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<!-- navigation toc: --> <li><a href="._odenn-bs001.html#___sec0" style="font-size: 80%;">Differential equations</a></li>
<!-- 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-bs009.html#___sec8" style="font-size: 80%;">Reformulating the problem</a></li>
<!-- navigation toc: --> <li><a href="._odenn-bs010.html#___sec9" style="font-size: 80%;">A possible implementation of a neural network using Autograd</a></li>
<!-- 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-bs013.html#___sec12" style="font-size: 80%;">The network with one input, hidden, and output layer</a></li>
<!-- 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-bs017.html#___sec16" style="font-size: 80%;">The trial solution</a></li>
<!-- navigation toc: --> <li><a href="._odenn-bs018.html#___sec17" style="font-size: 80%;">The program using Autograd</a></li>
<!-- navigation toc: --> <li><a href="._odenn-bs019.html#___sec18" style="font-size: 80%;">Using forward Euler to solve the ODE</a></li>
<!-- navigation toc: --> <li><a href="._odenn-bs020.html#___sec19" style="font-size: 80%;">Using TensorFlow to model logistic population growth</a></li>
<!-- navigation toc: --> <li><a href="._odenn-bs021.html#___sec20" style="font-size: 80%;">The general program flow in TensorFlow</a></li>
<!-- 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#___sec23" style="font-size: 80%;">The full program modeling logistic population growth using TensorFlow</a></li>
<!-- 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>
<!-- navigation toc: --> <li><a href="._odenn-bs023.html#___sec25" style="font-size: 80%;">The specific equation to solve for</a></li>
<!-- navigation toc: --> <li><a href="._odenn-bs024.html#___sec26" style="font-size: 80%;">Solving the equation using Autograd</a></li>
<!-- navigation toc: --> <li><a href="._odenn-bs025.html#___sec27" style="font-size: 80%;">Comparing with a numerical scheme</a></li>
<!-- navigation toc: --> <li><a href="#___sec28" style="font-size: 80%;">Using gradient descent in TensorFlow to solve Poisson equation</a></li>
<!-- 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-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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<h2 id="___sec28" class="anchor">Using gradient descent in TensorFlow to solve Poisson equation </h2>
The program follows the similar idea as for the logistic population model.
<p>
What has changed, is what the cost function minimizes and the trial solution.
<p>
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
<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>
<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>
<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>
<span style="color: #408080; font-style: italic">## Construction phase</span>
<span style="color: #408080; font-style: italic"># Just to reset the graph such that it is possible to rerun this in a</span>
<span style="color: #408080; font-style: italic"># Jupyter cell without resetting the whole kernel.</span>
tf<span style="color: #666666">.</span>reset_default_graph()
tf<span style="color: #666666">.</span>set_random_seed(<span style="color: #666666">4155</span>)
<span style="color: #408080; font-style: italic"># Convert the values the trial solution is evaluated at to a tensor.</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)
x_tf <span style="color: #666666">=</span> tf<span style="color: #666666">.</span>convert_to_tensor(x<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)
num_iter <span style="color: #666666">=</span> <span style="color: #666666">10000</span>
<span style="color: #408080; font-style: italic"># Define the number of neurons at each hidden layer</span>
num_hidden_neurons <span style="color: #666666">=</span> [<span style="color: #666666">20</span>,<span style="color: #666666">10</span>]
num_hidden_layers <span style="color: #666666">=</span> np<span style="color: #666666">.</span>size(num_hidden_neurons)
<span style="color: #408080; font-style: italic"># Construct the network.</span>
<span style="color: #408080; font-style: italic"># tf.name_scope is used to group each step in the construction,</span>
<span style="color: #408080; font-style: italic"># just for a more organized visualization in TensorBoard</span>
<span style="color: #008000; font-weight: bold">with</span> tf<span style="color: #666666">.</span>name_scope(<span style="color: #BA2121">&#39;dnn&#39;</span>):
<span style="color: #408080; font-style: italic"># Input layer</span>
previous_layer <span style="color: #666666">=</span> x_tf
<span style="color: #408080; font-style: italic"># Hidden layers</span>
<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):
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">&#39;hidden</span><span style="color: #BB6688; font-weight: bold">%d</span><span style="color: #BA2121">&#39;</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)
previous_layer <span style="color: #666666">=</span> current_layer
<span style="color: #408080; font-style: italic"># Output layer</span>
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">&#39;output&#39;</span>)
<span style="color: #408080; font-style: italic"># Define the cost function</span>
<span style="color: #008000; font-weight: bold">with</span> tf<span style="color: #666666">.</span>name_scope(<span style="color: #BA2121">&#39;cost&#39;</span>):
g_trial <span style="color: #666666">=</span> x_tf<span style="color: #666666">*</span>(<span style="color: #666666">1-</span>x_tf)<span style="color: #666666">*</span>dnn_output
d_g_trial <span style="color: #666666">=</span> tf<span style="color: #666666">.</span>gradients(g_trial,x_tf)
d2_g_trial <span style="color: #666666">=</span> tf<span style="color: #666666">.</span>gradients(d_g_trial,x_tf)
right_side <span style="color: #666666">=</span> (<span style="color: #666666">3*</span>x_tf <span style="color: #666666">+</span> x_tf<span style="color: #666666">**2</span>)<span style="color: #666666">*</span>tf<span style="color: #666666">.</span>exp(x_tf)
err <span style="color: #666666">=</span> tf<span style="color: #666666">.</span>square( <span style="color: #666666">-</span>d2_g_trial[<span style="color: #666666">0</span>] <span style="color: #666666">-</span> right_side)
cost <span style="color: #666666">=</span> tf<span style="color: #666666">.</span>reduce_sum(err, name <span style="color: #666666">=</span> <span style="color: #BA2121">&#39;cost&#39;</span>)
<span style="color: #408080; font-style: italic"># Choose the method to minimize the cost function, along with a learning rate</span>
learning_rate <span style="color: #666666">=</span> <span style="color: #666666">1e-2</span>
<span style="color: #008000; font-weight: bold">with</span> tf<span style="color: #666666">.</span>name_scope(<span style="color: #BA2121">&#39;train&#39;</span>):
optimizer <span style="color: #666666">=</span> tf<span style="color: #666666">.</span>train<span style="color: #666666">.</span>GradientDescentOptimizer(learning_rate)
traning_op <span style="color: #666666">=</span> optimizer<span style="color: #666666">.</span>minimize(cost)
g_dnn_tf <span style="color: #666666">=</span> <span style="color: #008000">None</span>
<span style="color: #408080; font-style: italic"># Define a node that initializes all of the other nodes in the computational graph</span>
<span style="color: #408080; font-style: italic"># used by TensorFlow:</span>
init <span style="color: #666666">=</span> tf<span style="color: #666666">.</span>global_variables_initializer()
<span style="color: #408080; font-style: italic">## Execution phase</span>
<span style="color: #408080; font-style: italic"># Start a session where the graph defined from the construction phase can be evaluated at:</span>
<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:
<span style="color: #408080; font-style: italic"># Initialize the whole graph</span>
init<span style="color: #666666">.</span>run()
<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">&#39;Initial cost: </span><span style="color: #BB6688; font-weight: bold">%g</span><span style="color: #BA2121">&#39;</span><span style="color: #666666">%</span>cost<span style="color: #666666">.</span>eval())
<span style="color: #408080; font-style: italic"># The traning 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):
sess<span style="color: #666666">.</span>run(traning_op)
<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">&#39;Final cost: </span><span style="color: #BB6688; font-weight: bold">%g</span><span style="color: #BA2121">&#39;</span><span style="color: #666666">%</span>cost<span style="color: #666666">.</span>eval())
<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()
writer <span style="color: #666666">=</span> tf<span style="color: #666666">.</span>summary<span style="color: #666666">.</span>FileWriter(<span style="color: #BA2121">&quot;./output&quot;</span>, sess<span style="color: #666666">.</span>graph)
writer<span style="color: #666666">.</span>close()
<span style="color: #408080; font-style: italic"># Evaluate the analytical function to compare with</span>
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">g_analytic</span>(x):
<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)
g_analytical <span style="color: #666666">=</span> g_analytic(x)
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>)
<span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">&#39;</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">&#39;</span><span style="color: #666666">%</span>np<span style="color: #666666">.</span>max(np<span style="color: #666666">.</span>abs(diff_tf)))
<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>))
plt<span style="color: #666666">.</span>title(<span style="color: #BA2121">&#39;Numerical solutions of the ODE&#39;</span>)
plt<span style="color: #666666">.</span>plot(x, g_dnn_tf)
plt<span style="color: #666666">.</span>plot(x, g_analytical)
plt<span style="color: #666666">.</span>legend([<span style="color: #BA2121">&#39;dnn, tensorflow&#39;</span>,<span style="color: #BA2121">&#39;exact&#39;</span>])
plt<span style="color: #666666">.</span>xlabel(<span style="color: #BA2121">&#39;x&#39;</span>)
plt<span style="color: #666666">.</span>ylabel(<span style="color: #BA2121">&#39;g(x)&#39;</span>)
plt<span style="color: #666666">.</span>show()
</pre></div>
<p>
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