Morten Hjorth-Jensen
529 lines
33 KiB
HTML
529 lines
33 KiB
HTML
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<title>Week 41 Constructing a Neural Network code, Tensor flow and start Convolutional Neural Networks</title>
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{'highest level': 2,
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'sections': [('Plan for week 41', 2, None, 'plan-for-week-41'),
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('Videos on Neural Networks',
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2,
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None,
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'videos-on-neural-networks'),
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('Review of the back propagation algorithm',
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2,
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None,
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'review-of-the-back-propagation-algorithm'),
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('Setting up the Back propagation algorithm',
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2,
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None,
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'setting-up-the-back-propagation-algorithm'),
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('Setting up a Multi-layer perceptron model for classification',
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2,
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None,
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'setting-up-a-multi-layer-perceptron-model-for-classification'),
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('Defining the cost function',
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2,
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None,
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'defining-the-cost-function'),
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('Example: binary classification problem',
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2,
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None,
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'example-binary-classification-problem'),
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('The Softmax function', 2, None, 'the-softmax-function'),
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('Developing a code for doing neural networks with back '
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'propagation',
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2,
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None,
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'developing-a-code-for-doing-neural-networks-with-back-propagation'),
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('Collect and pre-process data',
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('Define model and architecture',
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('Weights and biases', 2, None, 'weights-and-biases'),
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('Feed-forward pass', 2, None, 'feed-forward-pass'),
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('Matrix multiplications', 2, None, 'matrix-multiplications'),
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('Choose cost function and optimizer',
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2,
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None,
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'choose-cost-function-and-optimizer'),
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('Optimizing the cost function',
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2,
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None,
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'optimizing-the-cost-function'),
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('Regularization', 2, None, 'regularization'),
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('Matrix multiplication', 2, None, 'matrix-multiplication'),
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('Improving performance', 2, None, 'improving-performance'),
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('Full object-oriented implementation',
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2,
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None,
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'full-object-oriented-implementation'),
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('Evaluate model performance on test data',
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2,
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None,
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'evaluate-model-performance-on-test-data'),
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('Adjust hyperparameters', 2, None, 'adjust-hyperparameters'),
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('Visualization', 2, None, 'visualization'),
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('scikit-learn implementation',
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2,
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None,
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'scikit-learn-implementation'),
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('Visualization', 2, None, 'visualization'),
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('Testing our code for the XOR, OR and AND gates',
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2,
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None,
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'testing-our-code-for-the-xor-or-and-and-gates'),
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('The AND and XOR Gates', 2, None, 'the-and-and-xor-gates'),
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('Representing the Data Sets',
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2,
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None,
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'representing-the-data-sets'),
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('Setting up the Neural Network',
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2,
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None,
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'setting-up-the-neural-network'),
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('The Code using Scikit-Learn',
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2,
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None,
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'the-code-using-scikit-learn'),
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('Building neural networks in Tensorflow and Keras',
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2,
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None,
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'building-neural-networks-in-tensorflow-and-keras'),
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('Tensorflow', 2, None, 'tensorflow'),
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('Using Keras', 2, None, 'using-keras'),
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('Collect and pre-process data',
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2,
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None,
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'collect-and-pre-process-data'),
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('The Breast Cancer Data, now with Keras',
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2,
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None,
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'the-breast-cancer-data-now-with-keras'),
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('The Mathematics of Neural Networks',
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2,
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None,
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'the-mathematics-of-neural-networks'),
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('Fine-tuning neural network hyperparameters',
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2,
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None,
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'fine-tuning-neural-network-hyperparameters'),
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('Hidden layers', 2, None, 'hidden-layers'),
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('Which activation function should I use?',
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2,
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None,
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'which-activation-function-should-i-use'),
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('Is the Logistic activation function (Sigmoid) our choice?',
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2,
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None,
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'is-the-logistic-activation-function-sigmoid-our-choice'),
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('The derivative of the Logistic funtion',
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2,
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None,
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'the-derivative-of-the-logistic-funtion'),
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('The RELU function family', 2, None, 'the-relu-function-family'),
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('Which activation function should we use?',
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2,
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None,
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'which-activation-function-should-we-use'),
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('More on activation functions, output layers',
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2,
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None,
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'more-on-activation-functions-output-layers'),
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('Batch Normalization', 2, None, 'batch-normalization'),
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('Dropout', 2, None, 'dropout'),
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('Gradient Clipping', 2, None, 'gradient-clipping'),
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('A very nice website on Neural Networks',
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2,
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None,
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'a-very-nice-website-on-neural-networks'),
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('A top-down perspective on Neural networks',
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2,
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None,
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'a-top-down-perspective-on-neural-networks'),
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('Limitations of supervised learning with deep networks',
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2,
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None,
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'limitations-of-supervised-learning-with-deep-networks'),
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('Overarching Views, a personal note',
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2,
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None,
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'overarching-views-a-personal-note'),
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('Using Automatic differentiation',
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2,
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None,
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'using-automatic-differentiation'),
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('Solving ODEs with Deep Learning',
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2,
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None,
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'solving-odes-with-deep-learning'),
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('Ordinary Differential Equations',
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2,
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None,
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'ordinary-differential-equations'),
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('The trial solution', 2, None, 'the-trial-solution'),
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('Minimization process', 2, None, 'minimization-process'),
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('Minimizing the cost function using gradient descent and '
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'automatic differentiation',
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2,
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None,
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'minimizing-the-cost-function-using-gradient-descent-and-automatic-differentiation'),
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('Example: Exponential decay',
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2,
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None,
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'example-exponential-decay'),
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('The function to solve for',
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2,
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None,
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'the-function-to-solve-for'),
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('The trial solution', 2, None, 'the-trial-solution'),
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('Setup of Network', 2, None, 'setup-of-network'),
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('Reformulating the problem',
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2,
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None,
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'reformulating-the-problem'),
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('More technicalities', 2, None, 'more-technicalities'),
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('More details', 2, None, 'more-details'),
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('A possible implementation of a neural network',
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2,
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None,
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'a-possible-implementation-of-a-neural-network'),
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('Technicalities', 2, None, 'technicalities'),
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('Final technicalities I', 2, None, 'final-technicalities-i'),
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('Final technicalities II', 2, None, 'final-technicalities-ii'),
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('Final technicalities III', 2, None, 'final-technicalities-iii'),
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('Final technicalities IV', 2, None, 'final-technicalities-iv'),
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('Back propagation', 2, None, 'back-propagation'),
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('Gradient descent', 2, None, 'gradient-descent'),
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('The code for solving the ODE',
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2,
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None,
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'the-code-for-solving-the-ode'),
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('The network with one input layer, specified number of hidden '
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'layers, and one output layer',
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2,
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None,
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'the-network-with-one-input-layer-specified-number-of-hidden-layers-and-one-output-layer'),
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('Example: Population growth',
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2,
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None,
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'example-population-growth'),
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('Setting up the problem', 2, None, 'setting-up-the-problem'),
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('The trial solution', 2, None, 'the-trial-solution'),
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('The program using Autograd',
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2,
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None,
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'the-program-using-autograd'),
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('Using forward Euler to solve the ODE',
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2,
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None,
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'using-forward-euler-to-solve-the-ode'),
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('Example: Solving the one dimensional Poisson equation',
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2,
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|
None,
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'example-solving-the-one-dimensional-poisson-equation'),
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('The specific equation to solve for',
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2,
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None,
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'the-specific-equation-to-solve-for'),
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('Solving the equation using Autograd',
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2,
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None,
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'solving-the-equation-using-autograd'),
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('Comparing with a numerical scheme',
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2,
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None,
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'comparing-with-a-numerical-scheme'),
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('Setting up the code', 2, None, 'setting-up-the-code'),
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('Partial Differential Equations',
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2,
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None,
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'partial-differential-equations'),
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('Type of problem', 2, None, 'type-of-problem'),
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|
('Network requirements', 2, None, 'network-requirements'),
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|
('More details', 2, None, 'more-details'),
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|
('Example: The diffusion equation',
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|
2,
|
|
None,
|
|
'example-the-diffusion-equation'),
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|
('Defining the problem', 2, None, 'defining-the-problem'),
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('Setting up the network using Autograd',
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2,
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|
None,
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|
'setting-up-the-network-using-autograd'),
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('Setting up the network using Autograd; The trial solution',
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|
2,
|
|
None,
|
|
'setting-up-the-network-using-autograd-the-trial-solution'),
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|
('Why the jacobian?', 2, None, 'why-the-jacobian'),
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('Setting up the network using Autograd; The full program',
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|
2,
|
|
None,
|
|
'setting-up-the-network-using-autograd-the-full-program'),
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|
('Example: Solving the wave equation with Neural Networks',
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|
2,
|
|
None,
|
|
'example-solving-the-wave-equation-with-neural-networks'),
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|
('The problem to solve for', 2, None, 'the-problem-to-solve-for'),
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|
('The trial solution', 2, None, 'the-trial-solution'),
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('The analytical solution', 2, None, 'the-analytical-solution'),
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('Solving the wave equation - the full program using Autograd',
|
|
2,
|
|
None,
|
|
'solving-the-wave-equation-the-full-program-using-autograd'),
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|
('Resources on differential equations and deep learning',
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2,
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None,
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'resources-on-differential-equations-and-deep-learning')]}
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<a class="navbar-brand" href="week41-bs.html">Week 41 Constructing a Neural Network code, Tensor flow and start Convolutional Neural Networks</a>
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<a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
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<ul class="dropdown-menu">
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<!-- navigation toc: --> <li><a href="._week41-bs001.html#plan-for-week-41" style="font-size: 80%;">Plan for week 41</a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs002.html#videos-on-neural-networks" style="font-size: 80%;">Videos on Neural Networks</a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs003.html#review-of-the-back-propagation-algorithm" style="font-size: 80%;">Review of the back propagation algorithm</a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs004.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;">Setting up the Back propagation algorithm</a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs005.html#setting-up-a-multi-layer-perceptron-model-for-classification" style="font-size: 80%;">Setting up a Multi-layer perceptron model for classification</a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs006.html#defining-the-cost-function" style="font-size: 80%;">Defining the cost function</a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs007.html#example-binary-classification-problem" style="font-size: 80%;">Example: binary classification problem</a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs008.html#the-softmax-function" style="font-size: 80%;">The Softmax function</a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs009.html#developing-a-code-for-doing-neural-networks-with-back-propagation" style="font-size: 80%;">Developing a code for doing neural networks with back propagation</a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs036.html#collect-and-pre-process-data" style="font-size: 80%;">Collect and pre-process data</a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs011.html#train-and-test-datasets" style="font-size: 80%;">Train and test datasets</a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs012.html#define-model-and-architecture" style="font-size: 80%;">Define model and architecture</a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs013.html#layers" style="font-size: 80%;">Layers</a></li>
|
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<!-- navigation toc: --> <li><a href="._week41-bs014.html#weights-and-biases" style="font-size: 80%;">Weights and biases</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week41-bs015.html#feed-forward-pass" style="font-size: 80%;">Feed-forward pass</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week41-bs016.html#matrix-multiplications" style="font-size: 80%;">Matrix multiplications</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week41-bs017.html#choose-cost-function-and-optimizer" style="font-size: 80%;">Choose cost function and optimizer</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week41-bs018.html#optimizing-the-cost-function" style="font-size: 80%;">Optimizing the cost function</a></li>
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|
<!-- navigation toc: --> <li><a href="._week41-bs019.html#regularization" style="font-size: 80%;">Regularization</a></li>
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|
<!-- navigation toc: --> <li><a href="._week41-bs020.html#matrix-multiplication" style="font-size: 80%;">Matrix multiplication</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week41-bs021.html#improving-performance" style="font-size: 80%;">Improving performance</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week41-bs022.html#full-object-oriented-implementation" style="font-size: 80%;">Full object-oriented implementation</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week41-bs023.html#evaluate-model-performance-on-test-data" style="font-size: 80%;">Evaluate model performance on test data</a></li>
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|
<!-- navigation toc: --> <li><a href="._week41-bs024.html#adjust-hyperparameters" style="font-size: 80%;">Adjust hyperparameters</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week41-bs027.html#visualization" style="font-size: 80%;">Visualization</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week41-bs026.html#scikit-learn-implementation" style="font-size: 80%;">scikit-learn implementation</a></li>
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|
<!-- navigation toc: --> <li><a href="._week41-bs027.html#visualization" style="font-size: 80%;">Visualization</a></li>
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|
<!-- navigation toc: --> <li><a href="._week41-bs028.html#testing-our-code-for-the-xor-or-and-and-gates" style="font-size: 80%;">Testing our code for the XOR, OR and AND gates</a></li>
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|
<!-- navigation toc: --> <li><a href="._week41-bs029.html#the-and-and-xor-gates" style="font-size: 80%;">The AND and XOR Gates</a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs030.html#representing-the-data-sets" style="font-size: 80%;">Representing the Data Sets</a></li>
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|
<!-- navigation toc: --> <li><a href="._week41-bs031.html#setting-up-the-neural-network" style="font-size: 80%;">Setting up the Neural Network</a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs032.html#the-code-using-scikit-learn" style="font-size: 80%;">The Code using Scikit-Learn</a></li>
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|
<!-- navigation toc: --> <li><a href="._week41-bs033.html#building-neural-networks-in-tensorflow-and-keras" style="font-size: 80%;">Building neural networks in Tensorflow and Keras</a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs034.html#tensorflow" style="font-size: 80%;">Tensorflow</a></li>
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|
<!-- navigation toc: --> <li><a href="._week41-bs035.html#using-keras" style="font-size: 80%;">Using Keras</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week41-bs036.html#collect-and-pre-process-data" style="font-size: 80%;">Collect and pre-process data</a></li>
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|
<!-- navigation toc: --> <li><a href="._week41-bs037.html#the-breast-cancer-data-now-with-keras" style="font-size: 80%;">The Breast Cancer Data, now with Keras</a></li>
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|
<!-- navigation toc: --> <li><a href="._week41-bs038.html#the-mathematics-of-neural-networks" style="font-size: 80%;">The Mathematics of Neural Networks</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week41-bs039.html#fine-tuning-neural-network-hyperparameters" style="font-size: 80%;">Fine-tuning neural network hyperparameters</a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs040.html#hidden-layers" style="font-size: 80%;">Hidden layers</a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs041.html#which-activation-function-should-i-use" style="font-size: 80%;">Which activation function should I use?</a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs042.html#is-the-logistic-activation-function-sigmoid-our-choice" style="font-size: 80%;">Is the Logistic activation function (Sigmoid) our choice?</a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs043.html#the-derivative-of-the-logistic-funtion" style="font-size: 80%;">The derivative of the Logistic funtion</a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs044.html#the-relu-function-family" style="font-size: 80%;">The RELU function family</a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs045.html#which-activation-function-should-we-use" style="font-size: 80%;">Which activation function should we use?</a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs046.html#more-on-activation-functions-output-layers" style="font-size: 80%;">More on activation functions, output layers</a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs050.html#a-very-nice-website-on-neural-networks" style="font-size: 80%;">A very nice website on Neural Networks</a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs051.html#a-top-down-perspective-on-neural-networks" style="font-size: 80%;">A top-down perspective on Neural networks</a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs052.html#limitations-of-supervised-learning-with-deep-networks" style="font-size: 80%;">Limitations of supervised learning with deep networks</a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs053.html#overarching-views-a-personal-note" style="font-size: 80%;">Overarching Views, a personal note</a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs059.html#minimizing-the-cost-function-using-gradient-descent-and-automatic-differentiation" style="font-size: 80%;">Minimizing the cost function using gradient descent and automatic differentiation</a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs060.html#example-exponential-decay" style="font-size: 80%;">Example: Exponential decay</a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs061.html#the-function-to-solve-for" style="font-size: 80%;">The function to solve for</a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs099.html#the-trial-solution" style="font-size: 80%;">The trial solution</a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs067.html#a-possible-implementation-of-a-neural-network" style="font-size: 80%;">A possible implementation of a neural network</a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs077.html#example-population-growth" style="font-size: 80%;">Example: Population growth</a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs084.html#solving-the-equation-using-autograd" style="font-size: 80%;">Solving the equation using Autograd</a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs092.html#defining-the-problem" style="font-size: 80%;">Defining the problem</a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs093.html#setting-up-the-network-using-autograd" style="font-size: 80%;">Setting up the network using Autograd</a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs094.html#setting-up-the-network-using-autograd-the-trial-solution" style="font-size: 80%;">Setting up the network using Autograd; The trial solution</a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs095.html#why-the-jacobian" style="font-size: 80%;">Why the jacobian?</a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs097.html#example-solving-the-wave-equation-with-neural-networks" style="font-size: 80%;">Example: Solving the wave equation with Neural Networks</a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs098.html#the-problem-to-solve-for" style="font-size: 80%;">The problem to solve for</a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs101.html#solving-the-wave-equation-the-full-program-using-autograd" style="font-size: 80%;">Solving the wave equation - the full program using Autograd</a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs102.html#resources-on-differential-equations-and-deep-learning" style="font-size: 80%;">Resources on differential equations and deep learning</a></li>
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<a name="part0065"></a>
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<!-- !split -->
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<h2 id="more-technicalities" class="anchor">More technicalities </h2>
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<p>The left hand side and right hand side of <a href="._week41-bs064.html#mjx-eqn-8">(8)</a> must be computed separately, and then the neural network must choose weights and biases, contained in \( P \), such that the sides are equal as best as possible.
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This means that the absolute or squared difference between the sides must be as close to zero, ideally equal to zero.
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In this case, the difference squared shows to be an appropriate measurement of how erroneous the trial solution is with respect to \( P \) of the neural network.
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</p>
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<p>This gives the following cost function our neural network must solve for:</p>
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$$
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\min_{P}\Big\{ \big(g_t'(x, P) - ( -\gamma g_t(x, P) \big)^2 \Big\}
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$$
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<p>(the notation \( \min_{P}\{ f(x, P) \} \) means that we desire to find \( P \) that yields the minimum of \( f(x, P) \))</p>
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<p>or, in terms of weights and biases for the hidden and output layer in our network:</p>
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$$
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\min_{P_{\text{hidden} }, \ P_{\text{output} }}\Big\{ \big(g_t'(x, \{ P_{\text{hidden} }, P_{\text{output} }\}) - ( -\gamma g_t(x, \{ P_{\text{hidden} }, P_{\text{output} }\}) \big)^2 \Big\}
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$$
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<p>for an input value \( x \).</p>
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