Morten Hjorth-Jensen
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Applied Data Analysis and Machine Learning
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About the course
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Teachers and Grading
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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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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: Statistical interpretation of Linear Regression and Resampling techniques
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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 introduction to Tensor flow
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Exercises weeks 43 and 44
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Week 43: Deep Learning: Constructing a Neural Network code and solving differential equations
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Projects
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Project 1 on Machine Learning, deadline October 9 (midnight), 2023
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Project 2 on Machine Learning, deadline November 13 (Midnight)
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Plans for week 43
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Using Automatic differentiation
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Back propagation and automatic differentiation
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Material for exercises week 43 and week 44
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Writing our first neural network code, testing it for the OR and XOR gates
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<a class="reference internal nav-link" href="#the-and-and-xor-gates">
|
||
The AND and XOR Gates
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#representing-the-data-sets">
|
||
Representing the Data Sets
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#setting-up-the-neural-network">
|
||
Setting up the Neural Network
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#the-code-using-scikit-learn">
|
||
The Code using Scikit-Learn
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#lecture-thursday-october-26">
|
||
Lecture Thursday October 26
|
||
</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="#fine-tuning-neural-network-hyperparameters">
|
||
Fine-tuning neural network hyperparameters
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#hidden-layers">
|
||
Hidden layers
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#which-activation-function-should-i-use">
|
||
Which activation function should I use?
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#is-the-logistic-activation-function-sigmoid-our-choice">
|
||
Is the Logistic activation function (Sigmoid) our choice?
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#the-derivative-of-the-logistic-funtion">
|
||
The derivative of the Logistic funtion
|
||
</a>
|
||
</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="#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="#batch-normalization">
|
||
Batch Normalization
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#dropout">
|
||
Dropout
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#gradient-clipping">
|
||
Gradient Clipping
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#a-very-nice-website-on-neural-networks">
|
||
A very nice website on Neural Networks
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#a-top-down-perspective-on-neural-networks">
|
||
A top-down perspective on Neural networks
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#limitations-of-supervised-learning-with-deep-networks">
|
||
Limitations of supervised learning with deep networks
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#solving-odes-with-deep-learning">
|
||
Solving ODEs with Deep Learning
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#ordinary-differential-equations">
|
||
Ordinary Differential Equations
|
||
</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="#id3">
|
||
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="#id4">
|
||
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="#id5">
|
||
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="#id6">
|
||
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="#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="#material-for-exercises-week-43-and-week-44">
|
||
Material for exercises week 43 and week 44
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#writing-our-first-neural-network-code-testing-it-for-the-or-and-xor-gates">
|
||
Writing our first neural network code, testing it for the OR and XOR gates
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#the-and-and-xor-gates">
|
||
The AND and XOR Gates
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#representing-the-data-sets">
|
||
Representing the Data Sets
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#setting-up-the-neural-network">
|
||
Setting up the Neural Network
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#the-code-using-scikit-learn">
|
||
The Code using Scikit-Learn
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#lecture-thursday-october-26">
|
||
Lecture Thursday October 26
|
||
</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="#fine-tuning-neural-network-hyperparameters">
|
||
Fine-tuning neural network hyperparameters
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#hidden-layers">
|
||
Hidden layers
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#which-activation-function-should-i-use">
|
||
Which activation function should I use?
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#is-the-logistic-activation-function-sigmoid-our-choice">
|
||
Is the Logistic activation function (Sigmoid) our choice?
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#the-derivative-of-the-logistic-funtion">
|
||
The derivative of the Logistic funtion
|
||
</a>
|
||
</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="#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="#batch-normalization">
|
||
Batch Normalization
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#dropout">
|
||
Dropout
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#gradient-clipping">
|
||
Gradient Clipping
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#a-very-nice-website-on-neural-networks">
|
||
A very nice website on Neural Networks
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#a-top-down-perspective-on-neural-networks">
|
||
A top-down perspective on Neural networks
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#limitations-of-supervised-learning-with-deep-networks">
|
||
Limitations of supervised learning with deep networks
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#solving-odes-with-deep-learning">
|
||
Solving ODEs with Deep Learning
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#ordinary-differential-equations">
|
||
Ordinary Differential Equations
|
||
</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="#id3">
|
||
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="#id4">
|
||
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="#id5">
|
||
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="#id6">
|
||
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>Oct 23, 2023</strong></p>
|
||
<p>Copyright 1999-2023, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license</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 active learning sessions on Tuesday and Wednesday.</strong></p>
|
||
<ul class="simple">
|
||
<li><p>Exercise on writing your own neural network code, application to the OR and XOR gates</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>
|
||
<p><strong>Material for the lecture on Thursday October 26, 2023.</strong></p>
|
||
<ul class="simple">
|
||
<li><p>Building our own Feed-forward Neural Network and discussion of project 2, continuation from last week</p></li>
|
||
<li><p>Solving differential equations with Neural Networks and intro to <strong>Tensorflow</strong> with examples.</p></li>
|
||
<li><p>Readings and Videos:</p>
|
||
<ul>
|
||
<li><p>These lecture notes</p></li>
|
||
<li><p><a class="reference external" href="https://github.com/CompPhysics/MachineLearning/blob/master/doc/Textbooks/TensorflowML.pdf">Aurelien Geron’s chapters 10-11</a></p></li>
|
||
<li><p>For a more in depth discussion on neural networks we recommend Goodfellow et al chapters 6 and 7.</p></li>
|
||
<li><p><a class="reference external" href="https://www.youtube.com/watch?v=bxe2T-V8XRs&list=PLiaHhY2iBX9hdHaRr6b7XevZtgZRa1PoU&ab_channel=WelchLabs">Neural Networks demystified</a></p></li>
|
||
<li><p><a class="reference external" href="https://www.youtube.com/watch?v=Wo5dMEP_BbI&list=PLQVvvaa0QuDcjD5BAw2DxE6OF2tius3V3&ab_channel=sentdex">Building Neural Networks from scratch</a></p></li>
|
||
<li><p><a class="reference external" href="https://www.youtube.com/watch?v=CqOfi41LfDw">Video on Neural Networks</a></p></li>
|
||
<li><p><a class="reference external" href="https://www.youtube.com/watch?v=Ilg3gGewQ5U">Video on the back propagation algorithm</a></p></li>
|
||
</ul>
|
||
</li>
|
||
</ul>
|
||
<p>I also recommend Michael Nielsen’s intuitive approach to the neural networks and the universal approximation theorem, see the slides at <a class="reference external" href="http://neuralnetworksanddeeplearning.com/chap4.html">http://neuralnetworksanddeeplearning.com/chap4.html</a>.</p>
|
||
</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>a
|
||
In our discussions of ordinary differential equations
|
||
we will also study the usage of <a class="reference external" href="https://www.youtube.com/watch?v=fRf4l5qaX1M&ab_channel=AlexSmola">Autograd</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 <a class="reference external" href="https://github.com/HIPS/autograd">Autograd documentation</a>.
|
||
t</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 URL”:<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="material-for-exercises-week-43-and-week-44">
|
||
<h2>Material for exercises week 43 and week 44<a class="headerlink" href="#material-for-exercises-week-43-and-week-44" title="Permalink to this headline">¶</a></h2>
|
||
</div>
|
||
<div class="section" id="writing-our-first-neural-network-code-testing-it-for-the-or-and-xor-gates">
|
||
<h2>Writing our first neural network code, testing it for the OR and XOR gates<a class="headerlink" href="#writing-our-first-neural-network-code-testing-it-for-the-or-and-xor-gates" title="Permalink to this headline">¶</a></h2>
|
||
<p>During week 41 we discussed three different types of gates, the
|
||
so-called XOR, the OR and the AND gates. In order to develop a code
|
||
for neural networks, it can be useful to set up a simpler system with
|
||
only two inputs and one output. This can make it easier to debug and
|
||
study the feed forward pass and the back propagation part. In the
|
||
exercise this and next week, we propose to study this system with just
|
||
one hidden layer and two hidden nodes. There is only one output node
|
||
and we can choose to use either a simple regression case (fitting a
|
||
line) or just a binary classification case with the corss-entropy as
|
||
cost function.</p>
|
||
<p>Their inputs and outputs can be
|
||
summarized using the following tables, first for the OR gate with
|
||
inputs <span class="math notranslate nohighlight">\(x_1\)</span> and <span class="math notranslate nohighlight">\(x_2\)</span> and outputs <span class="math notranslate nohighlight">\(y\)</span>:</p>
|
||
<table class="dotable" border="1">
|
||
<thead>
|
||
<tr><th align="center">$x_1$</th> <th align="center">$x_2$</th> <th align="center">$y$</th> </tr>
|
||
</thead>
|
||
<tbody>
|
||
<tr><td align="center"> 0 </td> <td align="center"> 0 </td> <td align="center"> 0 </td> </tr>
|
||
<tr><td align="center"> 0 </td> <td align="center"> 1 </td> <td align="center"> 1 </td> </tr>
|
||
<tr><td align="center"> 1 </td> <td align="center"> 0 </td> <td align="center"> 1 </td> </tr>
|
||
<tr><td align="center"> 1 </td> <td align="center"> 1 </td> <td align="center"> 1 </td> </tr>
|
||
</tbody>
|
||
</table></div>
|
||
<div class="section" id="the-and-and-xor-gates">
|
||
<h2>The AND and XOR Gates<a class="headerlink" href="#the-and-and-xor-gates" title="Permalink to this headline">¶</a></h2>
|
||
<p>The AND gate is defined as</p>
|
||
<table class="dotable" border="1">
|
||
<thead>
|
||
<tr><th align="center">$x_1$</th> <th align="center">$x_2$</th> <th align="center">$y$</th> </tr>
|
||
</thead>
|
||
<tbody>
|
||
<tr><td align="center"> 0 </td> <td align="center"> 0 </td> <td align="center"> 0 </td> </tr>
|
||
<tr><td align="center"> 0 </td> <td align="center"> 1 </td> <td align="center"> 0 </td> </tr>
|
||
<tr><td align="center"> 1 </td> <td align="center"> 0 </td> <td align="center"> 0 </td> </tr>
|
||
<tr><td align="center"> 1 </td> <td align="center"> 1 </td> <td align="center"> 1 </td> </tr>
|
||
</tbody>
|
||
</table>
|
||
<p>And finally we have the XOR gate</p>
|
||
<table class="dotable" border="1">
|
||
<thead>
|
||
<tr><th align="center">$x_1$</th> <th align="center">$x_2$</th> <th align="center">$y$</th> </tr>
|
||
</thead>
|
||
<tbody>
|
||
<tr><td align="center"> 0 </td> <td align="center"> 0 </td> <td align="center"> 0 </td> </tr>
|
||
<tr><td align="center"> 0 </td> <td align="center"> 1 </td> <td align="center"> 1 </td> </tr>
|
||
<tr><td align="center"> 1 </td> <td align="center"> 0 </td> <td align="center"> 1 </td> </tr>
|
||
<tr><td align="center"> 1 </td> <td align="center"> 1 </td> <td align="center"> 0 </td> </tr>
|
||
</tbody>
|
||
</table></div>
|
||
<div class="section" id="representing-the-data-sets">
|
||
<h2>Representing the Data Sets<a class="headerlink" href="#representing-the-data-sets" title="Permalink to this headline">¶</a></h2>
|
||
<p>Our design matrix is defined by the input values <span class="math notranslate nohighlight">\(x_1\)</span> and <span class="math notranslate nohighlight">\(x_2\)</span>. Since we have four possible outputs, our design matrix reads</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[\begin{split}
|
||
\boldsymbol{X}=\begin{bmatrix} 0 & 0 \\
|
||
0 & 1 \\
|
||
1 & 0 \\
|
||
1 & 1 \end{bmatrix},
|
||
\end{split}\]</div>
|
||
<p>while the vector of outputs is <span class="math notranslate nohighlight">\(\boldsymbol{y}^T=[0,1,1,0]\)</span> for the XOR gate, <span class="math notranslate nohighlight">\(\boldsymbol{y}^T=[0,0,0,1]\)</span> for the AND gate and <span class="math notranslate nohighlight">\(\boldsymbol{y}^T=[0,1,1,1]\)</span> for the OR gate.</p>
|
||
</div>
|
||
<div class="section" id="setting-up-the-neural-network">
|
||
<h2>Setting up the Neural Network<a class="headerlink" href="#setting-up-the-neural-network" title="Permalink to this headline">¶</a></h2>
|
||
<p>We define first our design matrix and the various output vectors for the different gates.</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="sd">"""</span>
|
||
<span class="sd">Simple code that tests XOR, OR and AND gates with linear regression</span>
|
||
<span class="sd">"""</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">from</span> <span class="nn">sklearn</span> <span class="kn">import</span> <span class="n">datasets</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">probabilities</span> <span class="o">=</span> <span class="n">sigmoid</span><span class="p">(</span><span class="n">z_o</span><span class="p">)</span>
|
||
<span class="k">return</span> <span class="n">probabilities</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="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"># Design matrix</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="mi">1</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="c1"># The OR gate</span>
|
||
<span class="n">yOR</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">1</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="c1"># The AND gate</span>
|
||
<span class="n">yAND</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="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span>
|
||
|
||
<span class="c1"># Defining the 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</span><span class="o">.</span><span class="n">shape</span>
|
||
<span class="n">n_hidden_neurons</span> <span class="o">=</span> <span class="mi">2</span>
|
||
<span class="n">n_categories</span> <span class="o">=</span> <span class="mi">2</span>
|
||
<span class="n">n_features</span> <span class="o">=</span> <span class="mi">2</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>
|
||
|
||
<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="nb">print</span><span class="p">(</span><span class="n">probabilities</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</span><span class="p">)</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="n">predictions</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>[[0.80625657 0.36420967]
|
||
[0.90297441 0.30170017]
|
||
[0.89823921 0.28566769]
|
||
[0.93420126 0.25920793]]
|
||
[0 0 0 0]
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<p>Not an impressive result, but this was our first forward pass with randomly assigned weights. Let us now add the full network with the back-propagation algorithm discussed above.</p>
|
||
</div>
|
||
<div class="section" id="the-code-using-scikit-learn">
|
||
<h2>The Code using Scikit-Learn<a class="headerlink" href="#the-code-using-scikit-learn" 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"># 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.neural_network</span> <span class="kn">import</span> <span class="n">MLPClassifier</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="kn">import</span> <span class="nn">seaborn</span> <span class="k">as</span> <span class="nn">sns</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"># Design matrix</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="mi">1</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="c1"># The OR gate</span>
|
||
<span class="n">yOR</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">1</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="c1"># The AND gate</span>
|
||
<span class="n">yAND</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="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span>
|
||
|
||
<span class="c1"># Defining the 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</span><span class="o">.</span><span class="n">shape</span>
|
||
<span class="n">n_hidden_neurons</span> <span class="o">=</span> <span class="mi">2</span>
|
||
<span class="n">n_categories</span> <span class="o">=</span> <span class="mi">2</span>
|
||
<span class="n">n_features</span> <span class="o">=</span> <span class="mi">2</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 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="n">epochs</span> <span class="o">=</span> <span class="mi">100</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</span><span class="p">,</span> <span class="n">yXOR</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 data 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</span><span class="p">,</span> <span class="n">yXOR</span><span class="p">))</span>
|
||
<span class="nb">print</span><span class="p">()</span>
|
||
|
||
<span class="n">sns</span><span class="o">.</span><span class="n">set</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">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</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">yXOR</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">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 stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Learning rate = 1e-05
|
||
Lambda = 1e-05
|
||
Accuracy score on data set: 0.5
|
||
|
||
Learning rate = 1e-05
|
||
Lambda = 0.0001
|
||
Accuracy score on data set: 0.5
|
||
|
||
Learning rate = 1e-05
|
||
Lambda = 0.001
|
||
Accuracy score on data set: 0.5
|
||
|
||
Learning rate = 1e-05
|
||
Lambda = 0.01
|
||
Accuracy score on data set: 0.5
|
||
|
||
Learning rate = 1e-05
|
||
Lambda = 0.1
|
||
Accuracy score on data set: 0.5
|
||
|
||
Learning rate = 1e-05
|
||
Lambda = 1.0
|
||
Accuracy score on data set: 0.5
|
||
|
||
Learning rate = 1e-05
|
||
Lambda = 10.0
|
||
Accuracy score on data set: 0.5
|
||
|
||
Learning rate = 0.0001
|
||
Lambda = 1e-05
|
||
Accuracy score on data set: 0.5
|
||
|
||
Learning rate = 0.0001
|
||
Lambda = 0.0001
|
||
Accuracy score on data set: 0.5
|
||
|
||
Learning rate = 0.0001
|
||
Lambda = 0.001
|
||
Accuracy score on data set: 0.5
|
||
|
||
Learning rate = 0.0001
|
||
Lambda = 0.01
|
||
Accuracy score on data set: 0.5
|
||
|
||
Learning rate = 0.0001
|
||
Lambda = 0.1
|
||
Accuracy score on data set: 0.5
|
||
|
||
Learning rate = 0.0001
|
||
Lambda = 1.0
|
||
Accuracy score on data set: 0.5
|
||
|
||
Learning rate = 0.0001
|
||
Lambda = 10.0
|
||
Accuracy score on data set: 0.5
|
||
|
||
Learning rate = 0.001
|
||
Lambda = 1e-05
|
||
Accuracy score on data set: 0.5
|
||
|
||
Learning rate = 0.001
|
||
Lambda = 0.0001
|
||
Accuracy score on data set: 0.5
|
||
|
||
Learning rate = 0.001
|
||
Lambda = 0.001
|
||
Accuracy score on data set: 0.5
|
||
|
||
Learning rate = 0.001
|
||
Lambda = 0.01
|
||
Accuracy score on data set: 0.5
|
||
|
||
Learning rate = 0.001
|
||
Lambda = 0.1
|
||
Accuracy score on data set: 0.5
|
||
|
||
Learning rate = 0.001
|
||
Lambda = 1.0
|
||
Accuracy score on data set: 0.5
|
||
|
||
Learning rate = 0.001
|
||
Lambda = 10.0
|
||
Accuracy score on data set: 0.5
|
||
|
||
Learning rate = 0.01
|
||
Lambda = 1e-05
|
||
Accuracy score on data set: 0.25
|
||
|
||
Learning rate = 0.01
|
||
Lambda = 0.0001
|
||
Accuracy score on data set: 0.75
|
||
|
||
Learning rate = 0.01
|
||
Lambda = 0.001
|
||
Accuracy score on data set: 0.5
|
||
|
||
Learning rate = 0.01
|
||
Lambda = 0.01
|
||
Accuracy score on data set: 0.75
|
||
|
||
Learning rate = 0.01
|
||
Lambda = 0.1
|
||
Accuracy score on data set: 0.5
|
||
|
||
Learning rate = 0.01
|
||
Lambda = 1.0
|
||
Accuracy score on data set: 0.5
|
||
|
||
Learning rate = 0.01
|
||
Lambda = 10.0
|
||
Accuracy score on data set: 0.5
|
||
|
||
Learning rate = 0.1
|
||
Lambda = 1e-05
|
||
Accuracy score on data set: 0.5
|
||
|
||
Learning rate = 0.1
|
||
Lambda = 0.0001
|
||
Accuracy score on data set: 0.5
|
||
|
||
Learning rate = 0.1
|
||
Lambda = 0.001
|
||
Accuracy score on data set: 1.0
|
||
|
||
Learning rate = 0.1
|
||
Lambda = 0.01
|
||
Accuracy score on data set: 1.0
|
||
|
||
Learning rate = 0.1
|
||
Lambda = 0.1
|
||
Accuracy score on data set: 0.5
|
||
|
||
Learning rate = 0.1
|
||
Lambda = 1.0
|
||
Accuracy score on data set: 0.5
|
||
|
||
Learning rate = 0.1
|
||
Lambda = 10.0
|
||
Accuracy score on data set: 0.5
|
||
|
||
Learning rate = 1.0
|
||
Lambda = 1e-05
|
||
Accuracy score on data set: 0.75
|
||
|
||
Learning rate = 1.0
|
||
Lambda = 0.0001
|
||
Accuracy score on data set: 0.75
|
||
|
||
Learning rate = 1.0
|
||
Lambda = 0.001
|
||
Accuracy score on data set: 0.75
|
||
|
||
Learning rate = 1.0
|
||
Lambda = 0.01
|
||
Accuracy score on data set: 0.5
|
||
|
||
Learning rate = 1.0
|
||
Lambda = 0.1
|
||
Accuracy score on data set: 0.5
|
||
|
||
Learning rate = 1.0
|
||
Lambda = 1.0
|
||
Accuracy score on data set: 0.5
|
||
|
||
Learning rate = 1.0
|
||
Lambda = 10.0
|
||
Accuracy score on data set: 0.5
|
||
|
||
Learning rate = 10.0
|
||
Lambda = 1e-05
|
||
Accuracy score on data set: 0.5
|
||
|
||
Learning rate = 10.0
|
||
Lambda = 0.0001
|
||
Accuracy score on data set: 0.5
|
||
|
||
Learning rate = 10.0
|
||
Lambda = 0.001
|
||
Accuracy score on data set: 0.5
|
||
|
||
Learning rate = 10.0
|
||
Lambda = 0.01
|
||
Accuracy score on data set: 0.5
|
||
|
||
Learning rate = 10.0
|
||
Lambda = 0.1
|
||
Accuracy score on data set: 0.5
|
||
|
||
Learning rate = 10.0
|
||
Lambda = 1.0
|
||
Accuracy score on data set: 0.5
|
||
|
||
Learning rate = 10.0
|
||
Lambda = 10.0
|
||
Accuracy score on data set: 0.5
|
||
</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:692: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
|
||
warnings.warn(
|
||
/Users/mhjensen/miniforge3/envs/myenv/lib/python3.9/site-packages/sklearn/neural_network/_multilayer_perceptron.py:692: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
|
||
warnings.warn(
|
||
/Users/mhjensen/miniforge3/envs/myenv/lib/python3.9/site-packages/sklearn/neural_network/_multilayer_perceptron.py:692: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
|
||
warnings.warn(
|
||
/Users/mhjensen/miniforge3/envs/myenv/lib/python3.9/site-packages/sklearn/neural_network/_multilayer_perceptron.py:692: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
|
||
warnings.warn(
|
||
/Users/mhjensen/miniforge3/envs/myenv/lib/python3.9/site-packages/sklearn/neural_network/_multilayer_perceptron.py:692: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
|
||
warnings.warn(
|
||
/Users/mhjensen/miniforge3/envs/myenv/lib/python3.9/site-packages/sklearn/neural_network/_multilayer_perceptron.py:692: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
|
||
warnings.warn(
|
||
/Users/mhjensen/miniforge3/envs/myenv/lib/python3.9/site-packages/sklearn/neural_network/_multilayer_perceptron.py:692: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
|
||
warnings.warn(
|
||
/Users/mhjensen/miniforge3/envs/myenv/lib/python3.9/site-packages/sklearn/neural_network/_multilayer_perceptron.py:692: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
|
||
warnings.warn(
|
||
/Users/mhjensen/miniforge3/envs/myenv/lib/python3.9/site-packages/sklearn/neural_network/_multilayer_perceptron.py:692: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
|
||
warnings.warn(
|
||
/Users/mhjensen/miniforge3/envs/myenv/lib/python3.9/site-packages/sklearn/neural_network/_multilayer_perceptron.py:692: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
|
||
warnings.warn(
|
||
</pre></div>
|
||
</div>
|
||
<img alt="_images/week43_15_2.png" src="_images/week43_15_2.png" />
|
||
</div>
|
||
</div>
|
||
<p>How do we interpret these results?</p>
|
||
</div>
|
||
<div class="section" id="lecture-thursday-october-26">
|
||
<h2>Lecture Thursday October 26<a class="headerlink" href="#lecture-thursday-october-26" title="Permalink to this headline">¶</a></h2>
|
||
</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>We repeat some of the elements discussed last week. The first part of
|
||
the material for Thursday was contained in the slides for last
|
||
week as well. We will repeat some of the topics here before we move into
|
||
applications to differential equations and other examples.</p>
|
||
<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="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_20_1.png" src="_images/week43_20_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_87501/953065564.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_87501/953065564.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_87501/953065564.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_87501/953065564.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_87501/953065564.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_87501/953065564.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_87501/953065564.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_87501/953065564.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_87501/953065564.py:4: RuntimeWarning: overflow encountered in exp
|
||
return 1/(1 + np.exp(-x))
|
||
/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_87501/1630775253.py:43: RuntimeWarning: overflow encountered in exp
|
||
exp_term = np.exp(self.z_o)
|
||
/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_87501/1630775253.py:44: RuntimeWarning: invalid value encountered in true_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_87501/953065564.py:4: RuntimeWarning: overflow encountered in exp
|
||
return 1/(1 + np.exp(-x))
|
||
/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_87501/1630775253.py:43: RuntimeWarning: overflow encountered in exp
|
||
exp_term = np.exp(self.z_o)
|
||
/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_87501/1630775253.py:44: RuntimeWarning: invalid value encountered in true_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_87501/953065564.py:4: RuntimeWarning: overflow encountered in exp
|
||
return 1/(1 + np.exp(-x))
|
||
/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_87501/1630775253.py:43: RuntimeWarning: overflow encountered in exp
|
||
exp_term = np.exp(self.z_o)
|
||
/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_87501/1630775253.py:44: RuntimeWarning: invalid value encountered in true_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_87501/953065564.py:4: RuntimeWarning: overflow encountered in exp
|
||
return 1/(1 + np.exp(-x))
|
||
/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_87501/1630775253.py:43: RuntimeWarning: overflow encountered in exp
|
||
exp_term = np.exp(self.z_o)
|
||
/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_87501/1630775253.py:44: RuntimeWarning: invalid value encountered in true_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_87501/953065564.py:4: RuntimeWarning: overflow encountered in exp
|
||
return 1/(1 + np.exp(-x))
|
||
/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_87501/1630775253.py:43: RuntimeWarning: overflow encountered in exp
|
||
exp_term = np.exp(self.z_o)
|
||
/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_87501/1630775253.py:44: RuntimeWarning: invalid value encountered in true_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_87501/953065564.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_87501/953065564.py:4: RuntimeWarning: overflow encountered in exp
|
||
return 1/(1 + np.exp(-x))
|
||
/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_87501/1630775253.py:43: RuntimeWarning: overflow encountered in exp
|
||
exp_term = np.exp(self.z_o)
|
||
/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_87501/1630775253.py:44: RuntimeWarning: invalid value encountered in true_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_87501/953065564.py:4: RuntimeWarning: overflow encountered in exp
|
||
return 1/(1 + np.exp(-x))
|
||
/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_87501/1630775253.py:43: RuntimeWarning: overflow encountered in exp
|
||
exp_term = np.exp(self.z_o)
|
||
/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_87501/1630775253.py:44: RuntimeWarning: invalid value encountered in true_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_87501/953065564.py:4: RuntimeWarning: overflow encountered in exp
|
||
return 1/(1 + np.exp(-x))
|
||
/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_87501/1630775253.py:43: RuntimeWarning: overflow encountered in exp
|
||
exp_term = np.exp(self.z_o)
|
||
/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_87501/1630775253.py:44: RuntimeWarning: invalid value encountered in true_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_87501/953065564.py:4: RuntimeWarning: overflow encountered in exp
|
||
return 1/(1 + np.exp(-x))
|
||
/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_87501/1630775253.py:43: RuntimeWarning: overflow encountered in exp
|
||
exp_term = np.exp(self.z_o)
|
||
/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_87501/1630775253.py:44: RuntimeWarning: invalid value encountered in true_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_87501/953065564.py:4: RuntimeWarning: overflow encountered in exp
|
||
return 1/(1 + np.exp(-x))
|
||
/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_87501/1630775253.py:43: RuntimeWarning: overflow encountered in exp
|
||
exp_term = np.exp(self.z_o)
|
||
/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_87501/1630775253.py:44: RuntimeWarning: invalid value encountered in true_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_87501/953065564.py:4: RuntimeWarning: overflow encountered in exp
|
||
return 1/(1 + np.exp(-x))
|
||
/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_87501/1630775253.py:43: RuntimeWarning: overflow encountered in exp
|
||
exp_term = np.exp(self.z_o)
|
||
/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_87501/1630775253.py:44: RuntimeWarning: invalid value encountered in true_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_87501/953065564.py:4: RuntimeWarning: overflow encountered in exp
|
||
return 1/(1 + np.exp(-x))
|
||
/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_87501/1630775253.py:43: RuntimeWarning: overflow encountered in exp
|
||
exp_term = np.exp(self.z_o)
|
||
/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_87501/1630775253.py:44: RuntimeWarning: invalid value encountered in true_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_87501/953065564.py:4: RuntimeWarning: overflow encountered in exp
|
||
return 1/(1 + np.exp(-x))
|
||
/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_87501/1630775253.py:43: RuntimeWarning: overflow encountered in exp
|
||
exp_term = np.exp(self.z_o)
|
||
/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_87501/1630775253.py:44: RuntimeWarning: invalid value encountered in true_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_87501/953065564.py:4: RuntimeWarning: overflow encountered in exp
|
||
return 1/(1 + np.exp(-x))
|
||
/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_87501/953065564.py:4: RuntimeWarning: overflow encountered in exp
|
||
return 1/(1 + np.exp(-x))
|
||
/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_87501/953065564.py:4: RuntimeWarning: overflow encountered in exp
|
||
return 1/(1 + np.exp(-x))
|
||
/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_87501/953065564.py:4: RuntimeWarning: overflow encountered in exp
|
||
return 1/(1 + np.exp(-x))
|
||
/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_87501/953065564.py:4: RuntimeWarning: overflow encountered in exp
|
||
return 1/(1 + np.exp(-x))
|
||
</pre></div>
|
||
</div>
|
||
<img alt="_images/week43_43_1.png" src="_images/week43_43_1.png" />
|
||
<img alt="_images/week43_43_2.png" src="_images/week43_43_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:692: 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:692: 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:692: 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:692: 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:692: 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:692: 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:692: 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:692: 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:692: 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:692: 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:692: 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:692: 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:692: 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:692: 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:692: 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:692: 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:692: 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:692: 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:692: 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:692: 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:692: 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:692: 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
|
||
|
||
Learning rate = 1.0
|
||
Lambda = 0.1
|
||
Accuracy score on test set: 0.08333333333333333
|
||
</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: 0.08888888888888889
|
||
|
||
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_47_0.png" src="_images/week43_47_0.png" />
|
||
<img alt="_images/week43_47_1.png" src="_images/week43_47_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">Input</span> <span class="n">In</span> <span class="p">[</span><span class="mi">14</span><span class="p">]</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 our examples..</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">lr</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>
|
||
<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">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">lr</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="fine-tuning-neural-network-hyperparameters">
|
||
<h2>Fine-tuning neural network hyperparameters<a class="headerlink" href="#fine-tuning-neural-network-hyperparameters" title="Permalink to this headline">¶</a></h2>
|
||
<p>The flexibility of neural networks is also one of their main
|
||
drawbacks: there are many hyperparameters to tweak. Not only can you
|
||
use any imaginable network topology (how neurons/nodes are interconnected),
|
||
but even in a simple FFNN you can change the number of layers, the
|
||
number of neurons per layer, the type of activation function to use in
|
||
each layer, the weight initialization logic, the stochastic gradient optmized and much more. How do you
|
||
know what combination of hyperparameters is the best for your task?</p>
|
||
<ul class="simple">
|
||
<li><p>You can use grid search with cross-validation to find the right hyperparameters.</p></li>
|
||
</ul>
|
||
<p>However,since there are many hyperparameters to tune, and since
|
||
training a neural network on a large dataset takes a lot of time, you
|
||
will only be able to explore a tiny part of the hyperparameter space.</p>
|
||
<ul class="simple">
|
||
<li><p>You can use randomized search.</p></li>
|
||
<li><p>Or use tools like <a class="reference external" href="http://oscar.calldesk.ai/">Oscar</a>, which implements more complex algorithms to help you find a good set of hyperparameters quickly.</p></li>
|
||
</ul>
|
||
</div>
|
||
<div class="section" id="hidden-layers">
|
||
<h2>Hidden layers<a class="headerlink" href="#hidden-layers" title="Permalink to this headline">¶</a></h2>
|
||
<p>For many problems you can start with just one or two hidden layers and it will work just fine.
|
||
For the MNIST data set you ca easily get a high accuracy using just one hidden layer with a
|
||
few hundred neurons.
|
||
You can reach for this data set above 98% accuracy using two hidden layers with the same total amount of
|
||
neurons, in roughly the same amount of training time.</p>
|
||
<p>For more complex problems, you can gradually
|
||
ramp up the number of hidden layers, until you start overfitting the training set. Very complex tasks, such
|
||
as large image classification or speech recognition, typically require networks with dozens of layers
|
||
and they need a huge amount
|
||
of training data. However, you will rarely have to train such networks from scratch: it is much more
|
||
common to reuse parts of a pretrained state-of-the-art network that performs a similar task.</p>
|
||
</div>
|
||
<div class="section" id="which-activation-function-should-i-use">
|
||
<h2>Which activation function should I use?<a class="headerlink" href="#which-activation-function-should-i-use" title="Permalink to this headline">¶</a></h2>
|
||
<p>The Back propagation algorithm we derived above works by going from
|
||
the output layer to the input layer, propagating the error gradient on
|
||
the way. Once the algorithm has computed the gradient of the cost
|
||
function with regards to each parameter in the network, it uses these
|
||
gradients to update each parameter with a Gradient Descent (GD) step.</p>
|
||
<p>Unfortunately for us, the gradients often get smaller and smaller as the
|
||
algorithm progresses down to the first hidden layers. As a result, the
|
||
GD update leaves the lower layer connection weights
|
||
virtually unchanged, and training never converges to a good
|
||
solution. This is known in the literature as
|
||
<strong>the vanishing gradients problem</strong>.</p>
|
||
<p>In other cases, the opposite can happen, namely the the gradients can grow bigger and
|
||
bigger. The result is that many of the layers get large updates of the
|
||
weights the
|
||
algorithm diverges. This is the <strong>exploding gradients problem</strong>, which is
|
||
mostly encountered in recurrent neural networks. More generally, deep
|
||
neural networks suffer from unstable gradients, different layers may
|
||
learn at widely different speeds</p>
|
||
</div>
|
||
<div class="section" id="is-the-logistic-activation-function-sigmoid-our-choice">
|
||
<h2>Is the Logistic activation function (Sigmoid) our choice?<a class="headerlink" href="#is-the-logistic-activation-function-sigmoid-our-choice" title="Permalink to this headline">¶</a></h2>
|
||
<p>Although this unfortunate behavior has been empirically observed for
|
||
quite a while (it was one of the reasons why deep neural networks were
|
||
mostly abandoned for a long time), it is only around 2010 that
|
||
significant progress was made in understanding it.</p>
|
||
<p>A paper titled <a class="reference external" href="http://proceedings.mlr.press/v9/glorot10a.html">Understanding the Difficulty of Training Deep
|
||
Feedforward Neural Networks by Xavier Glorot and Yoshua Bengio</a> found that
|
||
the problems with the popular logistic
|
||
sigmoid activation function and the weight initialization technique
|
||
that was most popular at the time, namely random initialization using
|
||
a normal distribution with a mean of 0 and a standard deviation of
|
||
1.</p>
|
||
<p>They showed that with this activation function and this
|
||
initialization scheme, the variance of the outputs of each layer is
|
||
much greater than the variance of its inputs. Going forward in the
|
||
network, the variance keeps increasing after each layer until the
|
||
activation function saturates at the top layers. This is actually made
|
||
worse by the fact that the logistic function has a mean of 0.5, not 0
|
||
(the hyperbolic tangent function has a mean of 0 and behaves slightly
|
||
better than the logistic function in deep networks).</p>
|
||
</div>
|
||
<div class="section" id="the-derivative-of-the-logistic-funtion">
|
||
<h2>The derivative of the Logistic funtion<a class="headerlink" href="#the-derivative-of-the-logistic-funtion" title="Permalink to this headline">¶</a></h2>
|
||
<p>Looking at the logistic activation function, when inputs become large
|
||
(negative or positive), the function saturates at 0 or 1, with a
|
||
derivative extremely close to 0. Thus when backpropagation kicks in,
|
||
it has virtually no gradient to propagate back through the network,
|
||
and what little gradient exists keeps getting diluted as
|
||
backpropagation progresses down through the top layers, so there is
|
||
really nothing left for the lower layers.</p>
|
||
<p>In their paper, Glorot and Bengio propose a way to significantly
|
||
alleviate this problem. We need the signal to flow properly in both
|
||
directions: in the forward direction when making predictions, and in
|
||
the reverse direction when backpropagating gradients. We don’t want
|
||
the signal to die out, nor do we want it to explode and saturate. For
|
||
the signal to flow properly, the authors argue that we need the
|
||
variance of the outputs of each layer to be equal to the variance of
|
||
its inputs, and we also need the gradients to have equal variance
|
||
before and after flowing through a layer in the reverse direction.</p>
|
||
<p>One of the insights in the 2010 paper by Glorot and Bengio was that
|
||
the vanishing/exploding gradients problems were in part due to a poor
|
||
choice of activation function. Until then most people had assumed that
|
||
if Nature had chosen to use roughly sigmoid activation functions in
|
||
biological neurons, they must be an excellent choice. But it turns out
|
||
that other activation functions behave much better in deep neural
|
||
networks, in particular the ReLU activation function, mostly because
|
||
it does not saturate for positive values (and also because it is quite
|
||
fast to compute).</p>
|
||
</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>
|
||
<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="batch-normalization">
|
||
<h2>Batch Normalization<a class="headerlink" href="#batch-normalization" title="Permalink to this headline">¶</a></h2>
|
||
<p>Batch Normalization
|
||
aims to address the vanishing/exploding gradients problems, and more generally the problem that the
|
||
distribution of each layer’s inputs changes during training, as the parameters of the previous layers change.</p>
|
||
<p>The technique consists of adding an operation in the model just before the activation function of each
|
||
layer, simply zero-centering and normalizing the inputs, then scaling and shifting the result using two new
|
||
parameters per layer (one for scaling, the other for shifting). In other words, this operation lets the model
|
||
learn the optimal scale and mean of the inputs for each layer.
|
||
In order to zero-center and normalize the inputs, the algorithm needs to estimate the inputs’ mean and
|
||
standard deviation. It does so by evaluating the mean and standard deviation of the inputs over the current
|
||
mini-batch, from this the name batch normalization.</p>
|
||
</div>
|
||
<div class="section" id="dropout">
|
||
<h2>Dropout<a class="headerlink" href="#dropout" title="Permalink to this headline">¶</a></h2>
|
||
<p>It is a fairly simple algorithm: at every training step, every neuron (including the input neurons but
|
||
excluding the output neurons) has a probability <span class="math notranslate nohighlight">\(p\)</span> of being temporarily dropped out, meaning it will be
|
||
entirely ignored during this training step, but it may be active during the next step.</p>
|
||
<p>The
|
||
hyperparameter <span class="math notranslate nohighlight">\(p\)</span> is called the dropout rate, and it is typically set to 50%. After training, the neurons are not dropped anymore.
|
||
It is viewed as one of the most popular regularization techniques.</p>
|
||
</div>
|
||
<div class="section" id="gradient-clipping">
|
||
<h2>Gradient Clipping<a class="headerlink" href="#gradient-clipping" title="Permalink to this headline">¶</a></h2>
|
||
<p>A popular technique to lessen the exploding gradients problem is to simply clip the gradients during
|
||
backpropagation so that they never exceed some threshold (this is mostly useful for recurrent neural
|
||
networks).</p>
|
||
<p>This technique is called Gradient Clipping.</p>
|
||
<p>In general however, Batch
|
||
Normalization is preferred.</p>
|
||
</div>
|
||
<div class="section" id="a-very-nice-website-on-neural-networks">
|
||
<h2>A very nice website on Neural Networks<a class="headerlink" href="#a-very-nice-website-on-neural-networks" title="Permalink to this headline">¶</a></h2>
|
||
<p>You may find this <a class="reference external" href="https://playground.tensorflow.org/#activation=tanh&batchSize=10&dataset=circle%C2%AEDataset=reg-plane&learningRate=0.03%C2%AEularizationRate=0&noise=0&networkShape=4,2&seed=0.29243&showTestData=false&discretize=false&percTrainData=50&x=true&y=true&xTimesY=false&xSquared=false&ySquared=false&cosX=false&sinX=false&cosY=false&sinY=false&collectStats=false&problem=classification&initZero=false&hideText=false">website</a> very useful.</p>
|
||
</div>
|
||
<div class="section" id="a-top-down-perspective-on-neural-networks">
|
||
<h2>A top-down perspective on Neural networks<a class="headerlink" href="#a-top-down-perspective-on-neural-networks" title="Permalink to this headline">¶</a></h2>
|
||
<p>The first thing we would like to do is divide the data into two or three
|
||
parts. A training set, a validation or dev (development) set, and a
|
||
test set. The test set is the data on which we want to make
|
||
predictions. The dev set is a subset of the training data we use to
|
||
check how well we are doing out-of-sample, after training the model on
|
||
the training dataset. We use the validation error as a proxy for the
|
||
test error in order to make tweaks to our model. It is crucial that we
|
||
do not use any of the test data to train the algorithm. This is a
|
||
cardinal sin in ML. Then:</p>
|
||
<ul class="simple">
|
||
<li><p>Estimate optimal error rate</p></li>
|
||
<li><p>Minimize underfitting (bias) on training data set.</p></li>
|
||
<li><p>Make sure you are not overfitting.</p></li>
|
||
</ul>
|
||
<p>If the validation and test sets are drawn from the same distributions,
|
||
then a good performance on the validation set should lead to similarly
|
||
good performance on the test set.</p>
|
||
<p>However, sometimes
|
||
the training data and test data differ in subtle ways because, for
|
||
example, they are collected using slightly different methods, or
|
||
because it is cheaper to collect data in one way versus another. In
|
||
this case, there can be a mismatch between the training and test
|
||
data. This can lead to the neural network overfitting these small
|
||
differences between the test and training sets, and a poor performance
|
||
on the test set despite having a good performance on the validation
|
||
set. To rectify this, Andrew Ng suggests making two validation or dev
|
||
sets, one constructed from the training data and one constructed from
|
||
the test data. The difference between the performance of the algorithm
|
||
on these two validation sets quantifies the train-test mismatch. This
|
||
can serve as another important diagnostic when using DNNs for
|
||
supervised learning.</p>
|
||
</div>
|
||
<div class="section" id="limitations-of-supervised-learning-with-deep-networks">
|
||
<h2>Limitations of supervised learning with deep networks<a class="headerlink" href="#limitations-of-supervised-learning-with-deep-networks" title="Permalink to this headline">¶</a></h2>
|
||
<p>Like all statistical methods, supervised learning using neural
|
||
networks has important limitations. This is especially important when
|
||
one seeks to apply these methods, especially to physics problems. Like
|
||
all tools, DNNs are not a universal solution. Often, the same or
|
||
better performance on a task can be achieved by using a few
|
||
hand-engineered features (or even a collection of random
|
||
features).</p>
|
||
<p>Here we list some of the important limitations of supervised neural network based models.</p>
|
||
<ul class="simple">
|
||
<li><p><strong>Need labeled data</strong>. All supervised learning methods, DNNs for supervised learning require labeled data. Often, labeled data is harder to acquire than unlabeled data (e.g. one must pay for human experts to label images).</p></li>
|
||
<li><p><strong>Supervised neural networks are extremely data intensive.</strong> DNNs are data hungry. They perform best when data is plentiful. This is doubly so for supervised methods where the data must also be labeled. The utility of DNNs is extremely limited if data is hard to acquire or the datasets are small (hundreds to a few thousand samples). In this case, the performance of other methods that utilize hand-engineered features can exceed that of DNNs.</p></li>
|
||
<li><p><strong>Homogeneous data.</strong> Almost all DNNs deal with homogeneous data of one type. It is very hard to design architectures that mix and match data types (i.e. some continuous variables, some discrete variables, some time series). In applications beyond images, video, and language, this is often what is required. In contrast, ensemble models like random forests or gradient-boosted trees have no difficulty handling mixed data types.</p></li>
|
||
<li><p><strong>Many problems are not about prediction.</strong> In natural science we are often interested in learning something about the underlying distribution that generates the data. In this case, it is often difficult to cast these ideas in a supervised learning setting. While the problems are related, it is possible to make good predictions with a <em>wrong</em> model. The model might or might not be useful for understanding the underlying science.</p></li>
|
||
</ul>
|
||
<p>Some of these remarks are particular to DNNs, others are shared by all supervised learning methods. This motivates the use of unsupervised methods which in part circumvent these problems.</p>
|
||
</div>
|
||
<div class="section" id="solving-odes-with-deep-learning">
|
||
<h2>Solving ODEs with Deep Learning<a class="headerlink" href="#solving-odes-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">
|
||
<h2>Ordinary Differential Equations<a class="headerlink" href="#ordinary-differential-equations" 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="id3">
|
||
<h2>The trial solution<a class="headerlink" href="#id3" 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="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">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="id4">
|
||
<h2>The trial solution<a class="headerlink" href="#id4" 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">P</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="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">P</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"># 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">P</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">P</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="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">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"># 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="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">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"># 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="id5">
|
||
<h2>More details<a class="headerlink" href="#id5" 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="n">np</span><span class="o">.</span><span class="n">size</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="n">np</span><span class="o">.</span><span class="n">size</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">gca</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">gca</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">gca</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="id6">
|
||
<h2>The trial solution<a class="headerlink" href="#id6" 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="n">np</span><span class="o">.</span><span class="n">size</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">gca</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">gca</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">gca</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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