mhjensen
446 lines
30 KiB
HTML
446 lines
30 KiB
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<a class="navbar-brand" href="odenn-bs.html">Data Analysis and Machine Learning: Using Neural networks to solve ODEs and PDEs</a>
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<!-- navigation toc: --> <li><a href="._odenn-bs001.html#___sec0" style="font-size: 80%;">Differential equations</a></li>
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<!-- navigation toc: --> <li><a href="._odenn-bs002.html#___sec1" style="font-size: 80%;">Description of the equation to solve for</a></li>
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<!-- navigation toc: --> <li><a href="._odenn-bs003.html#___sec2" style="font-size: 80%;">Ordinary Differential Equations</a></li>
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<!-- navigation toc: --> <li><a href="._odenn-bs004.html#___sec3" style="font-size: 80%;">The trial solution</a></li>
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<!-- navigation toc: --> <li><a href="._odenn-bs005.html#___sec4" style="font-size: 80%;">Minimizing the cost function using gradient descent and automatic differentiation</a></li>
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<!-- navigation toc: --> <li><a href="._odenn-bs006.html#___sec5" style="font-size: 80%;">Example: Exponential decay and setting up the network using Autograd</a></li>
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<!-- navigation toc: --> <li><a href="._odenn-bs007.html#___sec6" style="font-size: 80%;">The function to solve for</a></li>
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<!-- navigation toc: --> <li><a href="._odenn-bs008.html#___sec7" style="font-size: 80%;">The trial solution</a></li>
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<!-- navigation toc: --> <li><a href="._odenn-bs009.html#___sec8" style="font-size: 80%;">Reformulating the problem</a></li>
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<!-- navigation toc: --> <li><a href="._odenn-bs010.html#___sec9" style="font-size: 80%;">A possible implementation of a neural network using Autograd</a></li>
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<!-- navigation toc: --> <li><a href="._odenn-bs011.html#___sec10" style="font-size: 80%;">Backpropagation using Autograd</a></li>
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<!-- navigation toc: --> <li><a href="._odenn-bs012.html#___sec11" style="font-size: 80%;">Gradient descent</a></li>
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<!-- navigation toc: --> <li><a href="._odenn-bs013.html#___sec12" style="font-size: 80%;">The network with one input, hidden, and output layer</a></li>
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<!-- navigation toc: --> <li><a href="._odenn-bs014.html#___sec13" style="font-size: 80%;">The network with one input layer, specified number of hidden layers, and one output layer output layer</a></li>
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<!-- navigation toc: --> <li><a href="._odenn-bs015.html#___sec14" style="font-size: 80%;">Example: Population growth, comparing Autograd, TensorFlow, and Euler's scheme</a></li>
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<!-- navigation toc: --> <li><a href="._odenn-bs016.html#___sec15" style="font-size: 80%;">Setting up the problem</a></li>
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<!-- navigation toc: --> <li><a href="._odenn-bs017.html#___sec16" style="font-size: 80%;">The trial solution</a></li>
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<!-- navigation toc: --> <li><a href="._odenn-bs018.html#___sec17" style="font-size: 80%;">The program using Autograd</a></li>
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<!-- navigation toc: --> <li><a href="._odenn-bs019.html#___sec18" style="font-size: 80%;">Using forward Euler to solve the ODE</a></li>
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<!-- navigation toc: --> <li><a href="._odenn-bs020.html#___sec19" style="font-size: 80%;">Using TensorFlow to model logistic population growth</a></li>
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<!-- navigation toc: --> <li><a href="._odenn-bs021.html#___sec20" style="font-size: 80%;">The general program flow in TensorFlow</a></li>
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<!-- navigation toc: --> <li><a href="._odenn-bs021.html#___sec21" style="font-size: 80%;">Program flow in TensorFlow - Construction phase</a></li>
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<!-- navigation toc: --> <li><a href="._odenn-bs021.html#___sec22" style="font-size: 80%;">Program flow in TensorFlow - Execution phase</a></li>
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<!-- navigation toc: --> <li><a href="._odenn-bs021.html#___sec23" style="font-size: 80%;">The full program modeling logistic population growth using TensorFlow</a></li>
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|
<!-- navigation toc: --> <li><a href="._odenn-bs022.html#___sec24" style="font-size: 80%;">Example: Solving the one dimensional Poisson equation using Autograd and TensorFlow</a></li>
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<!-- navigation toc: --> <li><a href="._odenn-bs023.html#___sec25" style="font-size: 80%;">The specific equation to solve for</a></li>
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<!-- navigation toc: --> <li><a href="#___sec26" style="font-size: 80%;">Solving the equation using Autograd</a></li>
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<!-- navigation toc: --> <li><a href="._odenn-bs025.html#___sec27" style="font-size: 80%;">Comparing with a numerical scheme</a></li>
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<!-- navigation toc: --> <li><a href="._odenn-bs026.html#___sec28" style="font-size: 80%;">Using gradient descent in TensorFlow to solve Poisson equation</a></li>
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|
<!-- navigation toc: --> <li><a href="._odenn-bs027.html#___sec29" style="font-size: 80%;">Using a different optimization algorithm implemented in TensorFlow to solve Poisson equation</a></li>
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|
<!-- navigation toc: --> <li><a href="._odenn-bs028.html#___sec30" style="font-size: 80%;">Partial Differential Equations</a></li>
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|
<!-- navigation toc: --> <li><a href="._odenn-bs029.html#___sec31" style="font-size: 80%;">Example: The diffusion equation</a></li>
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<!-- navigation toc: --> <li><a href="._odenn-bs030.html#___sec32" style="font-size: 80%;">Defining the problem</a></li>
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<!-- navigation toc: --> <li><a href="._odenn-bs031.html#___sec33" style="font-size: 80%;">Setting up the network using Autograd</a></li>
|
|
<!-- navigation toc: --> <li><a href="._odenn-bs032.html#___sec34" style="font-size: 80%;">Setting up the network using Autograd; The trial solution</a></li>
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<!-- navigation toc: --> <li><a href="._odenn-bs033.html#___sec35" style="font-size: 80%;">Setting up the network using Autograd; The full program</a></li>
|
|
<!-- navigation toc: --> <li><a href="._odenn-bs034.html#___sec36" style="font-size: 80%;">Example: Solving the wave equation using Autograd and TensorFlow</a></li>
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<!-- navigation toc: --> <li><a href="._odenn-bs035.html#___sec37" style="font-size: 80%;">The problem to solve for</a></li>
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<!-- navigation toc: --> <li><a href="._odenn-bs036.html#___sec38" style="font-size: 80%;">The trial solution</a></li>
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<!-- navigation toc: --> <li><a href="._odenn-bs037.html#___sec39" style="font-size: 80%;">The analytical solution</a></li>
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|
<!-- navigation toc: --> <li><a href="._odenn-bs038.html#___sec40" style="font-size: 80%;">Solving the wave equation - the full program using Autograd</a></li>
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<!-- navigation toc: --> <li><a href="._odenn-bs039.html#___sec41" style="font-size: 80%;">Solving the wave equation - the full program using TensorFlow</a></li>
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<!-- navigation toc: --> <li><a href="._odenn-bs040.html#___sec42" style="font-size: 80%;">Resources</a></li>
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<p> </p><p> </p><p> </p> <!-- add vertical space -->
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<a name="part0024"></a>
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<!-- !split -->
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<h2 id="___sec26" class="anchor">Solving the equation using Autograd </h2>
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<p>
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<!-- code=python (!bc pycod) typeset with pygments style "default" -->
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<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">autograd.numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
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<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">autograd</span> <span style="color: #008000; font-weight: bold">import</span> grad, elementwise_grad
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<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">autograd.numpy.random</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">npr</span>
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<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">matplotlib</span> <span style="color: #008000; font-weight: bold">import</span> pyplot <span style="color: #008000; font-weight: bold">as</span> plt
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<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">sigmoid</span>(z):
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<span style="color: #008000; font-weight: bold">return</span> <span style="color: #666666">1/</span>(<span style="color: #666666">1</span> <span style="color: #666666">+</span> np<span style="color: #666666">.</span>exp(<span style="color: #666666">-</span>z))
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<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">deep_neural_network</span>(deep_params, x):
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<span style="color: #408080; font-style: italic"># N_hidden is the number of hidden layers</span>
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N_hidden <span style="color: #666666">=</span> np<span style="color: #666666">.</span>size(deep_params) <span style="color: #666666">-</span> <span style="color: #666666">1</span> <span style="color: #408080; font-style: italic"># -1 since params consist of parameters to all the hidden layers AND the output layer</span>
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<span style="color: #408080; font-style: italic"># Assumes input x being an one-dimensional array</span>
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num_values <span style="color: #666666">=</span> np<span style="color: #666666">.</span>size(x)
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x <span style="color: #666666">=</span> x<span style="color: #666666">.</span>reshape(<span style="color: #666666">-1</span>, num_values)
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<span style="color: #408080; font-style: italic"># Assume that the input layer does nothing to the input x</span>
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x_input <span style="color: #666666">=</span> x
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<span style="color: #408080; font-style: italic"># Due to multiple hidden layers, define a variable referencing to the</span>
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<span style="color: #408080; font-style: italic"># output of the previous layer:</span>
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x_prev <span style="color: #666666">=</span> x_input
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<span style="color: #408080; font-style: italic">## Hidden layers:</span>
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<span style="color: #008000; font-weight: bold">for</span> l <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(N_hidden):
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<span style="color: #408080; font-style: italic"># From the list of parameters P; find the correct weigths and bias for this layer</span>
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w_hidden <span style="color: #666666">=</span> deep_params[l]
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<span style="color: #408080; font-style: italic"># Add a row of ones to include bias</span>
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x_prev <span style="color: #666666">=</span> np<span style="color: #666666">.</span>concatenate((np<span style="color: #666666">.</span>ones((<span style="color: #666666">1</span>,num_values)), x_prev ), axis <span style="color: #666666">=</span> <span style="color: #666666">0</span>)
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z_hidden <span style="color: #666666">=</span> np<span style="color: #666666">.</span>matmul(w_hidden, x_prev)
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x_hidden <span style="color: #666666">=</span> sigmoid(z_hidden)
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<span style="color: #408080; font-style: italic"># Update x_prev such that next layer can use the output from this layer</span>
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x_prev <span style="color: #666666">=</span> x_hidden
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<span style="color: #408080; font-style: italic">## Output layer:</span>
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<span style="color: #408080; font-style: italic"># Get the weights and bias for this layer</span>
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w_output <span style="color: #666666">=</span> deep_params[<span style="color: #666666">-1</span>]
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<span style="color: #408080; font-style: italic"># Include bias:</span>
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x_prev <span style="color: #666666">=</span> np<span style="color: #666666">.</span>concatenate((np<span style="color: #666666">.</span>ones((<span style="color: #666666">1</span>,num_values)), x_prev), axis <span style="color: #666666">=</span> <span style="color: #666666">0</span>)
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z_output <span style="color: #666666">=</span> np<span style="color: #666666">.</span>matmul(w_output, x_prev)
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x_output <span style="color: #666666">=</span> z_output
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<span style="color: #008000; font-weight: bold">return</span> x_output
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<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">solve_ode_deep_neural_network</span>(x, num_neurons, num_iter, lmb):
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<span style="color: #408080; font-style: italic"># num_hidden_neurons is now a list of number of neurons within each hidden layer</span>
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<span style="color: #408080; font-style: italic"># Find the number of hidden layers:</span>
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N_hidden <span style="color: #666666">=</span> np<span style="color: #666666">.</span>size(num_neurons)
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<span style="color: #408080; font-style: italic">## Set up initial weigths and biases</span>
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<span style="color: #408080; font-style: italic"># Initialize the list of parameters:</span>
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P <span style="color: #666666">=</span> [<span style="color: #008000">None</span>]<span style="color: #666666">*</span>(N_hidden <span style="color: #666666">+</span> <span style="color: #666666">1</span>) <span style="color: #408080; font-style: italic"># + 1 to include the output layer</span>
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P[<span style="color: #666666">0</span>] <span style="color: #666666">=</span> npr<span style="color: #666666">.</span>randn(num_neurons[<span style="color: #666666">0</span>], <span style="color: #666666">2</span> )
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<span style="color: #008000; font-weight: bold">for</span> l <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(<span style="color: #666666">1</span>,N_hidden):
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P[l] <span style="color: #666666">=</span> npr<span style="color: #666666">.</span>randn(num_neurons[l], num_neurons[l<span style="color: #666666">-1</span>] <span style="color: #666666">+</span> <span style="color: #666666">1</span>) <span style="color: #408080; font-style: italic"># +1 to include bias</span>
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<span style="color: #408080; font-style: italic"># For the output layer</span>
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P[<span style="color: #666666">-1</span>] <span style="color: #666666">=</span> npr<span style="color: #666666">.</span>randn(<span style="color: #666666">1</span>, num_neurons[<span style="color: #666666">-1</span>] <span style="color: #666666">+</span> <span style="color: #666666">1</span> ) <span style="color: #408080; font-style: italic"># +1 since bias is included</span>
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<span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">'Initial cost: </span><span style="color: #BB6688; font-weight: bold">%g</span><span style="color: #BA2121">'</span><span style="color: #666666">%</span>cost_function_deep(P, x))
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<span style="color: #408080; font-style: italic">## Start finding the optimal weigths using gradient descent</span>
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<span style="color: #408080; font-style: italic"># Find the Python function that represents the gradient of the cost function</span>
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<span style="color: #408080; font-style: italic"># w.r.t the 0-th input argument -- that is the weights and biases in the hidden and output layer</span>
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cost_function_deep_grad <span style="color: #666666">=</span> grad(cost_function_deep,<span style="color: #666666">0</span>)
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<span style="color: #408080; font-style: italic"># Let the update be done num_iter times</span>
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<span style="color: #008000; font-weight: bold">for</span> i <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(num_iter):
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<span style="color: #408080; font-style: italic"># Evaluate the gradient at the current weights and biases in P.</span>
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<span style="color: #408080; font-style: italic"># The cost_grad consist now of N_hidden + 1 arrays; the gradient w.r.t the weights and biases</span>
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<span style="color: #408080; font-style: italic"># in the hidden layers and output layers evaluated at x.</span>
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cost_deep_grad <span style="color: #666666">=</span> cost_function_deep_grad(P, x)
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<span style="color: #008000; font-weight: bold">for</span> l <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(N_hidden<span style="color: #666666">+1</span>):
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P[l] <span style="color: #666666">=</span> P[l] <span style="color: #666666">-</span> lmb <span style="color: #666666">*</span> cost_deep_grad[l]
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<span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">'Final cost: </span><span style="color: #BB6688; font-weight: bold">%g</span><span style="color: #BA2121">'</span><span style="color: #666666">%</span>cost_function_deep(P, x))
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<span style="color: #008000; font-weight: bold">return</span> P
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<span style="color: #408080; font-style: italic">## Set up the cost function specified for this Poisson equation:</span>
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<span style="color: #408080; font-style: italic"># The right side of the ODE</span>
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<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">f</span>(x):
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<span style="color: #008000; font-weight: bold">return</span> (<span style="color: #666666">3*</span>x <span style="color: #666666">+</span> x<span style="color: #666666">**2</span>)<span style="color: #666666">*</span>np<span style="color: #666666">.</span>exp(x)
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<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">cost_function_deep</span>(P, x):
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<span style="color: #408080; font-style: italic"># Evaluate the trial function with the current parameters P</span>
|
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g_t <span style="color: #666666">=</span> g_trial_deep(x,P)
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<span style="color: #408080; font-style: italic"># Find the derivative w.r.t x of the trial function</span>
|
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d2_g_t <span style="color: #666666">=</span> elementwise_grad(elementwise_grad(g_trial_deep,<span style="color: #666666">0</span>))(x,P)
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right_side <span style="color: #666666">=</span> f(x)
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err_sqr <span style="color: #666666">=</span> (<span style="color: #666666">-</span>d2_g_t <span style="color: #666666">-</span> right_side)<span style="color: #666666">**2</span>
|
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cost_sum <span style="color: #666666">=</span> np<span style="color: #666666">.</span>sum(err_sqr)
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<span style="color: #008000; font-weight: bold">return</span> cost_sum<span style="color: #666666">/</span>np<span style="color: #666666">.</span>size(err_sqr)
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<span style="color: #408080; font-style: italic"># The trial solution:</span>
|
|
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">g_trial_deep</span>(x,P):
|
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<span style="color: #008000; font-weight: bold">return</span> x<span style="color: #666666">*</span>(<span style="color: #666666">1-</span>x)<span style="color: #666666">*</span>deep_neural_network(P,x)
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<span style="color: #408080; font-style: italic"># The analytic solution;</span>
|
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<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">g_analytic</span>(x):
|
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<span style="color: #008000; font-weight: bold">return</span> x<span style="color: #666666">*</span>(<span style="color: #666666">1-</span>x)<span style="color: #666666">*</span>np<span style="color: #666666">.</span>exp(x)
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<span style="color: #008000; font-weight: bold">if</span> <span style="color: #19177C">__name__</span> <span style="color: #666666">==</span> <span style="color: #BA2121">'__main__'</span>:
|
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npr<span style="color: #666666">.</span>seed(<span style="color: #666666">4155</span>)
|
|
|
|
<span style="color: #408080; font-style: italic">## Decide the vales of arguments to the function to solve</span>
|
|
Nx <span style="color: #666666">=</span> <span style="color: #666666">10</span>
|
|
x <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linspace(<span style="color: #666666">0</span>,<span style="color: #666666">1</span>, Nx)
|
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<span style="color: #408080; font-style: italic">## Set up the initial parameters</span>
|
|
num_hidden_neurons <span style="color: #666666">=</span> [<span style="color: #666666">200</span>,<span style="color: #666666">100</span>]
|
|
num_iter <span style="color: #666666">=</span> <span style="color: #666666">1000</span>
|
|
lmb <span style="color: #666666">=</span> <span style="color: #666666">1e-3</span>
|
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|
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P <span style="color: #666666">=</span> solve_ode_deep_neural_network(x, num_hidden_neurons, num_iter, lmb)
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g_dnn_ag <span style="color: #666666">=</span> g_trial_deep(x,P)
|
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g_analytical <span style="color: #666666">=</span> g_analytic(x)
|
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<span style="color: #408080; font-style: italic"># Find the maximum absolute difference between the solutons:</span>
|
|
max_diff <span style="color: #666666">=</span> np<span style="color: #666666">.</span>max(np<span style="color: #666666">.</span>abs(g_dnn_ag <span style="color: #666666">-</span> g_analytical))
|
|
<span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">"The max absolute difference between the solutions is: </span><span style="color: #BB6688; font-weight: bold">%g</span><span style="color: #BA2121">"</span><span style="color: #666666">%</span>max_diff)
|
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plt<span style="color: #666666">.</span>figure(figsize<span style="color: #666666">=</span>(<span style="color: #666666">10</span>,<span style="color: #666666">10</span>))
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plt<span style="color: #666666">.</span>title(<span style="color: #BA2121">'Performance of neural network solving an ODE compared to the analytical solution'</span>)
|
|
plt<span style="color: #666666">.</span>plot(x, g_analytical)
|
|
plt<span style="color: #666666">.</span>plot(x, g_dnn_ag[<span style="color: #666666">0</span>,:])
|
|
plt<span style="color: #666666">.</span>legend([<span style="color: #BA2121">'analytical'</span>,<span style="color: #BA2121">'nn'</span>])
|
|
plt<span style="color: #666666">.</span>xlabel(<span style="color: #BA2121">'x'</span>)
|
|
plt<span style="color: #666666">.</span>ylabel(<span style="color: #BA2121">'g(x)'</span>)
|
|
plt<span style="color: #666666">.</span>show()
|
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</pre></div>
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<p>
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