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439 lines
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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="#___sec20" style="font-size: 80%;">The general program flow in TensorFlow</a></li>
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<!-- navigation toc: --> <li><a href="#___sec21" style="font-size: 80%;">Program flow in TensorFlow - Construction phase</a></li>
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<!-- navigation toc: --> <li><a href="#___sec22" style="font-size: 80%;">Program flow in TensorFlow - Execution phase</a></li>
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<!-- navigation toc: --> <li><a href="#___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="._odenn-bs024.html#___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>
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<!-- 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>
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|
<!-- 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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</ul>
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</ul>
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</div>
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<p> </p><p> </p><p> </p> <!-- add vertical space -->
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<a name="part0021"></a>
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<!-- !split -->
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<h2 id="___sec20" class="anchor">The general program flow in TensorFlow </h2>
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<p>
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Usually, a program in TensorFlow is divided into two parts; the <em>construction phase</em> and the <em>execution phase</em>.
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In the construction phase, the computational graph that TensorFlow uses to perform its calculations are set up.
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In the execution phase, TensorFlow evaluates any procedure that was defined in the construction phase.
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<h2 id="___sec21" class="anchor">Program flow in TensorFlow - Construction phase </h2>
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<p>
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Here, the architecture for the neural network will be set up, along with the cost function and an optimizer class used during training of the network.
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Note that TensorFlow uses a different convention for the weighting done in each neuron in each layer within the network than in the implementation using Autograd.
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The matrix-vector multiplication between the input from the previous layer and the weighting at the neuron at current layer in the program using Autograd, is the transpose of the convention used in TensorFlow. But it will not affect that much our construction, as TensorFlow takes care of most of the computations. The only thing we have to be aware of, is how the dimensions are for our inputs.
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<h2 id="___sec22" class="anchor">Program flow in TensorFlow - Execution phase </h2>
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<p>
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The computation graph has been defined, and is ready to be evaluated.
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In order to get access to the graph, it has to be initialized and be runned within a Session.
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<h2 id="___sec23" class="anchor">The full program modeling logistic population growth using TensorFlow </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">tensorflow</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">tf</span>
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<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">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">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
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<span style="color: #408080; font-style: italic"># Just to reset the graph such that it is possible to rerun this in a</span>
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<span style="color: #408080; font-style: italic"># Jupyter cell without resetting the whole kernel.</span>
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tf<span style="color: #666666">.</span>reset_default_graph()
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<span style="color: #408080; font-style: italic"># Set a seed to ensure getting the same results from every run</span>
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tf<span style="color: #666666">.</span>set_random_seed(<span style="color: #666666">4155</span>)
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Nt <span style="color: #666666">=</span> <span style="color: #666666">10</span>
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T <span style="color: #666666">=</span> <span style="color: #666666">1</span>
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t <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linspace(<span style="color: #666666">0</span>,T, Nt)
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<span style="color: #408080; font-style: italic">## The construction phase</span>
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<span style="color: #408080; font-style: italic"># Convert the values the trial solution is evaluated at to a tensor.</span>
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t_tf <span style="color: #666666">=</span> tf<span style="color: #666666">.</span>convert_to_tensor(t<span style="color: #666666">.</span>reshape(<span style="color: #666666">-1</span>,<span style="color: #666666">1</span>),dtype<span style="color: #666666">=</span>tf<span style="color: #666666">.</span>float64)
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zeros <span style="color: #666666">=</span> tf<span style="color: #666666">.</span>reshape(tf<span style="color: #666666">.</span>convert_to_tensor(np<span style="color: #666666">.</span>zeros(t<span style="color: #666666">.</span>shape)),shape<span style="color: #666666">=</span>(<span style="color: #666666">-1</span>,<span style="color: #666666">1</span>))
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<span style="color: #408080; font-style: italic"># Define the parameters of the equation</span>
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alpha <span style="color: #666666">=</span> tf<span style="color: #666666">.</span>constant(<span style="color: #666666">2.</span>,dtype<span style="color: #666666">=</span>tf<span style="color: #666666">.</span>float64)
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A <span style="color: #666666">=</span> tf<span style="color: #666666">.</span>constant(<span style="color: #666666">1.</span>,dtype<span style="color: #666666">=</span>tf<span style="color: #666666">.</span>float64)
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g0 <span style="color: #666666">=</span> tf<span style="color: #666666">.</span>constant(<span style="color: #666666">1.2</span>,dtype<span style="color: #666666">=</span>tf<span style="color: #666666">.</span>float64)
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num_iter <span style="color: #666666">=</span> <span style="color: #666666">100000</span>
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<span style="color: #408080; font-style: italic"># Define the number of neurons at each hidden layer</span>
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num_hidden_neurons <span style="color: #666666">=</span> [<span style="color: #666666">100</span>,<span style="color: #666666">50</span>,<span style="color: #666666">25</span>]
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num_hidden_layers <span style="color: #666666">=</span> np<span style="color: #666666">.</span>size(num_hidden_neurons)
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<span style="color: #408080; font-style: italic"># Construct the network.</span>
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<span style="color: #408080; font-style: italic"># tf.name_scope is used to group each step in the construction,</span>
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<span style="color: #408080; font-style: italic"># just for a more organized visualization in TensorBoard</span>
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<span style="color: #008000; font-weight: bold">with</span> tf<span style="color: #666666">.</span>name_scope(<span style="color: #BA2121">'dnn'</span>):
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<span style="color: #408080; font-style: italic"># Input layer</span>
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previous_layer <span style="color: #666666">=</span> t_tf
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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>(num_hidden_layers):
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current_layer <span style="color: #666666">=</span> tf<span style="color: #666666">.</span>layers<span style="color: #666666">.</span>dense(previous_layer, num_hidden_neurons[l], name<span style="color: #666666">=</span><span style="color: #BA2121">'hidden</span><span style="color: #BB6688; font-weight: bold">%d</span><span style="color: #BA2121">'</span><span style="color: #666666">%</span>(l<span style="color: #666666">+1</span>), activation<span style="color: #666666">=</span>tf<span style="color: #666666">.</span>nn<span style="color: #666666">.</span>sigmoid)
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previous_layer <span style="color: #666666">=</span> current_layer
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<span style="color: #408080; font-style: italic"># Output layer</span>
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dnn_output <span style="color: #666666">=</span> tf<span style="color: #666666">.</span>layers<span style="color: #666666">.</span>dense(previous_layer, <span style="color: #666666">1</span>, name<span style="color: #666666">=</span><span style="color: #BA2121">'output'</span>)
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<span style="color: #408080; font-style: italic"># Define the cost function</span>
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<span style="color: #008000; font-weight: bold">with</span> tf<span style="color: #666666">.</span>name_scope(<span style="color: #BA2121">'cost'</span>):
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g_trial <span style="color: #666666">=</span> g0 <span style="color: #666666">+</span> t_tf<span style="color: #666666">*</span>dnn_output
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d_g_trial <span style="color: #666666">=</span> tf<span style="color: #666666">.</span>gradients(g_trial,t_tf)
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func <span style="color: #666666">=</span> alpha<span style="color: #666666">*</span>g_trial<span style="color: #666666">*</span>(A <span style="color: #666666">-</span> g_trial)
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cost <span style="color: #666666">=</span> tf<span style="color: #666666">.</span>losses<span style="color: #666666">.</span>mean_squared_error(zeros, d_g_trial[<span style="color: #666666">0</span>] <span style="color: #666666">-</span> func)
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<span style="color: #408080; font-style: italic"># Choose the method to minimize the cost function, along with a learning rate</span>
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learning_rate <span style="color: #666666">=</span> <span style="color: #666666">1e-2</span>
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<span style="color: #008000; font-weight: bold">with</span> tf<span style="color: #666666">.</span>name_scope(<span style="color: #BA2121">'train'</span>):
|
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optimizer <span style="color: #666666">=</span> tf<span style="color: #666666">.</span>train<span style="color: #666666">.</span>GradientDescentOptimizer(learning_rate)
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traning_op <span style="color: #666666">=</span> optimizer<span style="color: #666666">.</span>minimize(cost)
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<span style="color: #408080; font-style: italic"># Set up a referance to the result from the neural network:</span>
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g_dnn_tf <span style="color: #666666">=</span> <span style="color: #008000">None</span>
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<span style="color: #408080; font-style: italic"># Define a node that initializes all of the other nodes in the computational graph</span>
|
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<span style="color: #408080; font-style: italic"># used by TensorFlow:</span>
|
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init <span style="color: #666666">=</span> tf<span style="color: #666666">.</span>global_variables_initializer()
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<span style="color: #408080; font-style: italic">## Execution phase</span>
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<span style="color: #408080; font-style: italic"># Start a session where the graph defined from the construction phase can be evaluated at:</span>
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<span style="color: #008000; font-weight: bold">with</span> tf<span style="color: #666666">.</span>Session() <span style="color: #008000; font-weight: bold">as</span> sess:
|
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<span style="color: #408080; font-style: italic"># Initialize the whole graph</span>
|
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init<span style="color: #666666">.</span>run()
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<span style="color: #408080; font-style: italic"># Evaluate the initial cost:</span>
|
|
<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<span style="color: #666666">.</span>eval())
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<span style="color: #408080; font-style: italic"># The training of the network:</span>
|
|
<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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sess<span style="color: #666666">.</span>run(traning_op)
|
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<span style="color: #408080; font-style: italic"># If one desires to see how the cost function behaves for each iteration:</span>
|
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<span style="color: #408080; font-style: italic">#if i % 1000 == 0:</span>
|
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<span style="color: #408080; font-style: italic"># print(cost.eval())</span>
|
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<span style="color: #408080; font-style: italic"># Training is done, and we have an approximate solution to the ODE</span>
|
|
<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<span style="color: #666666">.</span>eval())
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<span style="color: #408080; font-style: italic"># Store the result</span>
|
|
g_dnn_tf <span style="color: #666666">=</span> g_trial<span style="color: #666666">.</span>eval()
|
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|
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<span style="color: #408080; font-style: italic"># Compare with analytical solution</span>
|
|
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">get_parameters</span>():
|
|
alpha <span style="color: #666666">=</span> <span style="color: #666666">2</span>
|
|
A <span style="color: #666666">=</span> <span style="color: #666666">1</span>
|
|
g0 <span style="color: #666666">=</span> <span style="color: #666666">1.2</span>
|
|
<span style="color: #008000; font-weight: bold">return</span> alpha, A, g0
|
|
|
|
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">g_analytic</span>(t):
|
|
alpha,A, g0 <span style="color: #666666">=</span> get_parameters()
|
|
<span style="color: #008000; font-weight: bold">return</span> A<span style="color: #666666">*</span>g0<span style="color: #666666">/</span>(g0 <span style="color: #666666">+</span> (A <span style="color: #666666">-</span> g0)<span style="color: #666666">*</span>np<span style="color: #666666">.</span>exp(<span style="color: #666666">-</span>alpha<span style="color: #666666">*</span>A<span style="color: #666666">*</span>t))
|
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|
|
g_analytical <span style="color: #666666">=</span> g_analytic(t)
|
|
diff_tf <span style="color: #666666">=</span> g_dnn_tf <span style="color: #666666">-</span> g_analytical<span style="color: #666666">.</span>reshape(<span style="color: #666666">-1</span>,<span style="color: #666666">1</span>)
|
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|
|
<span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">'</span><span style="color: #BB6622; font-weight: bold">\n</span><span style="color: #BA2121">Max absolute difference between the analytical solution and solution from TensorFlow DNN: </span><span style="color: #BB6688; font-weight: bold">%g</span><span style="color: #BA2121">'</span><span style="color: #666666">%</span>np<span style="color: #666666">.</span>max(np<span style="color: #666666">.</span>abs(diff_tf)))
|
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|
|
<span style="color: #408080; font-style: italic"># Plot the result</span>
|
|
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">'Numerical solutions of the ODE'</span>)
|
|
|
|
plt<span style="color: #666666">.</span>plot(t, g_dnn_tf)
|
|
plt<span style="color: #666666">.</span>plot(t, g_analytical)
|
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|
|
plt<span style="color: #666666">.</span>legend([<span style="color: #BA2121">'dnn, tensorflow'</span>, <span style="color: #BA2121">'exact'</span>])
|
|
plt<span style="color: #666666">.</span>xlabel(<span style="color: #BA2121">'Time t'</span>)
|
|
plt<span style="color: #666666">.</span>ylabel(<span style="color: #BA2121">'g(t)'</span>)
|
|
|
|
plt<span style="color: #666666">.</span>show()
|
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</pre></div>
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<p>
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