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<h1>Week 43: Deep Learning: Constructing a Neural Network code and solving differential equations</h1>
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<h2> Contents </h2>
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#plans-for-week-43">Plans for week 43</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#exercises-and-lab-session-week-43">Exercises and lab session week 43</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#using-automatic-differentiation">Using Automatic differentiation</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#back-propagation-and-automatic-differentiation">Back propagation and automatic differentiation</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#lecture-monday-october-20">Lecture Monday October 20</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#setting-up-the-back-propagation-algorithm-and-algorithm-for-a-feed-forward-nn-initalizations">Setting up the back propagation algorithm and algorithm for a feed forward NN, initalizations</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#setting-up-the-back-propagation-algorithm-part-1">Setting up the back propagation algorithm, part 1</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#setting-up-the-back-propagation-algorithm-part-2">Setting up the back propagation algorithm, part 2</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#setting-up-the-back-propagation-algorithm-part-3">Setting up the Back propagation algorithm, part 3</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#updating-the-gradients">Updating the gradients</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#activation-functions">Activation functions</a><ul class="nav section-nav flex-column">
<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#activation-functions-examples">Activation functions, examples</a></li>
</ul>
</li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#the-relu-function-family">The RELU function family</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#elu-function">ELU function</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#which-activation-function-should-we-use">Which activation function should we use?</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#more-on-activation-functions-output-layers">More on activation functions, output layers</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#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="#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="#using-pytorch-with-the-full-mnist-data-set">Using Pytorch with the full MNIST data set</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#and-a-similar-example-using-tensorflow-with-keras">And a similar example using Tensorflow with Keras</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#building-our-own-neural-network-code">Building our own neural network code</a><ul class="nav section-nav flex-column">
<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#learning-rate-methods">Learning rate methods</a></li>
<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#usage-of-the-above-learning-rate-schedulers">Usage of the above learning rate schedulers</a></li>
<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#cost-functions">Cost functions</a></li>
<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#id1">Activation functions</a></li>
<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#the-neural-network">The Neural Network</a></li>
<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#multiclass-classification">Multiclass classification</a></li>
</ul>
</li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#testing-the-xor-gate-and-other-gates">Testing the XOR gate and other gates</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#solving-differential-equations-with-deep-learning">Solving differential equations with Deep Learning</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#ordinary-differential-equations-first">Ordinary Differential Equations first</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#the-trial-solution">The trial solution</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#minimization-process">Minimization process</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#minimizing-the-cost-function-using-gradient-descent-and-automatic-differentiation">Minimizing the cost function using gradient descent and automatic differentiation</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#example-exponential-decay">Example: Exponential decay</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#the-function-to-solve-for">The function to solve for</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#id2">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="#id3">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="#id4">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="#id5">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="searchbox"></div>
<article class="bd-article">
<!-- HTML file automatically generated from DocOnce source (https://github.com/doconce/doconce/)
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<!-- dom:TITLE: Week 43: Deep Learning: Constructing a Neural Network code and solving differential equations --><section class="tex2jax_ignore mathjax_ignore" 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="Link to this heading">#</a></h1>
<p><strong>Morten Hjorth-Jensen</strong>, Department of Physics, University of Oslo, Norway</p>
<p>Date: <strong>October 20, 2025</strong></p>
<section id="plans-for-week-43">
<h2>Plans for week 43<a class="headerlink" href="#plans-for-week-43" title="Link to this heading">#</a></h2>
<p><strong>Material for the lecture on Monday October 20, 2025.</strong></p>
<ol class="arabic simple">
<li><p>Reminder from last week, see also lecture notes from week 42 at <a class="reference external" href="https://compphysics.github.io/MachineLearning/doc/LectureNotes/_build/html/week42.html">https://compphysics.github.io/MachineLearning/doc/LectureNotes/_build/html/week42.html</a> as well as those from week 41, see see <a class="reference external" href="https://compphysics.github.io/MachineLearning/doc/LectureNotes/_build/html/week41.html">https://compphysics.github.io/MachineLearning/doc/LectureNotes/_build/html/week41.html</a>.</p></li>
<li><p>Building our own Feed-forward Neural Network.</p></li>
<li><p>Coding examples using Tensorflow/Keras and Pytorch examples. The Pytorch examples are adapted from Rashcka’s text, see chapters 11-13..</p></li>
<li><p>Start discussions on how to use neural networks for solving differential equations (ordinary and partial ones). This topic continues next week as well.</p></li>
<li><p>Video of lecture at <a class="reference external" href="https://youtu.be/Gi6mzxAT0Ew">https://youtu.be/Gi6mzxAT0Ew</a></p></li>
<li><p>Whiteboard notes at <a class="github reference external" href="https://github.com/CompPhysics/MachineLearning/blob/master/doc/HandWrittenNotes/2025/FYSSTKweek43.pdf">CompPhysics/MachineLearning</a></p></li>
</ol>
</section>
<section id="exercises-and-lab-session-week-43">
<h2>Exercises and lab session week 43<a class="headerlink" href="#exercises-and-lab-session-week-43" title="Link to this heading">#</a></h2>
<p><strong>Lab sessions on Tuesday and Wednesday.</strong></p>
<ol class="arabic simple">
<li><p>Work on writing your own neural network code and discussions of project 2. If you didn’t get time to do the exercises from the two last weeks, we recommend doing so as these exercises give you the basic elements of a neural network code.</p></li>
<li><p>The exercises this week are tailored to the optional part of project 2, and deal with studying ways to display results from classification problems</p></li>
</ol>
</section>
<section id="using-automatic-differentiation">
<h2>Using Automatic differentiation<a class="headerlink" href="#using-automatic-differentiation" title="Link to this heading">#</a></h2>
<p>In our discussions of ordinary differential equations and neural network codes
we will also study the usage of Autograd, see for example <a class="reference external" href="https://www.youtube.com/watch?v=fRf4l5qaX1M&amp;amp;ab_channel=AlexSmola">https://www.youtube.com/watch?v=fRf4l5qaX1M&amp;ab_channel=AlexSmola</a> in computing gradients for deep learning. For the documentation of Autograd and examples see the Autograd documentation at <a class="github reference external" href="https://github.com/HIPS/autograd">HIPS/autograd</a> and the lecture slides from week 41, see <a class="reference external" href="https://compphysics.github.io/MachineLearning/doc/LectureNotes/_build/html/week41.html">https://compphysics.github.io/MachineLearning/doc/LectureNotes/_build/html/week41.html</a>.</p>
</section>
<section id="back-propagation-and-automatic-differentiation">
<h2>Back propagation and automatic differentiation<a class="headerlink" href="#back-propagation-and-automatic-differentiation" title="Link to this heading">#</a></h2>
<p>For more details on the back propagation algorithm and automatic differentiation see</p>
<ol class="arabic simple">
<li><p><a class="reference external" href="https://www.jmlr.org/papers/volume18/17-468/17-468.pdf">https://www.jmlr.org/papers/volume18/17-468/17-468.pdf</a></p></li>
<li><p><a class="reference external" href="https://deepimaging.github.io/lectures/lecture_11_Backpropagation.pdf">https://deepimaging.github.io/lectures/lecture_11_Backpropagation.pdf</a></p></li>
<li><p>Slides 12-44 at <a class="reference external" href="http://cs231n.stanford.edu/slides/2017/cs231n_2017_lecture4.pdf">http://cs231n.stanford.edu/slides/2017/cs231n_2017_lecture4.pdf</a></p></li>
</ol>
</section>
<section id="lecture-monday-october-20">
<h2>Lecture Monday October 20<a class="headerlink" href="#lecture-monday-october-20" title="Link to this heading">#</a></h2>
</section>
<section id="setting-up-the-back-propagation-algorithm-and-algorithm-for-a-feed-forward-nn-initalizations">
<h2>Setting up the back propagation algorithm and algorithm for a feed forward NN, initalizations<a class="headerlink" href="#setting-up-the-back-propagation-algorithm-and-algorithm-for-a-feed-forward-nn-initalizations" title="Link to this heading">#</a></h2>
<p>This is a reminder from last week.</p>
<p><strong>The architecture (our model).</strong></p>
<ol class="arabic simple">
<li><p>Set up your inputs and outputs (scalars, vectors, matrices or higher-order arrays)</p></li>
<li><p>Define the number of hidden layers and hidden nodes</p></li>
<li><p>Define activation functions for hidden layers and output layers</p></li>
<li><p>Define optimizer (plan learning rate, momentum, ADAgrad, RMSprop, ADAM etc) and array of initial learning rates</p></li>
<li><p>Define cost function and possible regularization terms with hyperparameters</p></li>
<li><p>Initialize weights and biases</p></li>
<li><p>Fix number of iterations for the feed forward part and back propagation part</p></li>
</ol>
</section>
<section id="setting-up-the-back-propagation-algorithm-part-1">
<h2>Setting up the back propagation algorithm, part 1<a class="headerlink" href="#setting-up-the-back-propagation-algorithm-part-1" title="Link to this heading">#</a></h2>
<p>Let us write this out in the form of an algorithm.</p>
<p><strong>First</strong>, we set up the input data <span class="math notranslate nohighlight">\(\boldsymbol{x}\)</span> and the activations
<span class="math notranslate nohighlight">\(\boldsymbol{z}_1\)</span> of the input layer and compute the activation function and
the pertinent outputs <span class="math notranslate nohighlight">\(\boldsymbol{a}^1\)</span>.</p>
<p><strong>Secondly</strong>, we perform then the feed forward till we reach the output
layer and compute all <span class="math notranslate nohighlight">\(\boldsymbol{z}_l\)</span> of the input layer and compute the
activation function and the pertinent outputs <span class="math notranslate nohighlight">\(\boldsymbol{a}^l\)</span> for
<span class="math notranslate nohighlight">\(l=1,2,3,\dots,L\)</span>.</p>
<p><strong>Notation</strong>: The first hidden layer has <span class="math notranslate nohighlight">\(l=1\)</span> as label and the final output layer has <span class="math notranslate nohighlight">\(l=L\)</span>.</p>
</section>
<section id="setting-up-the-back-propagation-algorithm-part-2">
<h2>Setting up the back propagation algorithm, part 2<a class="headerlink" href="#setting-up-the-back-propagation-algorithm-part-2" title="Link to this heading">#</a></h2>
<p>Thereafter we compute the ouput error <span class="math notranslate nohighlight">\(\boldsymbol{\delta}^L\)</span> by computing all</p>
<div class="math notranslate nohighlight">
\[
\delta_j^L = \sigma'(z_j^L)\frac{\partial {\cal C}}{\partial (a_j^L)}.
\]</div>
<p>Then we compute the back propagate error for each <span class="math notranslate nohighlight">\(l=L-1,L-2,\dots,1\)</span> as</p>
<div class="math notranslate nohighlight">
\[
\delta_j^l = \sum_k \delta_k^{l+1}w_{kj}^{l+1}\sigma'(z_j^l).
\]</div>
</section>
<section id="setting-up-the-back-propagation-algorithm-part-3">
<h2>Setting up the Back propagation algorithm, part 3<a class="headerlink" href="#setting-up-the-back-propagation-algorithm-part-3" title="Link to this heading">#</a></h2>
<p>Finally, we update the weights and the biases using gradient descent
for each <span class="math notranslate nohighlight">\(l=L-1,L-2,\dots,1\)</span> (the first hidden layer) and update the weights and biases
according to the rules</p>
<div class="math notranslate nohighlight">
\[
w_{ij}^l\leftarrow = w_{ij}^l- \eta \delta_j^la_i^{l-1},
\]</div>
<div class="math notranslate nohighlight">
\[
b_j^l \leftarrow b_j^l-\eta \frac{\partial {\cal C}}{\partial b_j^l}=b_j^l-\eta \delta_j^l,
\]</div>
<p>with <span class="math notranslate nohighlight">\(\eta\)</span> being the learning rate.</p>
</section>
<section id="updating-the-gradients">
<h2>Updating the gradients<a class="headerlink" href="#updating-the-gradients" title="Link to this heading">#</a></h2>
<p>With the back propagate error for each <span class="math notranslate nohighlight">\(l=L-1,L-2,\dots,1\)</span> as</p>
<div class="math notranslate nohighlight">
\[
\delta_j^l = \sum_k \delta_k^{l+1}w_{kj}^{l+1}\sigma'(z_j^l),
\]</div>
<p>we update the weights and the biases using gradient descent for each <span class="math notranslate nohighlight">\(l=L-1,L-2,\dots,1\)</span> and update the weights and biases according to the rules</p>
<div class="math notranslate nohighlight">
\[
w_{ij}^l\leftarrow = w_{ij}^l- \eta \delta_j^la_i^{l-1},
\]</div>
<div class="math notranslate nohighlight">
\[
b_j^l \leftarrow b_j^l-\eta \frac{\partial {\cal C}}{\partial b_j^l}=b_j^l-\eta \delta_j^l,
\]</div>
</section>
<section id="activation-functions">
<h2>Activation functions<a class="headerlink" href="#activation-functions" title="Link to this heading">#</a></h2>
<p>A property that characterizes a neural network, other than its
connectivity, is the choice of activation function(s). The following
restrictions are imposed on an activation function for an FFNN to
fulfill the universal approximation theorem</p>
<ul class="simple">
<li><p>Non-constant</p></li>
<li><p>Bounded</p></li>
<li><p>Monotonically-increasing</p></li>
<li><p>Continuous</p></li>
</ul>
<section id="activation-functions-examples">
<h3>Activation functions, examples<a class="headerlink" href="#activation-functions-examples" title="Link to this heading">#</a></h3>
<p>Typical examples are the logistic <em>Sigmoid</em></p>
<div class="math notranslate nohighlight">
\[
\sigma(x) = \frac{1}{1 + e^{-x}},
\]</div>
<p>and the <em>hyperbolic tangent</em> function</p>
<div class="math notranslate nohighlight">
\[
\sigma(x) = \tanh(x)
\]</div>
</section>
</section>
<section id="the-relu-function-family">
<h2>The RELU function family<a class="headerlink" href="#the-relu-function-family" title="Link to this heading">#</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>
</section>
<section id="elu-function">
<h2>ELU function<a class="headerlink" href="#elu-function" title="Link to this heading">#</a></h2>
<p>To solve this problem, nowadays practitioners use a variant of the
ReLU function, such as the leaky ReLU discussed above or the so-called
exponential linear unit (ELU) function</p>
<div class="math notranslate nohighlight">
\[\begin{split}
ELU(z) = \left\{\begin{array}{cc} \alpha\left( \exp{(z)}-1\right) &amp; z &lt; 0,\\ z &amp; z \ge 0.\end{array}\right.
\end{split}\]</div>
</section>
<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="Link to this heading">#</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>
</section>
<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="Link to this heading">#</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>
</section>
<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="Link to this heading">#</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>
</section>
<section id="tensorflow">
<h2>Tensorflow<a class="headerlink" href="#tensorflow" title="Link to this heading">#</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-none notranslate"><div class="highlight"><pre><span></span>pip3 install tensorflow
</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-none notranslate"><div class="highlight"><pre><span></span>conda create -n tf tensorflow
conda activate tf
</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-none notranslate"><div class="highlight"><pre><span></span>conda create -n tf-gpu tensorflow-gpu
conda activate tf-gpu
</pre></div>
</div>
</div>
</div>
</section>
<section id="using-keras">
<h2>Using Keras<a class="headerlink" href="#using-keras" title="Link to this heading">#</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-none notranslate"><div class="highlight"><pre><span></span>conda install keras
</pre></div>
</div>
</div>
</div>
<p>You can look up the <a class="reference external" href="https://keras.io/">instructions here</a> for more information.</p>
<p>We will to a large extent use <strong>keras</strong> in this course.</p>
</section>
<section id="collect-and-pre-process-data">
<h2>Collect and pre-process data<a class="headerlink" href="#collect-and-pre-process-data" title="Link to this heading">#</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-none notranslate"><div class="highlight"><pre><span></span>%matplotlib inline
# import necessary packages
import numpy as np
import matplotlib.pyplot as plt
import tensorflow as tf
from sklearn import datasets
# ensure the same random numbers appear every time
np.random.seed(0)
# display images in notebook
%matplotlib inline
plt.rcParams[&#39;figure.figsize&#39;] = (12,12)
# download MNIST dataset
digits = datasets.load_digits()
# define inputs and labels
inputs = digits.images
labels = digits.target
print(&quot;inputs = (n_inputs, pixel_width, pixel_height) = &quot; + str(inputs.shape))
print(&quot;labels = (n_inputs) = &quot; + str(labels.shape))
# flatten the image
# the value -1 means dimension is inferred from the remaining dimensions: 8x8 = 64
n_inputs = len(inputs)
inputs = inputs.reshape(n_inputs, -1)
print(&quot;X = (n_inputs, n_features) = &quot; + str(inputs.shape))
# choose some random images to display
indices = np.arange(n_inputs)
random_indices = np.random.choice(indices, size=5)
for i, image in enumerate(digits.images[random_indices]):
plt.subplot(1, 5, i+1)
plt.axis(&#39;off&#39;)
plt.imshow(image, cmap=plt.cm.gray_r, interpolation=&#39;nearest&#39;)
plt.title(&quot;Label: %d&quot; % digits.target[random_indices[i]])
plt.show()
</pre></div>
</div>
</div>
</div>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>from tensorflow.keras.layers import Input
from tensorflow.keras.models import Sequential #This allows appending layers to existing models
from tensorflow.keras.layers import Dense #This allows defining the characteristics of a particular layer
from tensorflow.keras import optimizers #This allows using whichever optimiser we want (sgd,adam,RMSprop)
from tensorflow.keras import regularizers #This allows using whichever regularizer we want (l1,l2,l1_l2)
from tensorflow.keras.utils import to_categorical #This allows using categorical cross entropy as the cost function
from sklearn.model_selection import train_test_split
# one-hot representation of labels
labels = to_categorical(labels)
# split into train and test data
train_size = 0.8
test_size = 1 - train_size
X_train, X_test, Y_train, Y_test = train_test_split(inputs, labels, train_size=train_size,
test_size=test_size)
</pre></div>
</div>
</div>
</div>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>
epochs = 100
batch_size = 100
n_neurons_layer1 = 100
n_neurons_layer2 = 50
n_categories = 10
eta_vals = np.logspace(-5, 1, 7)
lmbd_vals = np.logspace(-5, 1, 7)
def create_neural_network_keras(n_neurons_layer1, n_neurons_layer2, n_categories, eta, lmbd):
model = Sequential()
model.add(Dense(n_neurons_layer1, activation=&#39;sigmoid&#39;, kernel_regularizer=regularizers.l2(lmbd)))
model.add(Dense(n_neurons_layer2, activation=&#39;sigmoid&#39;, kernel_regularizer=regularizers.l2(lmbd)))
model.add(Dense(n_categories, activation=&#39;softmax&#39;))
sgd = optimizers.SGD(learning_rate=eta)
model.compile(loss=&#39;categorical_crossentropy&#39;, optimizer=sgd, metrics=[&#39;accuracy&#39;])
return model
</pre></div>
</div>
</div>
</div>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>DNN_keras = np.zeros((len(eta_vals), len(lmbd_vals)), dtype=object)
for i, eta in enumerate(eta_vals):
for j, lmbd in enumerate(lmbd_vals):
DNN = create_neural_network_keras(n_neurons_layer1, n_neurons_layer2, n_categories,
eta=eta, lmbd=lmbd)
DNN.fit(X_train, Y_train, epochs=epochs, batch_size=batch_size, verbose=0)
scores = DNN.evaluate(X_test, Y_test)
DNN_keras[i][j] = DNN
print(&quot;Learning rate = &quot;, eta)
print(&quot;Lambda = &quot;, lmbd)
print(&quot;Test accuracy: %.3f&quot; % scores[1])
print()
</pre></div>
</div>
</div>
</div>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span># optional
# visual representation of grid search
# uses seaborn heatmap, could probably do this in matplotlib
import seaborn as sns
sns.set()
train_accuracy = np.zeros((len(eta_vals), len(lmbd_vals)))
test_accuracy = np.zeros((len(eta_vals), len(lmbd_vals)))
for i in range(len(eta_vals)):
for j in range(len(lmbd_vals)):
DNN = DNN_keras[i][j]
train_accuracy[i][j] = DNN.evaluate(X_train, Y_train)[1]
test_accuracy[i][j] = DNN.evaluate(X_test, Y_test)[1]
fig, ax = plt.subplots(figsize = (10, 10))
sns.heatmap(train_accuracy, annot=True, ax=ax, cmap=&quot;viridis&quot;)
ax.set_title(&quot;Training Accuracy&quot;)
ax.set_ylabel(&quot;$\eta$&quot;)
ax.set_xlabel(&quot;$\lambda$&quot;)
plt.show()
fig, ax = plt.subplots(figsize = (10, 10))
sns.heatmap(test_accuracy, annot=True, ax=ax, cmap=&quot;viridis&quot;)
ax.set_title(&quot;Test Accuracy&quot;)
ax.set_ylabel(&quot;$\eta$&quot;)
ax.set_xlabel(&quot;$\lambda$&quot;)
plt.show()
</pre></div>
</div>
</div>
</div>
</section>
<section id="using-pytorch-with-the-full-mnist-data-set">
<h2>Using Pytorch with the full MNIST data set<a class="headerlink" href="#using-pytorch-with-the-full-mnist-data-set" title="Link to this heading">#</a></h2>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>import torch
import torch.nn as nn
import torch.optim as optim
import torchvision
import torchvision.transforms as transforms
# Device configuration: use GPU if available
device = torch.device(&quot;cuda&quot; if torch.cuda.is_available() else &quot;cpu&quot;)
# MNIST dataset (downloads if not already present)
transform = transforms.Compose([
transforms.ToTensor(),
transforms.Normalize((0.5,), (0.5,)) # normalize to mean=0.5, std=0.5 (approx. [-1,1] pixel range)
])
train_dataset = torchvision.datasets.MNIST(root=&#39;./data&#39;, train=True, download=True, transform=transform)
test_dataset = torchvision.datasets.MNIST(root=&#39;./data&#39;, train=False, download=True, transform=transform)
train_loader = torch.utils.data.DataLoader(train_dataset, batch_size=64, shuffle=True)
test_loader = torch.utils.data.DataLoader(test_dataset, batch_size=64, shuffle=False)
class NeuralNet(nn.Module):
def __init__(self):
super(NeuralNet, self).__init__()
self.fc1 = nn.Linear(28*28, 100) # first hidden layer (784 -&gt; 100)
self.fc2 = nn.Linear(100, 100) # second hidden layer (100 -&gt; 100)
self.fc3 = nn.Linear(100, 10) # output layer (100 -&gt; 10 classes)
def forward(self, x):
x = x.view(x.size(0), -1) # flatten images into vectors of size 784
x = torch.relu(self.fc1(x)) # hidden layer 1 + ReLU activation
x = torch.relu(self.fc2(x)) # hidden layer 2 + ReLU activation
x = self.fc3(x) # output layer (logits for 10 classes)
return x
model = NeuralNet().to(device)
criterion = nn.CrossEntropyLoss()
optimizer = optim.SGD(model.parameters(), lr=0.01, weight_decay=1e-4)
num_epochs = 10
for epoch in range(num_epochs):
model.train() # set model to training mode
running_loss = 0.0
for images, labels in train_loader:
# Move data to device (GPU if available, else CPU)
images, labels = images.to(device), labels.to(device)
optimizer.zero_grad() # reset gradients to zero
outputs = model(images) # forward pass: compute predictions
loss = criterion(outputs, labels) # compute cross-entropy loss
loss.backward() # backpropagate to compute gradients
optimizer.step() # update weights using SGD step
running_loss += loss.item()
# Compute average loss over all batches in this epoch
avg_loss = running_loss / len(train_loader)
print(f&quot;Epoch {epoch+1}/{num_epochs}, Loss: {avg_loss:.4f}&quot;)
#Evaluation on the Test Set
model.eval() # set model to evaluation mode
correct = 0
total = 0
with torch.no_grad(): # disable gradient calculation for evaluation
for images, labels in test_loader:
images, labels = images.to(device), labels.to(device)
outputs = model(images)
_, predicted = torch.max(outputs, dim=1) # class with highest score
total += labels.size(0)
correct += (predicted == labels).sum().item()
accuracy = 100 * correct / total
print(f&quot;Test Accuracy: {accuracy:.2f}%&quot;)
</pre></div>
</div>
</div>
</div>
</section>
<section id="and-a-similar-example-using-tensorflow-with-keras">
<h2>And a similar example using Tensorflow with Keras<a class="headerlink" href="#and-a-similar-example-using-tensorflow-with-keras" title="Link to this heading">#</a></h2>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>
import tensorflow as tf
from tensorflow import keras
from tensorflow.keras import layers, regularizers
# Check for GPU (TensorFlow will use it automatically if available)
gpus = tf.config.list_physical_devices(&#39;GPU&#39;)
print(f&quot;GPUs available: {gpus}&quot;)
# 1) Load and preprocess MNIST
(x_train, y_train), (x_test, y_test) = keras.datasets.mnist.load_data()
# Normalize to [0, 1]
x_train = (x_train.astype(&quot;float32&quot;) / 255.0)
x_test = (x_test.astype(&quot;float32&quot;) / 255.0)
# 2) Build the model: 784 -&gt; 100 -&gt; 100 -&gt; 10
l2_reg = 1e-4 # L2 regularization strength
model = keras.Sequential([
layers.Input(shape=(28, 28)),
layers.Flatten(),
layers.Dense(100, activation=&quot;relu&quot;,
kernel_regularizer=regularizers.l2(l2_reg)),
layers.Dense(100, activation=&quot;relu&quot;,
kernel_regularizer=regularizers.l2(l2_reg)),
layers.Dense(10, activation=&quot;softmax&quot;) # output probabilities for 10 classes
])
# 3) Compile with SGD + weight decay via L2 regularizers
model.compile(
optimizer=keras.optimizers.SGD(learning_rate=0.01),
loss=&quot;sparse_categorical_crossentropy&quot;,
metrics=[&quot;accuracy&quot;],
)
model.summary()
# 4) Train
history = model.fit(
x_train, y_train,
epochs=10,
batch_size=64,
validation_split=0.1, # optional: monitor validation during training
verbose=1
)
# 5) Evaluate on test set
test_loss, test_acc = model.evaluate(x_test, y_test, verbose=0)
print(f&quot;Test accuracy: {test_acc:.4f}, Test loss: {test_loss:.4f}&quot;)
</pre></div>
</div>
</div>
</div>
</section>
<section id="building-our-own-neural-network-code">
<h2>Building our own neural network code<a class="headerlink" href="#building-our-own-neural-network-code" title="Link to this heading">#</a></h2>
<p>Here we present a flexible object oriented codebase
for a feed forward neural network, along with a demonstration of how
to use it. Before we get into the details of the neural network, we
will first present some implementations of various schedulers, cost
functions and activation functions that can be used together with the
neural network.</p>
<p>The codes here were developed by Eric Reber and Gregor Kajda during spring 2023.</p>
<section id="learning-rate-methods">
<h3>Learning rate methods<a class="headerlink" href="#learning-rate-methods" title="Link to this heading">#</a></h3>
<p>The code below shows object oriented implementations of the Constant,
Momentum, Adagrad, AdagradMomentum, RMS prop and Adam schedulers. All
of the classes belong to the shared abstract Scheduler class, and
share the update_change() and reset() methods allowing for any of the
schedulers to be seamlessly used during the training stage, as will
later be shown in the fit() method of the neural
network. Update_change() only has one parameter, the gradient
(<span class="math notranslate nohighlight">\(δ^l_ja^{l−1}_k\)</span>), and returns the change which will be subtracted
from the weights. The reset() function takes no parameters, and resets
the desired variables. For Constant and Momentum, reset does nothing.</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>import autograd.numpy as np
class Scheduler:
&quot;&quot;&quot;
Abstract class for Schedulers
&quot;&quot;&quot;
def __init__(self, eta):
self.eta = eta
# should be overwritten
def update_change(self, gradient):
raise NotImplementedError
# overwritten if needed
def reset(self):
pass
class Constant(Scheduler):
def __init__(self, eta):
super().__init__(eta)
def update_change(self, gradient):
return self.eta * gradient
def reset(self):
pass
class Momentum(Scheduler):
def __init__(self, eta: float, momentum: float):
super().__init__(eta)
self.momentum = momentum
self.change = 0
def update_change(self, gradient):
self.change = self.momentum * self.change + self.eta * gradient
return self.change
def reset(self):
pass
class Adagrad(Scheduler):
def __init__(self, eta):
super().__init__(eta)
self.G_t = None
def update_change(self, gradient):
delta = 1e-8 # avoid division ny zero
if self.G_t is None:
self.G_t = np.zeros((gradient.shape[0], gradient.shape[0]))
self.G_t += gradient @ gradient.T
G_t_inverse = 1 / (
delta + np.sqrt(np.reshape(np.diagonal(self.G_t), (self.G_t.shape[0], 1)))
)
return self.eta * gradient * G_t_inverse
def reset(self):
self.G_t = None
class AdagradMomentum(Scheduler):
def __init__(self, eta, momentum):
super().__init__(eta)
self.G_t = None
self.momentum = momentum
self.change = 0
def update_change(self, gradient):
delta = 1e-8 # avoid division ny zero
if self.G_t is None:
self.G_t = np.zeros((gradient.shape[0], gradient.shape[0]))
self.G_t += gradient @ gradient.T
G_t_inverse = 1 / (
delta + np.sqrt(np.reshape(np.diagonal(self.G_t), (self.G_t.shape[0], 1)))
)
self.change = self.change * self.momentum + self.eta * gradient * G_t_inverse
return self.change
def reset(self):
self.G_t = None
class RMS_prop(Scheduler):
def __init__(self, eta, rho):
super().__init__(eta)
self.rho = rho
self.second = 0.0
def update_change(self, gradient):
delta = 1e-8 # avoid division ny zero
self.second = self.rho * self.second + (1 - self.rho) * gradient * gradient
return self.eta * gradient / (np.sqrt(self.second + delta))
def reset(self):
self.second = 0.0
class Adam(Scheduler):
def __init__(self, eta, rho, rho2):
super().__init__(eta)
self.rho = rho
self.rho2 = rho2
self.moment = 0
self.second = 0
self.n_epochs = 1
def update_change(self, gradient):
delta = 1e-8 # avoid division ny zero
self.moment = self.rho * self.moment + (1 - self.rho) * gradient
self.second = self.rho2 * self.second + (1 - self.rho2) * gradient * gradient
moment_corrected = self.moment / (1 - self.rho**self.n_epochs)
second_corrected = self.second / (1 - self.rho2**self.n_epochs)
return self.eta * moment_corrected / (np.sqrt(second_corrected + delta))
def reset(self):
self.n_epochs += 1
self.moment = 0
self.second = 0
</pre></div>
</div>
</div>
</div>
</section>
<section id="usage-of-the-above-learning-rate-schedulers">
<h3>Usage of the above learning rate schedulers<a class="headerlink" href="#usage-of-the-above-learning-rate-schedulers" title="Link to this heading">#</a></h3>
<p>To initalize a scheduler, simply create the object and pass in the
necessary parameters such as the learning rate and the momentum as
shown below. As the Scheduler class is an abstract class it should not
called directly, and will raise an error upon usage.</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>momentum_scheduler = Momentum(eta=1e-3, momentum=0.9)
adam_scheduler = Adam(eta=1e-3, rho=0.9, rho2=0.999)
</pre></div>
</div>
</div>
</div>
<p>Here is a small example for how a segment of code using schedulers
could look. Switching out the schedulers is simple.</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>weights = np.ones((3,3))
print(f&quot;Before scheduler:\n{weights=}&quot;)
epochs = 10
for e in range(epochs):
gradient = np.random.rand(3, 3)
change = adam_scheduler.update_change(gradient)
weights = weights - change
adam_scheduler.reset()
print(f&quot;\nAfter scheduler:\n{weights=}&quot;)
</pre></div>
</div>
</div>
</div>
</section>
<section id="cost-functions">
<h3>Cost functions<a class="headerlink" href="#cost-functions" title="Link to this heading">#</a></h3>
<p>Here we discuss cost functions that can be used when creating the
neural network. Every cost function takes the target vector as its
parameter, and returns a function valued only at <span class="math notranslate nohighlight">\(x\)</span> such that it may
easily be differentiated.</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>import autograd.numpy as np
def CostOLS(target):
def func(X):
return (1.0 / target.shape[0]) * np.sum((target - X) ** 2)
return func
def CostLogReg(target):
def func(X):
return -(1.0 / target.shape[0]) * np.sum(
(target * np.log(X + 10e-10)) + ((1 - target) * np.log(1 - X + 10e-10))
)
return func
def CostCrossEntropy(target):
def func(X):
return -(1.0 / target.size) * np.sum(target * np.log(X + 10e-10))
return func
</pre></div>
</div>
</div>
</div>
<p>Below we give a short example of how these cost function may be used
to obtain results if you wish to test them out on your own using
AutoGrad’s automatics differentiation.</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>from autograd import grad
target = np.array([[1, 2, 3]]).T
a = np.array([[4, 5, 6]]).T
cost_func = CostCrossEntropy
cost_func_derivative = grad(cost_func(target))
valued_at_a = cost_func_derivative(a)
print(f&quot;Derivative of cost function {cost_func.__name__} valued at a:\n{valued_at_a}&quot;)
</pre></div>
</div>
</div>
</div>
</section>
<section id="id1">
<h3>Activation functions<a class="headerlink" href="#id1" title="Link to this heading">#</a></h3>
<p>Finally, before we look at the neural network, we will look at the
activation functions which can be specified between the hidden layers
and as the output function. Each function can be valued for any given
vector or matrix X, and can be differentiated via derivate().</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>import autograd.numpy as np
from autograd import elementwise_grad
def identity(X):
return X
def sigmoid(X):
try:
return 1.0 / (1 + np.exp(-X))
except FloatingPointError:
return np.where(X &gt; np.zeros(X.shape), np.ones(X.shape), np.zeros(X.shape))
def softmax(X):
X = X - np.max(X, axis=-1, keepdims=True)
delta = 10e-10
return np.exp(X) / (np.sum(np.exp(X), axis=-1, keepdims=True) + delta)
def RELU(X):
return np.where(X &gt; np.zeros(X.shape), X, np.zeros(X.shape))
def LRELU(X):
delta = 10e-4
return np.where(X &gt; np.zeros(X.shape), X, delta * X)
def derivate(func):
if func.__name__ == &quot;RELU&quot;:
def func(X):
return np.where(X &gt; 0, 1, 0)
return func
elif func.__name__ == &quot;LRELU&quot;:
def func(X):
delta = 10e-4
return np.where(X &gt; 0, 1, delta)
return func
else:
return elementwise_grad(func)
</pre></div>
</div>
</div>
</div>
<p>Below follows a short demonstration of how to use an activation
function. The derivative of the activation function will be important
when calculating the output delta term during backpropagation. Note
that derivate() can also be used for cost functions for a more
generalized approach.</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>z = np.array([[4, 5, 6]]).T
print(f&quot;Input to activation function:\n{z}&quot;)
act_func = sigmoid
a = act_func(z)
print(f&quot;\nOutput from {act_func.__name__} activation function:\n{a}&quot;)
act_func_derivative = derivate(act_func)
valued_at_z = act_func_derivative(a)
print(f&quot;\nDerivative of {act_func.__name__} activation function valued at z:\n{valued_at_z}&quot;)
</pre></div>
</div>
</div>
</div>
</section>
<section id="the-neural-network">
<h3>The Neural Network<a class="headerlink" href="#the-neural-network" title="Link to this heading">#</a></h3>
<p>Now that we have gotten a good understanding of the implementation of
some important components, we can take a look at an object oriented
implementation of a feed forward neural network. The feed forward
neural network has been implemented as a class named FFNN, which can
be initiated as a regressor or classifier dependant on the choice of
cost function. The FFNN can have any number of input nodes, hidden
layers with any amount of hidden nodes, and any amount of output nodes
meaning it can perform multiclass classification as well as binary
classification and regression problems. Although there is a lot of
code present, it makes for an easy to use and generalizeable interface
for creating many types of neural networks as will be demonstrated
below.</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>import math
import autograd.numpy as np
import sys
import warnings
from autograd import grad, elementwise_grad
from random import random, seed
from copy import deepcopy, copy
from typing import Tuple, Callable
from sklearn.utils import resample
warnings.simplefilter(&quot;error&quot;)
class FFNN:
&quot;&quot;&quot;
Description:
------------
Feed Forward Neural Network with interface enabling flexible design of a
nerual networks architecture and the specification of activation function
in the hidden layers and output layer respectively. This model can be used
for both regression and classification problems, depending on the output function.
Attributes:
------------
I dimensions (tuple[int]): A list of positive integers, which specifies the
number of nodes in each of the networks layers. The first integer in the array
defines the number of nodes in the input layer, the second integer defines number
of nodes in the first hidden layer and so on until the last number, which
specifies the number of nodes in the output layer.
II hidden_func (Callable): The activation function for the hidden layers
III output_func (Callable): The activation function for the output layer
IV cost_func (Callable): Our cost function
V seed (int): Sets random seed, makes results reproducible
&quot;&quot;&quot;
def __init__(
self,
dimensions: tuple[int],
hidden_func: Callable = sigmoid,
output_func: Callable = lambda x: x,
cost_func: Callable = CostOLS,
seed: int = None,
):
self.dimensions = dimensions
self.hidden_func = hidden_func
self.output_func = output_func
self.cost_func = cost_func
self.seed = seed
self.weights = list()
self.schedulers_weight = list()
self.schedulers_bias = list()
self.a_matrices = list()
self.z_matrices = list()
self.classification = None
self.reset_weights()
self._set_classification()
def fit(
self,
X: np.ndarray,
t: np.ndarray,
scheduler: Scheduler,
batches: int = 1,
epochs: int = 100,
lam: float = 0,
X_val: np.ndarray = None,
t_val: np.ndarray = None,
):
&quot;&quot;&quot;
Description:
------------
This function performs the training the neural network by performing the feedforward and backpropagation
algorithm to update the networks weights.
Parameters:
------------
I X (np.ndarray) : training data
II t (np.ndarray) : target data
III scheduler (Scheduler) : specified scheduler (algorithm for optimization of gradient descent)
IV scheduler_args (list[int]) : list of all arguments necessary for scheduler
Optional Parameters:
------------
V batches (int) : number of batches the datasets are split into, default equal to 1
VI epochs (int) : number of iterations used to train the network, default equal to 100
VII lam (float) : regularization hyperparameter lambda
VIII X_val (np.ndarray) : validation set
IX t_val (np.ndarray) : validation target set
Returns:
------------
I scores (dict) : A dictionary containing the performance metrics of the model.
The number of the metrics depends on the parameters passed to the fit-function.
&quot;&quot;&quot;
# setup
if self.seed is not None:
np.random.seed(self.seed)
val_set = False
if X_val is not None and t_val is not None:
val_set = True
# creating arrays for score metrics
train_errors = np.empty(epochs)
train_errors.fill(np.nan)
val_errors = np.empty(epochs)
val_errors.fill(np.nan)
train_accs = np.empty(epochs)
train_accs.fill(np.nan)
val_accs = np.empty(epochs)
val_accs.fill(np.nan)
self.schedulers_weight = list()
self.schedulers_bias = list()
batch_size = X.shape[0] // batches
X, t = resample(X, t)
# this function returns a function valued only at X
cost_function_train = self.cost_func(t)
if val_set:
cost_function_val = self.cost_func(t_val)
# create schedulers for each weight matrix
for i in range(len(self.weights)):
self.schedulers_weight.append(copy(scheduler))
self.schedulers_bias.append(copy(scheduler))
print(f&quot;{scheduler.__class__.__name__}: Eta={scheduler.eta}, Lambda={lam}&quot;)
try:
for e in range(epochs):
for i in range(batches):
# allows for minibatch gradient descent
if i == batches - 1:
# If the for loop has reached the last batch, take all thats left
X_batch = X[i * batch_size :, :]
t_batch = t[i * batch_size :, :]
else:
X_batch = X[i * batch_size : (i + 1) * batch_size, :]
t_batch = t[i * batch_size : (i + 1) * batch_size, :]
self._feedforward(X_batch)
self._backpropagate(X_batch, t_batch, lam)
# reset schedulers for each epoch (some schedulers pass in this call)
for scheduler in self.schedulers_weight:
scheduler.reset()
for scheduler in self.schedulers_bias:
scheduler.reset()
# computing performance metrics
pred_train = self.predict(X)
train_error = cost_function_train(pred_train)
train_errors[e] = train_error
if val_set:
pred_val = self.predict(X_val)
val_error = cost_function_val(pred_val)
val_errors[e] = val_error
if self.classification:
train_acc = self._accuracy(self.predict(X), t)
train_accs[e] = train_acc
if val_set:
val_acc = self._accuracy(pred_val, t_val)
val_accs[e] = val_acc
# printing progress bar
progression = e / epochs
print_length = self._progress_bar(
progression,
train_error=train_errors[e],
train_acc=train_accs[e],
val_error=val_errors[e],
val_acc=val_accs[e],
)
except KeyboardInterrupt:
# allows for stopping training at any point and seeing the result
pass
# visualization of training progression (similiar to tensorflow progression bar)
sys.stdout.write(&quot;\r&quot; + &quot; &quot; * print_length)
sys.stdout.flush()
self._progress_bar(
1,
train_error=train_errors[e],
train_acc=train_accs[e],
val_error=val_errors[e],
val_acc=val_accs[e],
)
sys.stdout.write(&quot;&quot;)
# return performance metrics for the entire run
scores = dict()
scores[&quot;train_errors&quot;] = train_errors
if val_set:
scores[&quot;val_errors&quot;] = val_errors
if self.classification:
scores[&quot;train_accs&quot;] = train_accs
if val_set:
scores[&quot;val_accs&quot;] = val_accs
return scores
def predict(self, X: np.ndarray, *, threshold=0.5):
&quot;&quot;&quot;
Description:
------------
Performs prediction after training of the network has been finished.
Parameters:
------------
I X (np.ndarray): The design matrix, with n rows of p features each
Optional Parameters:
------------
II threshold (float) : sets minimal value for a prediction to be predicted as the positive class
in classification problems
Returns:
------------
I z (np.ndarray): A prediction vector (row) for each row in our design matrix
This vector is thresholded if regression=False, meaning that classification results
in a vector of 1s and 0s, while regressions in an array of decimal numbers
&quot;&quot;&quot;
predict = self._feedforward(X)
if self.classification:
return np.where(predict &gt; threshold, 1, 0)
else:
return predict
def reset_weights(self):
&quot;&quot;&quot;
Description:
------------
Resets/Reinitializes the weights in order to train the network for a new problem.
&quot;&quot;&quot;
if self.seed is not None:
np.random.seed(self.seed)
self.weights = list()
for i in range(len(self.dimensions) - 1):
weight_array = np.random.randn(
self.dimensions[i] + 1, self.dimensions[i + 1]
)
weight_array[0, :] = np.random.randn(self.dimensions[i + 1]) * 0.01
self.weights.append(weight_array)
def _feedforward(self, X: np.ndarray):
&quot;&quot;&quot;
Description:
------------
Calculates the activation of each layer starting at the input and ending at the output.
Each following activation is calculated from a weighted sum of each of the preceeding
activations (except in the case of the input layer).
Parameters:
------------
I X (np.ndarray): The design matrix, with n rows of p features each
Returns:
------------
I z (np.ndarray): A prediction vector (row) for each row in our design matrix
&quot;&quot;&quot;
# reset matrices
self.a_matrices = list()
self.z_matrices = list()
# if X is just a vector, make it into a matrix
if len(X.shape) == 1:
X = X.reshape((1, X.shape[0]))
# Add a coloumn of zeros as the first coloumn of the design matrix, in order
# to add bias to our data
bias = np.ones((X.shape[0], 1)) * 0.01
X = np.hstack([bias, X])
# a^0, the nodes in the input layer (one a^0 for each row in X - where the
# exponent indicates layer number).
a = X
self.a_matrices.append(a)
self.z_matrices.append(a)
# The feed forward algorithm
for i in range(len(self.weights)):
if i &lt; len(self.weights) - 1:
z = a @ self.weights[i]
self.z_matrices.append(z)
a = self.hidden_func(z)
# bias column again added to the data here
bias = np.ones((a.shape[0], 1)) * 0.01
a = np.hstack([bias, a])
self.a_matrices.append(a)
else:
try:
# a^L, the nodes in our output layers
z = a @ self.weights[i]
a = self.output_func(z)
self.a_matrices.append(a)
self.z_matrices.append(z)
except Exception as OverflowError:
print(
&quot;OverflowError in fit() in FFNN\nHOW TO DEBUG ERROR: Consider lowering your learning rate or scheduler specific parameters such as momentum, or check if your input values need scaling&quot;
)
# this will be a^L
return a
def _backpropagate(self, X, t, lam):
&quot;&quot;&quot;
Description:
------------
Performs the backpropagation algorithm. In other words, this method
calculates the gradient of all the layers starting at the
output layer, and moving from right to left accumulates the gradient until
the input layer is reached. Each layers respective weights are updated while
the algorithm propagates backwards from the output layer (auto-differentation in reverse mode).
Parameters:
------------
I X (np.ndarray): The design matrix, with n rows of p features each.
II t (np.ndarray): The target vector, with n rows of p targets.
III lam (float32): regularization parameter used to punish the weights in case of overfitting
Returns:
------------
No return value.
&quot;&quot;&quot;
out_derivative = derivate(self.output_func)
hidden_derivative = derivate(self.hidden_func)
for i in range(len(self.weights) - 1, -1, -1):
# delta terms for output
if i == len(self.weights) - 1:
# for multi-class classification
if (
self.output_func.__name__ == &quot;softmax&quot;
):
delta_matrix = self.a_matrices[i + 1] - t
# for single class classification
else:
cost_func_derivative = grad(self.cost_func(t))
delta_matrix = out_derivative(
self.z_matrices[i + 1]
) * cost_func_derivative(self.a_matrices[i + 1])
# delta terms for hidden layer
else:
delta_matrix = (
self.weights[i + 1][1:, :] @ delta_matrix.T
).T * hidden_derivative(self.z_matrices[i + 1])
# calculate gradient
gradient_weights = self.a_matrices[i][:, 1:].T @ delta_matrix
gradient_bias = np.sum(delta_matrix, axis=0).reshape(
1, delta_matrix.shape[1]
)
# regularization term
gradient_weights += self.weights[i][1:, :] * lam
# use scheduler
update_matrix = np.vstack(
[
self.schedulers_bias[i].update_change(gradient_bias),
self.schedulers_weight[i].update_change(gradient_weights),
]
)
# update weights and bias
self.weights[i] -= update_matrix
def _accuracy(self, prediction: np.ndarray, target: np.ndarray):
&quot;&quot;&quot;
Description:
------------
Calculates accuracy of given prediction to target
Parameters:
------------
I prediction (np.ndarray): vector of predicitons output network
(1s and 0s in case of classification, and real numbers in case of regression)
II target (np.ndarray): vector of true values (What the network ideally should predict)
Returns:
------------
A floating point number representing the percentage of correctly classified instances.
&quot;&quot;&quot;
assert prediction.size == target.size
return np.average((target == prediction))
def _set_classification(self):
&quot;&quot;&quot;
Description:
------------
Decides if FFNN acts as classifier (True) og regressor (False),
sets self.classification during init()
&quot;&quot;&quot;
self.classification = False
if (
self.cost_func.__name__ == &quot;CostLogReg&quot;
or self.cost_func.__name__ == &quot;CostCrossEntropy&quot;
):
self.classification = True
def _progress_bar(self, progression, **kwargs):
&quot;&quot;&quot;
Description:
------------
Displays progress of training
&quot;&quot;&quot;
print_length = 40
num_equals = int(progression * print_length)
num_not = print_length - num_equals
arrow = &quot;&gt;&quot; if num_equals &gt; 0 else &quot;&quot;
bar = &quot;[&quot; + &quot;=&quot; * (num_equals - 1) + arrow + &quot;-&quot; * num_not + &quot;]&quot;
perc_print = self._format(progression * 100, decimals=5)
line = f&quot; {bar} {perc_print}% &quot;
for key in kwargs:
if not np.isnan(kwargs[key]):
value = self._format(kwargs[key], decimals=4)
line += f&quot;| {key}: {value} &quot;
sys.stdout.write(&quot;\r&quot; + line)
sys.stdout.flush()
return len(line)
def _format(self, value, decimals=4):
&quot;&quot;&quot;
Description:
------------
Formats decimal numbers for progress bar
&quot;&quot;&quot;
if value &gt; 0:
v = value
elif value &lt; 0:
v = -10 * value
else:
v = 1
n = 1 + math.floor(math.log10(v))
if n &gt;= decimals - 1:
return str(round(value))
return f&quot;{value:.{decimals-n-1}f}&quot;
</pre></div>
</div>
</div>
</div>
<p>Before we make a model, we will quickly generate a dataset we can use
for our linear regression problem as shown below</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>import autograd.numpy as np
from sklearn.model_selection import train_test_split
def SkrankeFunction(x, y):
return np.ravel(0 + 1*x + 2*y + 3*x**2 + 4*x*y + 5*y**2)
def create_X(x, y, n):
if len(x.shape) &gt; 1:
x = np.ravel(x)
y = np.ravel(y)
N = len(x)
l = int((n + 1) * (n + 2) / 2) # Number of elements in beta
X = np.ones((N, l))
for i in range(1, n + 1):
q = int((i) * (i + 1) / 2)
for k in range(i + 1):
X[:, q + k] = (x ** (i - k)) * (y**k)
return X
step=0.5
x = np.arange(0, 1, step)
y = np.arange(0, 1, step)
x, y = np.meshgrid(x, y)
target = SkrankeFunction(x, y)
target = target.reshape(target.shape[0], 1)
poly_degree=3
X = create_X(x, y, poly_degree)
X_train, X_test, t_train, t_test = train_test_split(X, target)
</pre></div>
</div>
</div>
</div>
<p>Now that we have our dataset ready for the regression, we can create
our regressor. Note that with the seed parameter, we can make sure our
results stay the same every time we run the neural network. For
inititialization, we simply specify the dimensions (we wish the amount
of input nodes to be equal to the datapoints, and the output to
predict one value).</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>input_nodes = X_train.shape[1]
output_nodes = 1
linear_regression = FFNN((input_nodes, output_nodes), output_func=identity, cost_func=CostOLS, seed=2023)
</pre></div>
</div>
</div>
</div>
<p>We then fit our model with our training data using the scheduler of our choice.</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>linear_regression.reset_weights() # reset weights such that previous runs or reruns don&#39;t affect the weights
scheduler = Constant(eta=1e-3)
scores = linear_regression.fit(X_train, t_train, scheduler)
</pre></div>
</div>
</div>
</div>
<p>Due to the progress bar we can see the MSE (train_error) throughout
the FFNN’s training. Note that the fit() function has some optional
parameters with defualt arguments. For example, the regularization
hyperparameter can be left ignored if not needed, and equally the FFNN
will by default run for 100 epochs. These can easily be changed, such
as for example:</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>linear_regression.reset_weights() # reset weights such that previous runs or reruns don&#39;t affect the weights
scores = linear_regression.fit(X_train, t_train, scheduler, lam=1e-4, epochs=1000)
</pre></div>
</div>
</div>
</div>
<p>We see that given more epochs to train on, the regressor reaches a lower MSE.</p>
<p>Let us then switch to a binary classification. We use a binary
classification dataset, and follow a similar setup to the regression
case.</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>from sklearn.datasets import load_breast_cancer
from sklearn.preprocessing import MinMaxScaler
wisconsin = load_breast_cancer()
X = wisconsin.data
target = wisconsin.target
target = target.reshape(target.shape[0], 1)
X_train, X_val, t_train, t_val = train_test_split(X, target)
scaler = MinMaxScaler()
scaler.fit(X_train)
X_train = scaler.transform(X_train)
X_val = scaler.transform(X_val)
</pre></div>
</div>
</div>
</div>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>input_nodes = X_train.shape[1]
output_nodes = 1
logistic_regression = FFNN((input_nodes, output_nodes), output_func=sigmoid, cost_func=CostLogReg, seed=2023)
</pre></div>
</div>
</div>
</div>
<p>We will now make use of our validation data by passing it into our fit function as a keyword argument</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>logistic_regression.reset_weights() # reset weights such that previous runs or reruns don&#39;t affect the weights
scheduler = Adam(eta=1e-3, rho=0.9, rho2=0.999)
scores = logistic_regression.fit(X_train, t_train, scheduler, epochs=1000, X_val=X_val, t_val=t_val)
</pre></div>
</div>
</div>
</div>
<p>Finally, we will create a neural network with 2 hidden layers with activation functions.</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>input_nodes = X_train.shape[1]
hidden_nodes1 = 100
hidden_nodes2 = 30
output_nodes = 1
dims = (input_nodes, hidden_nodes1, hidden_nodes2, output_nodes)
neural_network = FFNN(dims, hidden_func=RELU, output_func=sigmoid, cost_func=CostLogReg, seed=2023)
</pre></div>
</div>
</div>
</div>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>neural_network.reset_weights() # reset weights such that previous runs or reruns don&#39;t affect the weights
scheduler = Adam(eta=1e-4, rho=0.9, rho2=0.999)
scores = neural_network.fit(X_train, t_train, scheduler, epochs=1000, X_val=X_val, t_val=t_val)
</pre></div>
</div>
</div>
</div>
</section>
<section id="multiclass-classification">
<h3>Multiclass classification<a class="headerlink" href="#multiclass-classification" title="Link to this heading">#</a></h3>
<p>Finally, we will demonstrate the use case of multiclass classification
using our FFNN with the famous MNIST dataset, which contain images of
digits between the range of 0 to 9.</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>from sklearn.datasets import load_digits
def onehot(target: np.ndarray):
onehot = np.zeros((target.size, target.max() + 1))
onehot[np.arange(target.size), target] = 1
return onehot
digits = load_digits()
X = digits.data
target = digits.target
target = onehot(target)
input_nodes = 64
hidden_nodes1 = 100
hidden_nodes2 = 30
output_nodes = 10
dims = (input_nodes, hidden_nodes1, hidden_nodes2, output_nodes)
multiclass = FFNN(dims, hidden_func=LRELU, output_func=softmax, cost_func=CostCrossEntropy)
multiclass.reset_weights() # reset weights such that previous runs or reruns don&#39;t affect the weights
scheduler = Adam(eta=1e-4, rho=0.9, rho2=0.999)
scores = multiclass.fit(X, target, scheduler, epochs=1000)
</pre></div>
</div>
</div>
</div>
</section>
</section>
<section id="testing-the-xor-gate-and-other-gates">
<h2>Testing the XOR gate and other gates<a class="headerlink" href="#testing-the-xor-gate-and-other-gates" title="Link to this heading">#</a></h2>
<p>Let us now use our code to test the XOR gate.</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>X = np.array([ [0, 0], [0, 1], [1, 0],[1, 1]],dtype=np.float64)
# The XOR gate
yXOR = np.array( [[ 0], [1] ,[1], [0]])
input_nodes = X.shape[1]
output_nodes = 1
logistic_regression = FFNN((input_nodes, output_nodes), output_func=sigmoid, cost_func=CostLogReg, seed=2023)
logistic_regression.reset_weights() # reset weights such that previous runs or reruns don&#39;t affect the weights
scheduler = Adam(eta=1e-1, rho=0.9, rho2=0.999)
scores = logistic_regression.fit(X, yXOR, scheduler, epochs=1000)
</pre></div>
</div>
</div>
</div>
<p>Not bad, but the results depend strongly on the learning reate. Try different learning rates.</p>
</section>
<section id="solving-differential-equations-with-deep-learning">
<h2>Solving differential equations with Deep Learning<a class="headerlink" href="#solving-differential-equations-with-deep-learning" title="Link to this heading">#</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>Physics informed neural networks.</strong></p>
<p><a class="reference external" href="https://link.springer.com/article/10.1007/s10915-022-01939-z">Scientific Machine Learning Through Physics–Informed Neural Networks: Where we are and What’s Next</a>, by Cuomo et al</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>
</section>
<section id="ordinary-differential-equations-first">
<h2>Ordinary Differential Equations first<a class="headerlink" href="#ordinary-differential-equations-first" title="Link to this heading">#</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 internal" href="#ode"><span class="xref myst">1</span></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 internal" href="#ode"><span class="xref myst">1</span></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>
</section>
<section id="the-trial-solution">
<h2>The trial solution<a class="headerlink" href="#the-trial-solution" title="Link to this heading">#</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>
</section>
<section id="minimization-process">
<h2>Minimization process<a class="headerlink" href="#minimization-process" title="Link to this heading">#</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 internal" href="#ode"><span class="xref myst">1</span></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 internal" href="#cost"><span class="xref myst">3</span></a>) for a set of <span class="math notranslate nohighlight">\(N\)</span> training samples <span class="math notranslate nohighlight">\(x_i\)</span>.</p>
</section>
<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="Link to this heading">#</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 internal" href="#cost"><span class="xref myst">3</span></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>
</section>
<section id="example-exponential-decay">
<h2>Example: Exponential decay<a class="headerlink" href="#example-exponential-decay" title="Link to this heading">#</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 internal" href="#solve_expdec"><span class="xref myst">4</span></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 internal" href="#solve_expdec"><span class="xref myst">4</span></a>).</p>
</section>
<section id="the-function-to-solve-for">
<h2>The function to solve for<a class="headerlink" href="#the-function-to-solve-for" title="Link to this heading">#</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>
</section>
<section id="id2">
<h2>The trial solution<a class="headerlink" href="#id2" title="Link to this heading">#</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>
</section>
<section id="setup-of-network">
<h2>Setup of Network<a class="headerlink" href="#setup-of-network" title="Link to this heading">#</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 internal" href="#solveode"><span class="xref myst">6</span></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>
</section>
<section id="reformulating-the-problem">
<h2>Reformulating the problem<a class="headerlink" href="#reformulating-the-problem" title="Link to this heading">#</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 internal" href="#solveode"><span class="xref myst">6</span></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 internal" href="#trial"><span class="xref myst">7</span></a>) satisfies (<a class="reference internal" href="#solveode"><span class="xref myst">6</span></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>
</section>
<section id="more-technicalities">
<h2>More technicalities<a class="headerlink" href="#more-technicalities" title="Link to this heading">#</a></h2>
<p>The left hand side and right hand side of (<a class="reference internal" href="#nnmin"><span class="xref myst">8</span></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>
</section>
<section id="more-details">
<h2>More details<a class="headerlink" href="#more-details" title="Link to this heading">#</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>
</section>
<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="Link to this heading">#</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 internal" href="#min"><span class="xref myst">9</span></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>
</section>
<section id="technicalities">
<h2>Technicalities<a class="headerlink" href="#technicalities" title="Link to this heading">#</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">
\[\begin{split}
\begin{aligned}
z_{i,j}^{\text{hidden}} &amp;= b_i^{\text{hidden}} + w_i^{\text{hidden}}x_j \\
&amp;=
\begin{pmatrix}
b_i^{\text{hidden}} &amp; w_i^{\text{hidden}}
\end{pmatrix}
\begin{pmatrix}
1 \\
x_j
\end{pmatrix}
\end{aligned}
\end{split}\]</div>
</section>
<section id="final-technicalities-i">
<h2>Final technicalities I<a class="headerlink" href="#final-technicalities-i" title="Link to this heading">#</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">
\[\begin{split}
\begin{aligned}
\boldsymbol{z}_{i}^{\text{hidden}} &amp;= \Big( b_i^{\text{hidden}} + w_i^{\text{hidden}}x_1 , \ b_i^{\text{hidden}} + w_i^{\text{hidden}} x_2, \ \dots \, , \ b_i^{\text{hidden}} + w_i^{\text{hidden}} x_N\Big) \\
&amp;=
\begin{pmatrix}
b_i^{\text{hidden}} &amp; w_i^{\text{hidden}}
\end{pmatrix}
\begin{pmatrix}
1 &amp; 1 &amp; \dots &amp; 1 \\
x_1 &amp; x_2 &amp; \dots &amp; x_N
\end{pmatrix} \\
&amp;= \boldsymbol{p}_{i, \text{hidden}}^T X
\end{aligned}
\end{split}\]</div>
</section>
<section id="final-technicalities-ii">
<h2>Final technicalities II<a class="headerlink" href="#final-technicalities-ii" title="Link to this heading">#</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 internal" href="#min"><span class="xref myst">9</span></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>
</section>
<section id="final-technicalities-iii">
<h2>Final technicalities III<a class="headerlink" href="#final-technicalities-iii" title="Link to this heading">#</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">
\[\begin{split}
\begin{aligned}
z_{1,j}^{\text{output}} &amp; =
\begin{pmatrix}
b_1^{\text{output}} &amp; \boldsymbol{w}_1^{\text{output}}
\end{pmatrix}
\begin{pmatrix}
1 \\
\boldsymbol{x}_j^{\text{hidden}}
\end{pmatrix}
\end{aligned}
\end{split}\]</div>
</section>
<section id="final-technicalities-iv">
<h2>Final technicalities IV<a class="headerlink" href="#final-technicalities-iv" title="Link to this heading">#</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">
\[\begin{split}
\boldsymbol{z}_{1}^{\text{output}} =
\begin{pmatrix}
b_1^{\text{output}} &amp; \boldsymbol{w}_1^{\text{output}}
\end{pmatrix}
\begin{pmatrix}
1 &amp; 1 &amp; \dots &amp; 1 \\
\boldsymbol{x}_1^{\text{hidden}} &amp; \boldsymbol{x}_2^{\text{hidden}} &amp; \dots &amp; \boldsymbol{x}_N^{\text{hidden}}
\end{pmatrix}
\end{split}\]</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>
</section>
<section id="back-propagation">
<h2>Back propagation<a class="headerlink" href="#back-propagation" title="Link to this heading">#</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>
</section>
<section id="gradient-descent">
<h2>Gradient descent<a class="headerlink" href="#gradient-descent" title="Link to this heading">#</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">
\[\begin{split}
\begin{aligned}
P_{\text{hidden},\text{new}} &amp;= P_{\text{hidden}} - \lambda \nabla_{P_{\text{hidden}}} C(\boldsymbol{x}, P) \\
P_{\text{output},\text{new}} &amp;= P_{\text{output}} - \lambda \nabla_{P_{\text{output}}} C(\boldsymbol{x}, P)
\end{aligned}
\end{split}\]</div>
</section>
<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="Link to this heading">#</a></h2>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>import autograd.numpy as np
from autograd import grad, elementwise_grad
import autograd.numpy.random as npr
from matplotlib import pyplot as plt
def sigmoid(z):
return 1/(1 + np.exp(-z))
# Assuming one input, hidden, and output layer
def neural_network(params, x):
# Find the weights (including and biases) for the hidden and output layer.
# Assume that params is a list of parameters for each layer.
# The biases are the first element for each array in params,
# and the weights are the remaning elements in each array in params.
w_hidden = params[0]
w_output = params[1]
# Assumes input x being an one-dimensional array
num_values = np.size(x)
x = x.reshape(-1, num_values)
# Assume that the input layer does nothing to the input x
x_input = x
## Hidden layer:
# Add a row of ones to include bias
x_input = np.concatenate((np.ones((1,num_values)), x_input ), axis = 0)
z_hidden = np.matmul(w_hidden, x_input)
x_hidden = sigmoid(z_hidden)
## Output layer:
# Include bias:
x_hidden = np.concatenate((np.ones((1,num_values)), x_hidden ), axis = 0)
z_output = np.matmul(w_output, x_hidden)
x_output = z_output
return x_output
# The trial solution using the deep neural network:
def g_trial(x,params, g0 = 10):
return g0 + x*neural_network(params,x)
# The right side of the ODE:
def g(x, g_trial, gamma = 2):
return -gamma*g_trial
# The cost function:
def cost_function(P, x):
# Evaluate the trial function with the current parameters P
g_t = g_trial(x,P)
# Find the derivative w.r.t x of the neural network
d_net_out = elementwise_grad(neural_network,1)(P,x)
# Find the derivative w.r.t x of the trial function
d_g_t = elementwise_grad(g_trial,0)(x,P)
# The right side of the ODE
func = g(x, g_t)
err_sqr = (d_g_t - func)**2
cost_sum = np.sum(err_sqr)
return cost_sum / np.size(err_sqr)
# Solve the exponential decay ODE using neural network with one input, hidden, and output layer
def solve_ode_neural_network(x, num_neurons_hidden, num_iter, lmb):
## Set up initial weights and biases
# For the hidden layer
p0 = npr.randn(num_neurons_hidden, 2 )
# For the output layer
p1 = npr.randn(1, num_neurons_hidden + 1 ) # +1 since bias is included
P = [p0, p1]
print(&#39;Initial cost: %g&#39;%cost_function(P, x))
## Start finding the optimal weights using gradient descent
# Find the Python function that represents the gradient of the cost function
# w.r.t the 0-th input argument -- that is the weights and biases in the hidden and output layer
cost_function_grad = grad(cost_function,0)
# Let the update be done num_iter times
for i in range(num_iter):
# Evaluate the gradient at the current weights and biases in P.
# The cost_grad consist now of two arrays;
# one for the gradient w.r.t P_hidden and
# one for the gradient w.r.t P_output
cost_grad = cost_function_grad(P, x)
P[0] = P[0] - lmb * cost_grad[0]
P[1] = P[1] - lmb * cost_grad[1]
print(&#39;Final cost: %g&#39;%cost_function(P, x))
return P
def g_analytic(x, gamma = 2, g0 = 10):
return g0*np.exp(-gamma*x)
# Solve the given problem
if __name__ == &#39;__main__&#39;:
# Set seed such that the weight are initialized
# with same weights and biases for every run.
npr.seed(15)
## Decide the vales of arguments to the function to solve
N = 10
x = np.linspace(0, 1, N)
## Set up the initial parameters
num_hidden_neurons = 10
num_iter = 10000
lmb = 0.001
# Use the network
P = solve_ode_neural_network(x, num_hidden_neurons, num_iter, lmb)
# Print the deviation from the trial solution and true solution
res = g_trial(x,P)
res_analytical = g_analytic(x)
print(&#39;Max absolute difference: %g&#39;%np.max(np.abs(res - res_analytical)))
# Plot the results
plt.figure(figsize=(10,10))
plt.title(&#39;Performance of neural network solving an ODE compared to the analytical solution&#39;)
plt.plot(x, res_analytical)
plt.plot(x, res[0,:])
plt.legend([&#39;analytical&#39;,&#39;nn&#39;])
plt.xlabel(&#39;x&#39;)
plt.ylabel(&#39;g(x)&#39;)
plt.show()
</pre></div>
</div>
</div>
</div>
</section>
<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="Link to this heading">#</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-none notranslate"><div class="highlight"><pre><span></span>import autograd.numpy as np
from autograd import grad, elementwise_grad
import autograd.numpy.random as npr
from matplotlib import pyplot as plt
def sigmoid(z):
return 1/(1 + np.exp(-z))
# The neural network with one input layer and one output layer,
# but with number of hidden layers specified by the user.
def deep_neural_network(deep_params, x):
# N_hidden is the number of hidden layers
# deep_params is a list, len() should be used
N_hidden = len(deep_params) - 1 # -1 since params consists of
# parameters to all the hidden
# layers AND the output layer.
# Assumes input x being an one-dimensional array
num_values = np.size(x)
x = x.reshape(-1, num_values)
# Assume that the input layer does nothing to the input x
x_input = x
# Due to multiple hidden layers, define a variable referencing to the
# output of the previous layer:
x_prev = x_input
## Hidden layers:
for l in range(N_hidden):
# From the list of parameters P; find the correct weigths and bias for this layer
w_hidden = deep_params[l]
# Add a row of ones to include bias
x_prev = np.concatenate((np.ones((1,num_values)), x_prev ), axis = 0)
z_hidden = np.matmul(w_hidden, x_prev)
x_hidden = sigmoid(z_hidden)
# Update x_prev such that next layer can use the output from this layer
x_prev = x_hidden
## Output layer:
# Get the weights and bias for this layer
w_output = deep_params[-1]
# Include bias:
x_prev = np.concatenate((np.ones((1,num_values)), x_prev), axis = 0)
z_output = np.matmul(w_output, x_prev)
x_output = z_output
return x_output
# The trial solution using the deep neural network:
def g_trial_deep(x,params, g0 = 10):
return g0 + x*deep_neural_network(params, x)
# The right side of the ODE:
def g(x, g_trial, gamma = 2):
return -gamma*g_trial
# The same cost function as before, but calls deep_neural_network instead.
def cost_function_deep(P, x):
# Evaluate the trial function with the current parameters P
g_t = g_trial_deep(x,P)
# Find the derivative w.r.t x of the neural network
d_net_out = elementwise_grad(deep_neural_network,1)(P,x)
# Find the derivative w.r.t x of the trial function
d_g_t = elementwise_grad(g_trial_deep,0)(x,P)
# The right side of the ODE
func = g(x, g_t)
err_sqr = (d_g_t - func)**2
cost_sum = np.sum(err_sqr)
return cost_sum / np.size(err_sqr)
# Solve the exponential decay ODE using neural network with one input and one output layer,
# but with specified number of hidden layers from the user.
def solve_ode_deep_neural_network(x, num_neurons, num_iter, lmb):
# num_hidden_neurons is now a list of number of neurons within each hidden layer
# The number of elements in the list num_hidden_neurons thus represents
# the number of hidden layers.
# Find the number of hidden layers:
N_hidden = np.size(num_neurons)
## Set up initial weights and biases
# Initialize the list of parameters:
P = [None]*(N_hidden + 1) # + 1 to include the output layer
P[0] = npr.randn(num_neurons[0], 2 )
for l in range(1,N_hidden):
P[l] = npr.randn(num_neurons[l], num_neurons[l-1] + 1) # +1 to include bias
# For the output layer
P[-1] = npr.randn(1, num_neurons[-1] + 1 ) # +1 since bias is included
print(&#39;Initial cost: %g&#39;%cost_function_deep(P, x))
## Start finding the optimal weights using gradient descent
# Find the Python function that represents the gradient of the cost function
# w.r.t the 0-th input argument -- that is the weights and biases in the hidden and output layer
cost_function_deep_grad = grad(cost_function_deep,0)
# Let the update be done num_iter times
for i in range(num_iter):
# Evaluate the gradient at the current weights and biases in P.
# The cost_grad consist now of N_hidden + 1 arrays; the gradient w.r.t the weights and biases
# in the hidden layers and output layers evaluated at x.
cost_deep_grad = cost_function_deep_grad(P, x)
for l in range(N_hidden+1):
P[l] = P[l] - lmb * cost_deep_grad[l]
print(&#39;Final cost: %g&#39;%cost_function_deep(P, x))
return P
def g_analytic(x, gamma = 2, g0 = 10):
return g0*np.exp(-gamma*x)
# Solve the given problem
if __name__ == &#39;__main__&#39;:
npr.seed(15)
## Decide the vales of arguments to the function to solve
N = 10
x = np.linspace(0, 1, N)
## Set up the initial parameters
num_hidden_neurons = np.array([10,10])
num_iter = 10000
lmb = 0.001
P = solve_ode_deep_neural_network(x, num_hidden_neurons, num_iter, lmb)
res = g_trial_deep(x,P)
res_analytical = g_analytic(x)
plt.figure(figsize=(10,10))
plt.title(&#39;Performance of a deep neural network solving an ODE compared to the analytical solution&#39;)
plt.plot(x, res_analytical)
plt.plot(x, res[0,:])
plt.legend([&#39;analytical&#39;,&#39;dnn&#39;])
plt.ylabel(&#39;g(x)&#39;)
plt.show()
</pre></div>
</div>
</div>
</div>
</section>
<section id="example-population-growth">
<h2>Example: Population growth<a class="headerlink" href="#example-population-growth" title="Link to this heading">#</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">\(\alpha &gt; 0\)</span> the growth rate and <span class="math notranslate nohighlight">\(A &gt; 0\)</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>
</section>
<section id="setting-up-the-problem">
<h2>Setting up the problem<a class="headerlink" href="#setting-up-the-problem" title="Link to this heading">#</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>
</section>
<section id="id3">
<h2>The trial solution<a class="headerlink" href="#id3" title="Link to this heading">#</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>
</section>
<section id="the-program-using-autograd">
<h2>The program using Autograd<a class="headerlink" href="#the-program-using-autograd" title="Link to this heading">#</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-none notranslate"><div class="highlight"><pre><span></span>import autograd.numpy as np
from autograd import grad, elementwise_grad
import autograd.numpy.random as npr
from matplotlib import pyplot as plt
def sigmoid(z):
return 1/(1 + np.exp(-z))
# Function to get the parameters.
# Done such that one can easily change the paramaters after one&#39;s liking.
def get_parameters():
alpha = 2
A = 1
g0 = 1.2
return alpha, A, g0
def deep_neural_network(deep_params, x):
# N_hidden is the number of hidden layers
# deep_params is a list, len() should be used
N_hidden = len(deep_params) - 1 # -1 since params consists of
# parameters to all the hidden
# layers AND the output layer.
# Assumes input x being an one-dimensional array
num_values = np.size(x)
x = x.reshape(-1, num_values)
# Assume that the input layer does nothing to the input x
x_input = x
# Due to multiple hidden layers, define a variable referencing to the
# output of the previous layer:
x_prev = x_input
## Hidden layers:
for l in range(N_hidden):
# From the list of parameters P; find the correct weigths and bias for this layer
w_hidden = deep_params[l]
# Add a row of ones to include bias
x_prev = np.concatenate((np.ones((1,num_values)), x_prev ), axis = 0)
z_hidden = np.matmul(w_hidden, x_prev)
x_hidden = sigmoid(z_hidden)
# Update x_prev such that next layer can use the output from this layer
x_prev = x_hidden
## Output layer:
# Get the weights and bias for this layer
w_output = deep_params[-1]
# Include bias:
x_prev = np.concatenate((np.ones((1,num_values)), x_prev), axis = 0)
z_output = np.matmul(w_output, x_prev)
x_output = z_output
return x_output
def cost_function_deep(P, x):
# Evaluate the trial function with the current parameters P
g_t = g_trial_deep(x,P)
# Find the derivative w.r.t x of the trial function
d_g_t = elementwise_grad(g_trial_deep,0)(x,P)
# The right side of the ODE
func = f(x, g_t)
err_sqr = (d_g_t - func)**2
cost_sum = np.sum(err_sqr)
return cost_sum / np.size(err_sqr)
# The right side of the ODE:
def f(x, g_trial):
alpha,A, g0 = get_parameters()
return alpha*g_trial*(A - g_trial)
# The trial solution using the deep neural network:
def g_trial_deep(x, params):
alpha,A, g0 = get_parameters()
return g0 + x*deep_neural_network(params,x)
# The analytical solution:
def g_analytic(t):
alpha,A, g0 = get_parameters()
return A*g0/(g0 + (A - g0)*np.exp(-alpha*A*t))
def solve_ode_deep_neural_network(x, num_neurons, num_iter, lmb):
# num_hidden_neurons is now a list of number of neurons within each hidden layer
# Find the number of hidden layers:
N_hidden = np.size(num_neurons)
## Set up initial weigths and biases
# Initialize the list of parameters:
P = [None]*(N_hidden + 1) # + 1 to include the output layer
P[0] = npr.randn(num_neurons[0], 2 )
for l in range(1,N_hidden):
P[l] = npr.randn(num_neurons[l], num_neurons[l-1] + 1) # +1 to include bias
# For the output layer
P[-1] = npr.randn(1, num_neurons[-1] + 1 ) # +1 since bias is included
print(&#39;Initial cost: %g&#39;%cost_function_deep(P, x))
## Start finding the optimal weigths using gradient descent
# Find the Python function that represents the gradient of the cost function
# w.r.t the 0-th input argument -- that is the weights and biases in the hidden and output layer
cost_function_deep_grad = grad(cost_function_deep,0)
# Let the update be done num_iter times
for i in range(num_iter):
# Evaluate the gradient at the current weights and biases in P.
# The cost_grad consist now of N_hidden + 1 arrays; the gradient w.r.t the weights and biases
# in the hidden layers and output layers evaluated at x.
cost_deep_grad = cost_function_deep_grad(P, x)
for l in range(N_hidden+1):
P[l] = P[l] - lmb * cost_deep_grad[l]
print(&#39;Final cost: %g&#39;%cost_function_deep(P, x))
return P
if __name__ == &#39;__main__&#39;:
npr.seed(4155)
## Decide the vales of arguments to the function to solve
Nt = 10
T = 1
t = np.linspace(0,T, Nt)
## Set up the initial parameters
num_hidden_neurons = [100, 50, 25]
num_iter = 1000
lmb = 1e-3
P = solve_ode_deep_neural_network(t, num_hidden_neurons, num_iter, lmb)
g_dnn_ag = g_trial_deep(t,P)
g_analytical = g_analytic(t)
# Find the maximum absolute difference between the solutons:
diff_ag = np.max(np.abs(g_dnn_ag - g_analytical))
print(&quot;The max absolute difference between the solutions is: %g&quot;%diff_ag)
plt.figure(figsize=(10,10))
plt.title(&#39;Performance of neural network solving an ODE compared to the analytical solution&#39;)
plt.plot(t, g_analytical)
plt.plot(t, g_dnn_ag[0,:])
plt.legend([&#39;analytical&#39;,&#39;nn&#39;])
plt.xlabel(&#39;t&#39;)
plt.ylabel(&#39;g(t)&#39;)
plt.show()
</pre></div>
</div>
</div>
</div>
</section>
<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="Link to this heading">#</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">
\[\begin{split}
\begin{aligned}
g(t + \Delta t) &amp;\approx g(t) + \Delta t g'(t) \\
&amp;= g(t) + \Delta t \big(\alpha g(t)(A - g(t))\big)
\end{aligned}
\end{split}\]</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">
\[\begin{split}
\begin{aligned}
t_i &amp;= i\Delta t \\
&amp;= (i - 1)\Delta t + \Delta t \\
&amp;= t_{i-1} + \Delta t
\end{aligned}
\end{split}\]</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">
\[\begin{split}
\begin{equation}
\begin{aligned}
g_i &amp;= g(t_i) \\
&amp;= g(t_{i-1} + \Delta t) \\
&amp;\approx g(t_{i-1}) + \Delta t \big(\alpha g(t_{i-1})(A - g(t_{i-1}))\big) \\
&amp;= g_{i-1} + \Delta t \big(\alpha g_{i-1}(A - g_{i-1})\big)
\end{aligned}
\end{equation} \label{odenum} \tag{12}
\end{split}\]</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 internal" href="#odenum"><span class="xref myst">12</span></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-none notranslate"><div class="highlight"><pre><span></span># Assume that all function definitions from the example program using Autograd
# are located here.
if __name__ == &#39;__main__&#39;:
npr.seed(4155)
## Decide the vales of arguments to the function to solve
Nt = 10
T = 1
t = np.linspace(0,T, Nt)
## Set up the initial parameters
num_hidden_neurons = [100,50,25]
num_iter = 1000
lmb = 1e-3
P = solve_ode_deep_neural_network(t, num_hidden_neurons, num_iter, lmb)
g_dnn_ag = g_trial_deep(t,P)
g_analytical = g_analytic(t)
# Find the maximum absolute difference between the solutons:
diff_ag = np.max(np.abs(g_dnn_ag - g_analytical))
print(&quot;The max absolute difference between the solutions is: %g&quot;%diff_ag)
plt.figure(figsize=(10,10))
plt.title(&#39;Performance of neural network solving an ODE compared to the analytical solution&#39;)
plt.plot(t, g_analytical)
plt.plot(t, g_dnn_ag[0,:])
plt.legend([&#39;analytical&#39;,&#39;nn&#39;])
plt.xlabel(&#39;t&#39;)
plt.ylabel(&#39;g(t)&#39;)
## Find an approximation to the funtion using forward Euler
alpha, A, g0 = get_parameters()
dt = T/(Nt - 1)
# Perform forward Euler to solve the ODE
g_euler = np.zeros(Nt)
g_euler[0] = g0
for i in range(1,Nt):
g_euler[i] = g_euler[i-1] + dt*(alpha*g_euler[i-1]*(A - g_euler[i-1]))
# Print the errors done by each method
diff1 = np.max(np.abs(g_euler - g_analytical))
diff2 = np.max(np.abs(g_dnn_ag[0,:] - g_analytical))
print(&#39;Max absolute difference between Euler method and analytical: %g&#39;%diff1)
print(&#39;Max absolute difference between deep neural network and analytical: %g&#39;%diff2)
# Plot results
plt.figure(figsize=(10,10))
plt.plot(t,g_euler)
plt.plot(t,g_analytical)
plt.plot(t,g_dnn_ag[0,:])
plt.legend([&#39;euler&#39;,&#39;analytical&#39;,&#39;dnn&#39;])
plt.xlabel(&#39;Time t&#39;)
plt.ylabel(&#39;g(t)&#39;)
plt.show()
</pre></div>
</div>
</div>
</div>
</section>
<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="Link to this heading">#</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">
\[\begin{split}
\begin{align*}
g(0) &amp;= 0 \\
g(1) &amp;= 0
\end{align*}
\end{split}\]</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>
</section>
<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="Link to this heading">#</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>
</section>
<section id="solving-the-equation-using-autograd">
<h2>Solving the equation using Autograd<a class="headerlink" href="#solving-the-equation-using-autograd" title="Link to this heading">#</a></h2>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>import autograd.numpy as np
from autograd import grad, elementwise_grad
import autograd.numpy.random as npr
from matplotlib import pyplot as plt
def sigmoid(z):
return 1/(1 + np.exp(-z))
def deep_neural_network(deep_params, x):
# N_hidden is the number of hidden layers
# deep_params is a list, len() should be used
N_hidden = len(deep_params) - 1 # -1 since params consists of
# parameters to all the hidden
# layers AND the output layer.
# Assumes input x being an one-dimensional array
num_values = np.size(x)
x = x.reshape(-1, num_values)
# Assume that the input layer does nothing to the input x
x_input = x
# Due to multiple hidden layers, define a variable referencing to the
# output of the previous layer:
x_prev = x_input
## Hidden layers:
for l in range(N_hidden):
# From the list of parameters P; find the correct weigths and bias for this layer
w_hidden = deep_params[l]
# Add a row of ones to include bias
x_prev = np.concatenate((np.ones((1,num_values)), x_prev ), axis = 0)
z_hidden = np.matmul(w_hidden, x_prev)
x_hidden = sigmoid(z_hidden)
# Update x_prev such that next layer can use the output from this layer
x_prev = x_hidden
## Output layer:
# Get the weights and bias for this layer
w_output = deep_params[-1]
# Include bias:
x_prev = np.concatenate((np.ones((1,num_values)), x_prev), axis = 0)
z_output = np.matmul(w_output, x_prev)
x_output = z_output
return x_output
def solve_ode_deep_neural_network(x, num_neurons, num_iter, lmb):
# num_hidden_neurons is now a list of number of neurons within each hidden layer
# Find the number of hidden layers:
N_hidden = np.size(num_neurons)
## Set up initial weigths and biases
# Initialize the list of parameters:
P = [None]*(N_hidden + 1) # + 1 to include the output layer
P[0] = npr.randn(num_neurons[0], 2 )
for l in range(1,N_hidden):
P[l] = npr.randn(num_neurons[l], num_neurons[l-1] + 1) # +1 to include bias
# For the output layer
P[-1] = npr.randn(1, num_neurons[-1] + 1 ) # +1 since bias is included
print(&#39;Initial cost: %g&#39;%cost_function_deep(P, x))
## Start finding the optimal weigths using gradient descent
# Find the Python function that represents the gradient of the cost function
# w.r.t the 0-th input argument -- that is the weights and biases in the hidden and output layer
cost_function_deep_grad = grad(cost_function_deep,0)
# Let the update be done num_iter times
for i in range(num_iter):
# Evaluate the gradient at the current weights and biases in P.
# The cost_grad consist now of N_hidden + 1 arrays; the gradient w.r.t the weights and biases
# in the hidden layers and output layers evaluated at x.
cost_deep_grad = cost_function_deep_grad(P, x)
for l in range(N_hidden+1):
P[l] = P[l] - lmb * cost_deep_grad[l]
print(&#39;Final cost: %g&#39;%cost_function_deep(P, x))
return P
## Set up the cost function specified for this Poisson equation:
# The right side of the ODE
def f(x):
return (3*x + x**2)*np.exp(x)
def cost_function_deep(P, x):
# Evaluate the trial function with the current parameters P
g_t = g_trial_deep(x,P)
# Find the derivative w.r.t x of the trial function
d2_g_t = elementwise_grad(elementwise_grad(g_trial_deep,0))(x,P)
right_side = f(x)
err_sqr = (-d2_g_t - right_side)**2
cost_sum = np.sum(err_sqr)
return cost_sum/np.size(err_sqr)
# The trial solution:
def g_trial_deep(x,P):
return x*(1-x)*deep_neural_network(P,x)
# The analytic solution;
def g_analytic(x):
return x*(1-x)*np.exp(x)
if __name__ == &#39;__main__&#39;:
npr.seed(4155)
## Decide the vales of arguments to the function to solve
Nx = 10
x = np.linspace(0,1, Nx)
## Set up the initial parameters
num_hidden_neurons = [200,100]
num_iter = 1000
lmb = 1e-3
P = solve_ode_deep_neural_network(x, num_hidden_neurons, num_iter, lmb)
g_dnn_ag = g_trial_deep(x,P)
g_analytical = g_analytic(x)
# Find the maximum absolute difference between the solutons:
max_diff = np.max(np.abs(g_dnn_ag - g_analytical))
print(&quot;The max absolute difference between the solutions is: %g&quot;%max_diff)
plt.figure(figsize=(10,10))
plt.title(&#39;Performance of neural network solving an ODE compared to the analytical solution&#39;)
plt.plot(x, g_analytical)
plt.plot(x, g_dnn_ag[0,:])
plt.legend([&#39;analytical&#39;,&#39;nn&#39;])
plt.xlabel(&#39;x&#39;)
plt.ylabel(&#39;g(x)&#39;)
plt.show()
</pre></div>
</div>
</div>
</div>
</section>
<section id="comparing-with-a-numerical-scheme">
<h2>Comparing with a numerical scheme<a class="headerlink" href="#comparing-with-a-numerical-scheme" title="Link to this heading">#</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 internal" href="#approx"><span class="xref myst">15</span></a>) becomes</p>
<div class="math notranslate nohighlight">
\[\begin{split}
\begin{aligned}
g''(x_i) &amp;\approx \frac{g(x_i + \Delta x) - 2g(x_i) + g(x_i -\Delta x)}{\Delta x^2} \\
&amp;= \frac{g_{i+1} - 2g_i + g_{i-1}}{\Delta x^2}
\end{aligned}
\end{split}\]</div>
<p>Since we know from our problem that</p>
<div class="math notranslate nohighlight">
\[\begin{split}
\begin{aligned}
-g''(x) &amp;= f(x) \\
&amp;= (3x + x^2)\exp(x)
\end{aligned}
\end{split}\]</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">
\[\begin{split}
\begin{equation}
\begin{aligned}
-\Big( \frac{g_{i+1} - 2g_i + g_{i-1}}{\Delta x^2} \Big) &amp;= f(x_i) \\
-g_{i+1} + 2g_i - g_{i-1} &amp;= \Delta x^2 f(x_i)
\end{aligned}
\end{equation} \label{odesys} \tag{16}
\end{split}\]</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">
\[\begin{split}
\begin{aligned}
\begin{pmatrix}
2 &amp; -1 &amp; 0 &amp; \dots &amp; 0 \\
-1 &amp; 2 &amp; -1 &amp; \dots &amp; 0 \\
\vdots &amp; &amp; \ddots &amp; &amp; \vdots \\
0 &amp; \dots &amp; -1 &amp; 2 &amp; -1 \\
0 &amp; \dots &amp; 0 &amp; -1 &amp; 2\\
\end{pmatrix}
\begin{pmatrix}
g_1 \\
g_2 \\
\vdots \\
g_{N_x - 3} \\
g_{N_x - 2}
\end{pmatrix}
&amp;=
\Delta x^2
\begin{pmatrix}
f(x_1) \\
f(x_2) \\
\vdots \\
f(x_{N_x - 3}) \\
f(x_{N_x - 2})
\end{pmatrix} \\
\boldsymbol{A}\boldsymbol{g} &amp;= \boldsymbol{f},
\end{aligned}
\end{split}\]</div>
<p>which makes it possible to solve for the vector <span class="math notranslate nohighlight">\(\boldsymbol{g}\)</span>.</p>
</section>
<section id="setting-up-the-code">
<h2>Setting up the code<a class="headerlink" href="#setting-up-the-code" title="Link to this heading">#</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-none notranslate"><div class="highlight"><pre><span></span>import autograd.numpy as np
from autograd import grad, elementwise_grad
import autograd.numpy.random as npr
from matplotlib import pyplot as plt
def sigmoid(z):
return 1/(1 + np.exp(-z))
def deep_neural_network(deep_params, x):
# N_hidden is the number of hidden layers
# deep_params is a list, len() should be used
N_hidden = len(deep_params) - 1 # -1 since params consists of
# parameters to all the hidden
# layers AND the output layer.
# Assumes input x being an one-dimensional array
num_values = np.size(x)
x = x.reshape(-1, num_values)
# Assume that the input layer does nothing to the input x
x_input = x
# Due to multiple hidden layers, define a variable referencing to the
# output of the previous layer:
x_prev = x_input
## Hidden layers:
for l in range(N_hidden):
# From the list of parameters P; find the correct weigths and bias for this layer
w_hidden = deep_params[l]
# Add a row of ones to include bias
x_prev = np.concatenate((np.ones((1,num_values)), x_prev ), axis = 0)
z_hidden = np.matmul(w_hidden, x_prev)
x_hidden = sigmoid(z_hidden)
# Update x_prev such that next layer can use the output from this layer
x_prev = x_hidden
## Output layer:
# Get the weights and bias for this layer
w_output = deep_params[-1]
# Include bias:
x_prev = np.concatenate((np.ones((1,num_values)), x_prev), axis = 0)
z_output = np.matmul(w_output, x_prev)
x_output = z_output
return x_output
def solve_ode_deep_neural_network(x, num_neurons, num_iter, lmb):
# num_hidden_neurons is now a list of number of neurons within each hidden layer
# Find the number of hidden layers:
N_hidden = np.size(num_neurons)
## Set up initial weigths and biases
# Initialize the list of parameters:
P = [None]*(N_hidden + 1) # + 1 to include the output layer
P[0] = npr.randn(num_neurons[0], 2 )
for l in range(1,N_hidden):
P[l] = npr.randn(num_neurons[l], num_neurons[l-1] + 1) # +1 to include bias
# For the output layer
P[-1] = npr.randn(1, num_neurons[-1] + 1 ) # +1 since bias is included
print(&#39;Initial cost: %g&#39;%cost_function_deep(P, x))
## Start finding the optimal weigths using gradient descent
# Find the Python function that represents the gradient of the cost function
# w.r.t the 0-th input argument -- that is the weights and biases in the hidden and output layer
cost_function_deep_grad = grad(cost_function_deep,0)
# Let the update be done num_iter times
for i in range(num_iter):
# Evaluate the gradient at the current weights and biases in P.
# The cost_grad consist now of N_hidden + 1 arrays; the gradient w.r.t the weights and biases
# in the hidden layers and output layers evaluated at x.
cost_deep_grad = cost_function_deep_grad(P, x)
for l in range(N_hidden+1):
P[l] = P[l] - lmb * cost_deep_grad[l]
print(&#39;Final cost: %g&#39;%cost_function_deep(P, x))
return P
## Set up the cost function specified for this Poisson equation:
# The right side of the ODE
def f(x):
return (3*x + x**2)*np.exp(x)
def cost_function_deep(P, x):
# Evaluate the trial function with the current parameters P
g_t = g_trial_deep(x,P)
# Find the derivative w.r.t x of the trial function
d2_g_t = elementwise_grad(elementwise_grad(g_trial_deep,0))(x,P)
right_side = f(x)
err_sqr = (-d2_g_t - right_side)**2
cost_sum = np.sum(err_sqr)
return cost_sum/np.size(err_sqr)
# The trial solution:
def g_trial_deep(x,P):
return x*(1-x)*deep_neural_network(P,x)
# The analytic solution;
def g_analytic(x):
return x*(1-x)*np.exp(x)
if __name__ == &#39;__main__&#39;:
npr.seed(4155)
## Decide the vales of arguments to the function to solve
Nx = 10
x = np.linspace(0,1, Nx)
## Set up the initial parameters
num_hidden_neurons = [200,100]
num_iter = 1000
lmb = 1e-3
P = solve_ode_deep_neural_network(x, num_hidden_neurons, num_iter, lmb)
g_dnn_ag = g_trial_deep(x,P)
g_analytical = g_analytic(x)
# Find the maximum absolute difference between the solutons:
plt.figure(figsize=(10,10))
plt.title(&#39;Performance of neural network solving an ODE compared to the analytical solution&#39;)
plt.plot(x, g_analytical)
plt.plot(x, g_dnn_ag[0,:])
plt.legend([&#39;analytical&#39;,&#39;nn&#39;])
plt.xlabel(&#39;x&#39;)
plt.ylabel(&#39;g(x)&#39;)
## Perform the computation using the numerical scheme
dx = 1/(Nx - 1)
# Set up the matrix A
A = np.zeros((Nx-2,Nx-2))
A[0,0] = 2
A[0,1] = -1
for i in range(1,Nx-3):
A[i,i-1] = -1
A[i,i] = 2
A[i,i+1] = -1
A[Nx - 3, Nx - 4] = -1
A[Nx - 3, Nx - 3] = 2
# Set up the vector f
f_vec = dx**2 * f(x[1:-1])
# Solve the equation
g_res = np.linalg.solve(A,f_vec)
g_vec = np.zeros(Nx)
g_vec[1:-1] = g_res
# Print the differences between each method
max_diff1 = np.max(np.abs(g_dnn_ag - g_analytical))
max_diff2 = np.max(np.abs(g_vec - g_analytical))
print(&quot;The max absolute difference between the analytical solution and DNN Autograd: %g&quot;%max_diff1)
print(&quot;The max absolute difference between the analytical solution and numerical scheme: %g&quot;%max_diff2)
# Plot the results
plt.figure(figsize=(10,10))
plt.plot(x,g_vec)
plt.plot(x,g_analytical)
plt.plot(x,g_dnn_ag[0,:])
plt.legend([&#39;numerical scheme&#39;,&#39;analytical&#39;,&#39;dnn&#39;])
plt.show()
</pre></div>
</div>
</div>
</div>
</section>
<section id="partial-differential-equations">
<h2>Partial Differential Equations<a class="headerlink" href="#partial-differential-equations" title="Link to this heading">#</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>
</section>
<section id="type-of-problem">
<h2>Type of problem<a class="headerlink" href="#type-of-problem" title="Link to this heading">#</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>
</section>
<section id="network-requirements">
<h2>Network requirements<a class="headerlink" href="#network-requirements" title="Link to this heading">#</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 internal" href="#PDE"><span class="xref myst">17</span></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>
</section>
<section id="id4">
<h2>More details<a class="headerlink" href="#id4" title="Link to this heading">#</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>
</section>
<section id="example-the-diffusion-equation">
<h2>Example: The diffusion equation<a class="headerlink" href="#example-the-diffusion-equation" title="Link to this heading">#</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">
\[\begin{split}
\begin{align*}
g(0,t) &amp;= 0 ,\qquad t \geq 0 \\
g(1,t) &amp;= 0, \qquad t \geq 0 \\
g(x,0) &amp;= u(x),\qquad x\in [0,1]
\end{align*}
\end{split}\]</div>
<p>with <span class="math notranslate nohighlight">\(u(x)\)</span> being some given function.</p>
</section>
<section id="defining-the-problem">
<h2>Defining the problem<a class="headerlink" href="#defining-the-problem" title="Link to this heading">#</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">
\[\begin{split}
\begin{align*}
g(0,t) &amp;= 0 ,\qquad t \geq 0 \\
g(1,t) &amp;= 0, \qquad t \geq 0 \\
g(x,0) &amp;= u(x),\qquad x\in [0,1]
\end{align*}
\end{split}\]</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>
</section>
<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="Link to this heading">#</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-none notranslate"><div class="highlight"><pre><span></span>def sigmoid(z):
return 1/(1 + np.exp(-z))
def deep_neural_network(deep_params, x):
# x is now a point and a 1D numpy array; make it a column vector
num_coordinates = np.size(x,0)
x = x.reshape(num_coordinates,-1)
num_points = np.size(x,1)
# N_hidden is the number of hidden layers
N_hidden = len(deep_params) - 1 # -1 since params consist of parameters to all the hidden layers AND the output layer
# Assume that the input layer does nothing to the input x
x_input = x
x_prev = x_input
## Hidden layers:
for l in range(N_hidden):
# From the list of parameters P; find the correct weigths and bias for this layer
w_hidden = deep_params[l]
# Add a row of ones to include bias
x_prev = np.concatenate((np.ones((1,num_points)), x_prev ), axis = 0)
z_hidden = np.matmul(w_hidden, x_prev)
x_hidden = sigmoid(z_hidden)
# Update x_prev such that next layer can use the output from this layer
x_prev = x_hidden
## Output layer:
# Get the weights and bias for this layer
w_output = deep_params[-1]
# Include bias:
x_prev = np.concatenate((np.ones((1,num_points)), x_prev), axis = 0)
z_output = np.matmul(w_output, x_prev)
x_output = z_output
return x_output[0][0]
</pre></div>
</div>
</div>
</div>
</section>
<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="Link to this heading">#</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>
</section>
<section id="why-the-jacobian">
<h2>Why the jacobian?<a class="headerlink" href="#why-the-jacobian" title="Link to this heading">#</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-none notranslate"><div class="highlight"><pre><span></span># Set up the trial function:
def u(x):
return np.sin(np.pi*x)
def g_trial(point,P):
x,t = point
return (1-t)*u(x) + x*(1-x)*t*deep_neural_network(P,point)
# The right side of the ODE:
def f(point):
return 0.
# The cost function:
def cost_function(P, x, t):
cost_sum = 0
g_t_jacobian_func = jacobian(g_trial)
g_t_hessian_func = hessian(g_trial)
for x_ in x:
for t_ in t:
point = np.array([x_,t_])
g_t = g_trial(point,P)
g_t_jacobian = g_t_jacobian_func(point,P)
g_t_hessian = g_t_hessian_func(point,P)
g_t_dt = g_t_jacobian[1]
g_t_d2x = g_t_hessian[0][0]
func = f(point)
err_sqr = ( (g_t_dt - g_t_d2x) - func)**2
cost_sum += err_sqr
return cost_sum
</pre></div>
</div>
</div>
</div>
</section>
<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="Link to this heading">#</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-none notranslate"><div class="highlight"><pre><span></span>import autograd.numpy as np
from autograd import jacobian,hessian,grad
import autograd.numpy.random as npr
from matplotlib import cm
from matplotlib import pyplot as plt
from mpl_toolkits.mplot3d import axes3d
## Set up the network
def sigmoid(z):
return 1/(1 + np.exp(-z))
def deep_neural_network(deep_params, x):
# x is now a point and a 1D numpy array; make it a column vector
num_coordinates = np.size(x,0)
x = x.reshape(num_coordinates,-1)
num_points = np.size(x,1)
# N_hidden is the number of hidden layers
N_hidden = len(deep_params) - 1 # -1 since params consist of parameters to all the hidden layers AND the output layer
# Assume that the input layer does nothing to the input x
x_input = x
x_prev = x_input
## Hidden layers:
for l in range(N_hidden):
# From the list of parameters P; find the correct weigths and bias for this layer
w_hidden = deep_params[l]
# Add a row of ones to include bias
x_prev = np.concatenate((np.ones((1,num_points)), x_prev ), axis = 0)
z_hidden = np.matmul(w_hidden, x_prev)
x_hidden = sigmoid(z_hidden)
# Update x_prev such that next layer can use the output from this layer
x_prev = x_hidden
## Output layer:
# Get the weights and bias for this layer
w_output = deep_params[-1]
# Include bias:
x_prev = np.concatenate((np.ones((1,num_points)), x_prev), axis = 0)
z_output = np.matmul(w_output, x_prev)
x_output = z_output
return x_output[0][0]
## Define the trial solution and cost function
def u(x):
return np.sin(np.pi*x)
def g_trial(point,P):
x,t = point
return (1-t)*u(x) + x*(1-x)*t*deep_neural_network(P,point)
# The right side of the ODE:
def f(point):
return 0.
# The cost function:
def cost_function(P, x, t):
cost_sum = 0
g_t_jacobian_func = jacobian(g_trial)
g_t_hessian_func = hessian(g_trial)
for x_ in x:
for t_ in t:
point = np.array([x_,t_])
g_t = g_trial(point,P)
g_t_jacobian = g_t_jacobian_func(point,P)
g_t_hessian = g_t_hessian_func(point,P)
g_t_dt = g_t_jacobian[1]
g_t_d2x = g_t_hessian[0][0]
func = f(point)
err_sqr = ( (g_t_dt - g_t_d2x) - func)**2
cost_sum += err_sqr
return cost_sum /( np.size(x)*np.size(t) )
## For comparison, define the analytical solution
def g_analytic(point):
x,t = point
return np.exp(-np.pi**2*t)*np.sin(np.pi*x)
## Set up a function for training the network to solve for the equation
def solve_pde_deep_neural_network(x,t, num_neurons, num_iter, lmb):
## Set up initial weigths and biases
N_hidden = np.size(num_neurons)
## Set up initial weigths and biases
# Initialize the list of parameters:
P = [None]*(N_hidden + 1) # + 1 to include the output layer
P[0] = npr.randn(num_neurons[0], 2 + 1 ) # 2 since we have two points, +1 to include bias
for l in range(1,N_hidden):
P[l] = npr.randn(num_neurons[l], num_neurons[l-1] + 1) # +1 to include bias
# For the output layer
P[-1] = npr.randn(1, num_neurons[-1] + 1 ) # +1 since bias is included
print(&#39;Initial cost: &#39;,cost_function(P, x, t))
cost_function_grad = grad(cost_function,0)
# Let the update be done num_iter times
for i in range(num_iter):
cost_grad = cost_function_grad(P, x , t)
for l in range(N_hidden+1):
P[l] = P[l] - lmb * cost_grad[l]
print(&#39;Final cost: &#39;,cost_function(P, x, t))
return P
if __name__ == &#39;__main__&#39;:
### Use the neural network:
npr.seed(15)
## Decide the vales of arguments to the function to solve
Nx = 10; Nt = 10
x = np.linspace(0, 1, Nx)
t = np.linspace(0,1,Nt)
## Set up the parameters for the network
num_hidden_neurons = [100, 25]
num_iter = 250
lmb = 0.01
P = solve_pde_deep_neural_network(x,t, num_hidden_neurons, num_iter, lmb)
## Store the results
g_dnn_ag = np.zeros((Nx, Nt))
G_analytical = np.zeros((Nx, Nt))
for i,x_ in enumerate(x):
for j, t_ in enumerate(t):
point = np.array([x_, t_])
g_dnn_ag[i,j] = g_trial(point,P)
G_analytical[i,j] = g_analytic(point)
# Find the map difference between the analytical and the computed solution
diff_ag = np.abs(g_dnn_ag - G_analytical)
print(&#39;Max absolute difference between the analytical solution and the network: %g&#39;%np.max(diff_ag))
## Plot the solutions in two dimensions, that being in position and time
T,X = np.meshgrid(t,x)
fig = plt.figure(figsize=(10,10))
ax = fig.add_suplot(projection=&#39;3d&#39;)
ax.set_title(&#39;Solution from the deep neural network w/ %d layer&#39;%len(num_hidden_neurons))
s = ax.plot_surface(T,X,g_dnn_ag,linewidth=0,antialiased=False,cmap=cm.viridis)
ax.set_xlabel(&#39;Time $t$&#39;)
ax.set_ylabel(&#39;Position $x$&#39;);
fig = plt.figure(figsize=(10,10))
ax = fig.add_suplot(projection=&#39;3d&#39;)
ax.set_title(&#39;Analytical solution&#39;)
s = ax.plot_surface(T,X,G_analytical,linewidth=0,antialiased=False,cmap=cm.viridis)
ax.set_xlabel(&#39;Time $t$&#39;)
ax.set_ylabel(&#39;Position $x$&#39;);
fig = plt.figure(figsize=(10,10))
ax = fig.add_suplot(projection=&#39;3d&#39;)
ax.set_title(&#39;Difference&#39;)
s = ax.plot_surface(T,X,diff_ag,linewidth=0,antialiased=False,cmap=cm.viridis)
ax.set_xlabel(&#39;Time $t$&#39;)
ax.set_ylabel(&#39;Position $x$&#39;);
## Take some slices of the 3D plots just to see the solutions at particular times
indx1 = 0
indx2 = int(Nt/2)
indx3 = Nt-1
t1 = t[indx1]
t2 = t[indx2]
t3 = t[indx3]
# Slice the results from the DNN
res1 = g_dnn_ag[:,indx1]
res2 = g_dnn_ag[:,indx2]
res3 = g_dnn_ag[:,indx3]
# Slice the analytical results
res_analytical1 = G_analytical[:,indx1]
res_analytical2 = G_analytical[:,indx2]
res_analytical3 = G_analytical[:,indx3]
# Plot the slices
plt.figure(figsize=(10,10))
plt.title(&quot;Computed solutions at time = %g&quot;%t1)
plt.plot(x, res1)
plt.plot(x,res_analytical1)
plt.legend([&#39;dnn&#39;,&#39;analytical&#39;])
plt.figure(figsize=(10,10))
plt.title(&quot;Computed solutions at time = %g&quot;%t2)
plt.plot(x, res2)
plt.plot(x,res_analytical2)
plt.legend([&#39;dnn&#39;,&#39;analytical&#39;])
plt.figure(figsize=(10,10))
plt.title(&quot;Computed solutions at time = %g&quot;%t3)
plt.plot(x, res3)
plt.plot(x,res_analytical3)
plt.legend([&#39;dnn&#39;,&#39;analytical&#39;])
plt.show()
</pre></div>
</div>
</div>
</div>
</section>
<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="Link to this heading">#</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">
\[\begin{split}
\begin{align*}
g(0,t) &amp;= 0 \\
g(1,t) &amp;= 0 \\
g(x,0) &amp;= u(x) \\
\frac{\partial g(x,t)}{\partial t} \Big |_{t = 0} &amp;= v(x)
\end{align*}
\end{split}\]</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>
</section>
<section id="the-problem-to-solve-for">
<h2>The problem to solve for<a class="headerlink" href="#the-problem-to-solve-for" title="Link to this heading">#</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">
\[\begin{split}
\begin{aligned}
g(0,t) &amp;= 0, &amp;t \geq 0 \\
g(1,t) &amp;= 0, &amp;t \geq 0 \\
g(x,0) &amp;= u(x), &amp;x\in[0,1] \\
\frac{\partial g(x,t)}{\partial t}\Big |_{t = 0} &amp;= v(x), &amp;x \in [0,1]
\end{aligned} \label{condwave} \tag{20}
\end{split}\]</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>
</section>
<section id="id5">
<h2>The trial solution<a class="headerlink" href="#id5" title="Link to this heading">#</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 internal" href="#condwave"><span class="xref myst">20</span></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>
</section>
<section id="the-analytical-solution">
<h2>The analytical solution<a class="headerlink" href="#the-analytical-solution" title="Link to this heading">#</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>
</section>
<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="Link to this heading">#</a></h2>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>import autograd.numpy as np
from autograd import hessian,grad
import autograd.numpy.random as npr
from matplotlib import cm
from matplotlib import pyplot as plt
from mpl_toolkits.mplot3d import axes3d
## Set up the trial function:
def u(x):
return np.sin(np.pi*x)
def v(x):
return -np.pi*np.sin(np.pi*x)
def h1(point):
x,t = point
return (1 - t**2)*u(x) + t*v(x)
def g_trial(point,P):
x,t = point
return h1(point) + x*(1-x)*t**2*deep_neural_network(P,point)
## Define the cost function
def cost_function(P, x, t):
cost_sum = 0
g_t_hessian_func = hessian(g_trial)
for x_ in x:
for t_ in t:
point = np.array([x_,t_])
g_t_hessian = g_t_hessian_func(point,P)
g_t_d2x = g_t_hessian[0][0]
g_t_d2t = g_t_hessian[1][1]
err_sqr = ( (g_t_d2t - g_t_d2x) )**2
cost_sum += err_sqr
return cost_sum / (np.size(t) * np.size(x))
## The neural network
def sigmoid(z):
return 1/(1 + np.exp(-z))
def deep_neural_network(deep_params, x):
# x is now a point and a 1D numpy array; make it a column vector
num_coordinates = np.size(x,0)
x = x.reshape(num_coordinates,-1)
num_points = np.size(x,1)
# N_hidden is the number of hidden layers
N_hidden = len(deep_params) - 1 # -1 since params consist of parameters to all the hidden layers AND the output layer
# Assume that the input layer does nothing to the input x
x_input = x
x_prev = x_input
## Hidden layers:
for l in range(N_hidden):
# From the list of parameters P; find the correct weigths and bias for this layer
w_hidden = deep_params[l]
# Add a row of ones to include bias
x_prev = np.concatenate((np.ones((1,num_points)), x_prev ), axis = 0)
z_hidden = np.matmul(w_hidden, x_prev)
x_hidden = sigmoid(z_hidden)
# Update x_prev such that next layer can use the output from this layer
x_prev = x_hidden
## Output layer:
# Get the weights and bias for this layer
w_output = deep_params[-1]
# Include bias:
x_prev = np.concatenate((np.ones((1,num_points)), x_prev), axis = 0)
z_output = np.matmul(w_output, x_prev)
x_output = z_output
return x_output[0][0]
## The analytical solution
def g_analytic(point):
x,t = point
return np.sin(np.pi*x)*np.cos(np.pi*t) - np.sin(np.pi*x)*np.sin(np.pi*t)
def solve_pde_deep_neural_network(x,t, num_neurons, num_iter, lmb):
## Set up initial weigths and biases
N_hidden = np.size(num_neurons)
## Set up initial weigths and biases
# Initialize the list of parameters:
P = [None]*(N_hidden + 1) # + 1 to include the output layer
P[0] = npr.randn(num_neurons[0], 2 + 1 ) # 2 since we have two points, +1 to include bias
for l in range(1,N_hidden):
P[l] = npr.randn(num_neurons[l], num_neurons[l-1] + 1) # +1 to include bias
# For the output layer
P[-1] = npr.randn(1, num_neurons[-1] + 1 ) # +1 since bias is included
print(&#39;Initial cost: &#39;,cost_function(P, x, t))
cost_function_grad = grad(cost_function,0)
# Let the update be done num_iter times
for i in range(num_iter):
cost_grad = cost_function_grad(P, x , t)
for l in range(N_hidden+1):
P[l] = P[l] - lmb * cost_grad[l]
print(&#39;Final cost: &#39;,cost_function(P, x, t))
return P
if __name__ == &#39;__main__&#39;:
### Use the neural network:
npr.seed(15)
## Decide the vales of arguments to the function to solve
Nx = 10; Nt = 10
x = np.linspace(0, 1, Nx)
t = np.linspace(0,1,Nt)
## Set up the parameters for the network
num_hidden_neurons = [50,20]
num_iter = 1000
lmb = 0.01
P = solve_pde_deep_neural_network(x,t, num_hidden_neurons, num_iter, lmb)
## Store the results
res = np.zeros((Nx, Nt))
res_analytical = np.zeros((Nx, Nt))
for i,x_ in enumerate(x):
for j, t_ in enumerate(t):
point = np.array([x_, t_])
res[i,j] = g_trial(point,P)
res_analytical[i,j] = g_analytic(point)
diff = np.abs(res - res_analytical)
print(&quot;Max difference between analytical and solution from nn: %g&quot;%np.max(diff))
## Plot the solutions in two dimensions, that being in position and time
T,X = np.meshgrid(t,x)
fig = plt.figure(figsize=(10,10))
ax = fig.add_suplot(projection=&#39;3d&#39;)
ax.set_title(&#39;Solution from the deep neural network w/ %d layer&#39;%len(num_hidden_neurons))
s = ax.plot_surface(T,X,res,linewidth=0,antialiased=False,cmap=cm.viridis)
ax.set_xlabel(&#39;Time $t$&#39;)
ax.set_ylabel(&#39;Position $x$&#39;);
fig = plt.figure(figsize=(10,10))
ax = fig.add_suplot(projection=&#39;3d&#39;)
ax.set_title(&#39;Analytical solution&#39;)
s = ax.plot_surface(T,X,res_analytical,linewidth=0,antialiased=False,cmap=cm.viridis)
ax.set_xlabel(&#39;Time $t$&#39;)
ax.set_ylabel(&#39;Position $x$&#39;);
fig = plt.figure(figsize=(10,10))
ax = fig.add_suplot(projection=&#39;3d&#39;)
ax.set_title(&#39;Difference&#39;)
s = ax.plot_surface(T,X,diff,linewidth=0,antialiased=False,cmap=cm.viridis)
ax.set_xlabel(&#39;Time $t$&#39;)
ax.set_ylabel(&#39;Position $x$&#39;);
## Take some slices of the 3D plots just to see the solutions at particular times
indx1 = 0
indx2 = int(Nt/2)
indx3 = Nt-1
t1 = t[indx1]
t2 = t[indx2]
t3 = t[indx3]
# Slice the results from the DNN
res1 = res[:,indx1]
res2 = res[:,indx2]
res3 = res[:,indx3]
# Slice the analytical results
res_analytical1 = res_analytical[:,indx1]
res_analytical2 = res_analytical[:,indx2]
res_analytical3 = res_analytical[:,indx3]
# Plot the slices
plt.figure(figsize=(10,10))
plt.title(&quot;Computed solutions at time = %g&quot;%t1)
plt.plot(x, res1)
plt.plot(x,res_analytical1)
plt.legend([&#39;dnn&#39;,&#39;analytical&#39;])
plt.figure(figsize=(10,10))
plt.title(&quot;Computed solutions at time = %g&quot;%t2)
plt.plot(x, res2)
plt.plot(x,res_analytical2)
plt.legend([&#39;dnn&#39;,&#39;analytical&#39;])
plt.figure(figsize=(10,10))
plt.title(&quot;Computed solutions at time = %g&quot;%t3)
plt.plot(x, res3)
plt.plot(x,res_analytical3)
plt.legend([&#39;dnn&#39;,&#39;analytical&#39;])
plt.show()
</pre></div>
</div>
</div>
</div>
</section>
<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="Link to this heading">#</a></h2>
<ol class="arabic 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>
</section>
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#plans-for-week-43">Plans for week 43</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#exercises-and-lab-session-week-43">Exercises and lab session week 43</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#using-automatic-differentiation">Using Automatic differentiation</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#back-propagation-and-automatic-differentiation">Back propagation and automatic differentiation</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#lecture-monday-october-20">Lecture Monday October 20</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#setting-up-the-back-propagation-algorithm-and-algorithm-for-a-feed-forward-nn-initalizations">Setting up the back propagation algorithm and algorithm for a feed forward NN, initalizations</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#setting-up-the-back-propagation-algorithm-part-1">Setting up the back propagation algorithm, part 1</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#setting-up-the-back-propagation-algorithm-part-2">Setting up the back propagation algorithm, part 2</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#setting-up-the-back-propagation-algorithm-part-3">Setting up the Back propagation algorithm, part 3</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#updating-the-gradients">Updating the gradients</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#activation-functions">Activation functions</a><ul class="nav section-nav flex-column">
<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#activation-functions-examples">Activation functions, examples</a></li>
</ul>
</li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#the-relu-function-family">The RELU function family</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#elu-function">ELU function</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#which-activation-function-should-we-use">Which activation function should we use?</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#more-on-activation-functions-output-layers">More on activation functions, output layers</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#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="#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="#using-pytorch-with-the-full-mnist-data-set">Using Pytorch with the full MNIST data set</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#and-a-similar-example-using-tensorflow-with-keras">And a similar example using Tensorflow with Keras</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#building-our-own-neural-network-code">Building our own neural network code</a><ul class="nav section-nav flex-column">
<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#learning-rate-methods">Learning rate methods</a></li>
<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#usage-of-the-above-learning-rate-schedulers">Usage of the above learning rate schedulers</a></li>
<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#cost-functions">Cost functions</a></li>
<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#id1">Activation functions</a></li>
<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#the-neural-network">The Neural Network</a></li>
<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#multiclass-classification">Multiclass classification</a></li>
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#testing-the-xor-gate-and-other-gates">Testing the XOR gate and other gates</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#solving-differential-equations-with-deep-learning">Solving differential equations with Deep Learning</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#ordinary-differential-equations-first">Ordinary Differential Equations first</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#the-trial-solution">The trial solution</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#minimization-process">Minimization process</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#minimizing-the-cost-function-using-gradient-descent-and-automatic-differentiation">Minimizing the cost function using gradient descent and automatic differentiation</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#example-exponential-decay">Example: Exponential decay</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#the-function-to-solve-for">The function to solve for</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#id2">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="#id3">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="#id4">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="#id5">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>
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