span class="n">fig</span><span class="o">.</span><span class="n">add_suplot</span><span class="p">(</span><span class="n">projection</span><span class="o">=</span><span class="s1">'3d'</span><span class="p">)</span>
|
||
<span class="n">ax</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">'Solution from the deep neural network w/ </span><span class="si">%d</span><span class="s1"> layer'</span><span class="o">%</span><span class="k">len</span>(num_hidden_neurons))
|
||
<span class="n">s</span> <span class="o">=</span> <span class="n">ax</span><span class="o">.</span><span class="n">plot_surface</span><span class="p">(</span><span class="n">T</span><span class="p">,</span><span class="n">X</span><span class="p">,</span><span class="n">res</span><span class="p">,</span><span class="n">linewidth</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span><span class="n">antialiased</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span><span class="n">cmap</span><span class="o">=</span><span class="n">cm</span><span class="o">.</span><span class="n">viridis</span><span class="p">)</span>
|
||
<span class="n">ax</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s1">'Time $t$'</span><span class="p">)</span>
|
||
<span class="n">ax</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">'Position $x$'</span><span class="p">);</span>
|
||
|
||
|
||
<span class="n">fig</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span><span class="mi">10</span><span class="p">))</span>
|
||
<span class="n">ax</span> <span class="o">=</span> <span class="n">fig</span><span class="o">.</span><span class="n">add_suplot</span><span class="p">(</span><span class="n">projection</span><span class="o">=</span><span class="s1">'3d'</span><span class="p">)</span>
|
||
<span class="n">ax</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">'Analytical solution'</span><span class="p">)</span>
|
||
<span class="n">s</span> <span class="o">=</span> <span class="n">ax</span><span class="o">.</span><span class="n">plot_surface</span><span class="p">(</span><span class="n">T</span><span class="p">,</span><span class="n">X</span><span class="p">,</span><span class="n">res_analytical</span><span class="p">,</span><span class="n">linewidth</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span><span class="n">antialiased</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span><span class="n">cmap</span><span class="o">=</span><span class="n">cm</span><span class="o">.</span><span class="n">viridis</span><span class="p">)</span>
|
||
<span class="n">ax</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s1">'Time $t$'</span><span class="p">)</span>
|
||
<span class="n">ax</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">'Position $x$'</span><span class="p">);</span>
|
||
|
||
|
||
<span class="n">fig</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span><span class="mi">10</span><span class="p">))</span>
|
||
<span class="n">ax</span> <span class="o">=</span> <span class="n">fig</span><span class="o">.</span><span class="n">add_suplot</span><span class="p">(</span><span class="n">projection</span><span class="o">=</span><span class="s1">'3d'</span><span class="p">)</span>
|
||
<span class="n">ax</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">'Difference'</span><span class="p">)</span>
|
||
<span class="n">s</span> <span class="o">=</span> <span class="n">ax</span><span class="o">.</span><span class="n">plot_surface</span><span class="p">(</span><span class="n">T</span><span class="p">,</span><span class="n">X</span><span class="p">,</span><span class="n">diff</span><span class="p">,</span><span class="n">linewidth</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span><span class="n">antialiased</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span><span class="n">cmap</span><span class="o">=</span><span class="n">cm</span><span class="o">.</span><span class="n">viridis</span><span class="p">)</span>
|
||
<span class="n">ax</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s1">'Time $t$'</span><span class="p">)</span>
|
||
<span class="n">ax</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">'Position $x$'</span><span class="p">);</span>
|
||
|
||
<span class="c1">## Take some slices of the 3D plots just to see the solutions at particular times</span>
|
||
<span class="n">indx1</span> <span class="o">=</span> <span class="mi">0</span>
|
||
<span class="n">indx2</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">Nt</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span>
|
||
<span class="n">indx3</span> <span class="o">=</span> <span class="n">Nt</span><span class="o">-</span><span class="mi">1</span>
|
||
|
||
<span class="n">t1</span> <span class="o">=</span> <span class="n">t</span><span class="p">[</span><span class="n">indx1</span><span class="p">]</span>
|
||
<span class="n">t2</span> <span class="o">=</span> <span class="n">t</span><span class="p">[</span><span class="n">indx2</span><span class="p">]</span>
|
||
<span class="n">t3</span> <span class="o">=</span> <span class="n">t</span><span class="p">[</span><span class="n">indx3</span><span class="p">]</span>
|
||
|
||
<span class="c1"># Slice the results from the DNN</span>
|
||
<span class="n">res1</span> <span class="o">=</span> <span class="n">res</span><span class="p">[:,</span><span class="n">indx1</span><span class="p">]</span>
|
||
<span class="n">res2</span> <span class="o">=</span> <span class="n">res</span><span class="p">[:,</span><span class="n">indx2</span><span class="p">]</span>
|
||
<span class="n">res3</span> <span class="o">=</span> <span class="n">res</span><span class="p">[:,</span><span class="n">indx3</span><span class="p">]</span>
|
||
|
||
<span class="c1"># Slice the analytical results</span>
|
||
<span class="n">res_analytical1</span> <span class="o">=</span> <span class="n">res_analytical</span><span class="p">[:,</span><span class="n">indx1</span><span class="p">]</span>
|
||
<span class="n">res_analytical2</span> <span class="o">=</span> <span class="n">res_analytical</span><span class="p">[:,</span><span class="n">indx2</span><span class="p">]</span>
|
||
<span class="n">res_analytical3</span> <span class="o">=</span> <span class="n">res_analytical</span><span class="p">[:,</span><span class="n">indx3</span><span class="p">]</span>
|
||
|
||
<span class="c1"># Plot the slices</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span><span class="mi">10</span><span class="p">))</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">"Computed solutions at time = </span><span class="si">%g</span><span class="s2">"</span><span class="o">%</span><span class="k">t1</span>)
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">res1</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">res_analytical1</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">([</span><span class="s1">'dnn'</span><span class="p">,</span><span class="s1">'analytical'</span><span class="p">])</span>
|
||
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span><span class="mi">10</span><span class="p">))</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">"Computed solutions at time = </span><span class="si">%g</span><span class="s2">"</span><span class="o">%</span><span class="k">t2</span>)
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">res2</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">res_analytical2</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">([</span><span class="s1">'dnn'</span><span class="p">,</span><span class="s1">'analytical'</span><span class="p">])</span>
|
||
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span><span class="mi">10</span><span class="p">))</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">"Computed solutions at time = </span><span class="si">%g</span><span class="s2">"</span><span class="o">%</span><span class="k">t3</span>)
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">res3</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">res_analytical3</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">([</span><span class="s1">'dnn'</span><span class="p">,</span><span class="s1">'analytical'</span><span class="p">])</span>
|
||
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<div class="section" id="resources-on-differential-equations-and-deep-learning">
|
||
<h2>Resources on differential equations and deep learning<a class="headerlink" href="#resources-on-differential-equations-and-deep-learning" title="Permalink to this headline">¶</a></h2>
|
||
<ol class="simple">
|
||
<li><p><a class="reference external" href="https://pdfs.semanticscholar.org/d061/df393e0e8fbfd0ea24976458b7d42419040d.pdf">Artificial neural networks for solving ordinary and partial differential equations by I.E. Lagaris et al</a></p></li>
|
||
<li><p><a class="reference external" href="https://becominghuman.ai/neural-networks-for-solving-differential-equations-fa230ac5e04c">Neural networks for solving differential equations by A. Honchar</a></p></li>
|
||
<li><p><a class="reference external" href="http://cs229.stanford.edu/proj2013/ChiaramonteKiener-SolvingDifferentialEquationsUsingNeuralNetworks.pdf">Solving differential equations using neural networks by M.M Chiaramonte and M. Kiener</a></p></li>
|
||
<li><p><a class="reference external" href="https://www.springer.com/us/book/9783540225515">Introduction to Partial Differential Equations by A. Tveito, R. Winther</a></p></li>
|
||
</ol>
|
||
</div>
|
||
</div>
|
||
|
||
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