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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>
<!-- navigation toc: --> <li><a href="._odenn-bs012.html#___sec11" style="font-size: 80%;">Gradient descent</a></li>
<!-- 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>
<!-- 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-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>
<!-- navigation toc: --> <li><a href="._odenn-bs021.html#___sec22" style="font-size: 80%;">Program flow in TensorFlow - Execution phase</a></li>
<!-- 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="._odenn-bs026.html#___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>
<!-- 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-bs030.html#___sec32" style="font-size: 80%;">Defining the problem</a></li>
<!-- 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="#___sec41" style="font-size: 80%;">Solving the wave equation - the full program using TensorFlow</a></li>
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<h2 id="___sec41" class="anchor">Solving the wave equation - the full program using TensorFlow </h2>
As the program using Autograd is fairly slow, one could hope that using TensorFlow
could make a naive implementation faster, and more numerically robust.
<p>
In addition, having TensorFlow at hand, it could be easier to experiment with different
optimization algorithms, and other constructions of the network.
<p>
The following program solves the given wave equation much faster,
<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">from</span> <span style="color: #0000FF; font-weight: bold">matplotlib</span> <span style="color: #008000; font-weight: bold">import</span> cm
<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
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">mpl_toolkits.mplot3d</span> <span style="color: #008000; font-weight: bold">import</span> axes3d
Nx <span style="color: #666666">=</span> <span style="color: #666666">10</span>
x_np <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)
Nt <span style="color: #666666">=</span> <span style="color: #666666">10</span>
t_np <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linspace(<span style="color: #666666">0</span>,<span style="color: #666666">1</span>,Nt)
X,T <span style="color: #666666">=</span> np<span style="color: #666666">.</span>meshgrid(x_np, t_np)
x <span style="color: #666666">=</span> X<span style="color: #666666">.</span>ravel()
t <span style="color: #666666">=</span> T<span style="color: #666666">.</span>ravel()
<span style="color: #408080; font-style: italic">## The construction phase</span>
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(x<span style="color: #666666">.</span>shape)),shape<span style="color: #666666">=</span>(<span style="color: #666666">-1</span>,<span style="color: #666666">1</span>))
x <span style="color: #666666">=</span> tf<span style="color: #666666">.</span>reshape(tf<span style="color: #666666">.</span>convert_to_tensor(x),shape<span style="color: #666666">=</span>(<span style="color: #666666">-1</span>,<span style="color: #666666">1</span>))
t <span style="color: #666666">=</span> tf<span style="color: #666666">.</span>reshape(tf<span style="color: #666666">.</span>convert_to_tensor(t),shape<span style="color: #666666">=</span>(<span style="color: #666666">-1</span>,<span style="color: #666666">1</span>))
points <span style="color: #666666">=</span> tf<span style="color: #666666">.</span>concat([x,t],<span style="color: #666666">1</span>)
num_iter <span style="color: #666666">=</span> <span style="color: #666666">100000</span>
num_hidden_neurons <span style="color: #666666">=</span> [<span style="color: #666666">90</span>]
X <span style="color: #666666">=</span> tf<span style="color: #666666">.</span>convert_to_tensor(X)
T <span style="color: #666666">=</span> tf<span style="color: #666666">.</span>convert_to_tensor(T)
<span style="color: #008000; font-weight: bold">with</span> tf<span style="color: #666666">.</span>variable_scope(<span style="color: #BA2121">&#39;dnn&#39;</span>):
num_hidden_layers <span style="color: #666666">=</span> np<span style="color: #666666">.</span>size(num_hidden_neurons)
previous_layer <span style="color: #666666">=</span> points
<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],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
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>)
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">u</span>(x):
<span style="color: #008000; font-weight: bold">return</span> tf<span style="color: #666666">.</span>sin(np<span style="color: #666666">.</span>pi<span style="color: #666666">*</span>x)
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">v</span>(x):
<span style="color: #008000; font-weight: bold">return</span> <span style="color: #666666">-</span>np<span style="color: #666666">.</span>pi<span style="color: #666666">*</span>tf<span style="color: #666666">.</span>sin(np<span style="color: #666666">.</span>pi<span style="color: #666666">*</span>x)
<span style="color: #008000; font-weight: bold">with</span> tf<span style="color: #666666">.</span>name_scope(<span style="color: #BA2121">&#39;loss&#39;</span>):
g_trial <span style="color: #666666">=</span> (<span style="color: #666666">1</span> <span style="color: #666666">-</span> t<span style="color: #666666">**2</span>)<span style="color: #666666">*</span>u(x) <span style="color: #666666">+</span> t<span style="color: #666666">*</span>v(x) <span style="color: #666666">+</span> x<span style="color: #666666">*</span>(<span style="color: #666666">1-</span>x)<span style="color: #666666">*</span>t<span style="color: #666666">**2*</span>dnn_output
g_trial_d2t <span style="color: #666666">=</span> tf<span style="color: #666666">.</span>gradients(tf<span style="color: #666666">.</span>gradients(g_trial,t),t)
g_trial_d2x <span style="color: #666666">=</span> tf<span style="color: #666666">.</span>gradients(tf<span style="color: #666666">.</span>gradients(g_trial,x),x)
loss <span style="color: #666666">=</span> tf<span style="color: #666666">.</span>losses<span style="color: #666666">.</span>mean_squared_error(zeros, g_trial_d2t[<span style="color: #666666">0</span>] <span style="color: #666666">-</span> g_trial_d2x[<span style="color: #666666">0</span>])
learning_rate <span style="color: #666666">=</span> <span style="color: #666666">0.01</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(loss)
init <span style="color: #666666">=</span> tf<span style="color: #666666">.</span>global_variables_initializer()
g_analytic <span style="color: #666666">=</span> tf<span style="color: #666666">.</span>sin(np<span style="color: #666666">.</span>pi<span style="color: #666666">*</span>x)<span style="color: #666666">*</span>tf<span style="color: #666666">.</span>cos(np<span style="color: #666666">.</span>pi<span style="color: #666666">*</span>t) <span style="color: #666666">-</span> tf<span style="color: #666666">.</span>sin(np<span style="color: #666666">.</span>pi<span style="color: #666666">*</span>x)<span style="color: #666666">*</span>tf<span style="color: #666666">.</span>sin(np<span style="color: #666666">.</span>pi<span style="color: #666666">*</span>t)
g_dnn <span style="color: #666666">=</span> <span style="color: #008000">None</span>
<span style="color: #408080; font-style: italic">## The execution phase</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:
init<span style="color: #666666">.</span>run()
<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"># If one desires to see how the cost function behaves during training</span>
<span style="color: #408080; font-style: italic">#if i % 100 == 0:</span>
<span style="color: #408080; font-style: italic"># print(loss.eval())</span>
g_analytic <span style="color: #666666">=</span> g_analytic<span style="color: #666666">.</span>eval()
g_dnn <span style="color: #666666">=</span> g_trial<span style="color: #666666">.</span>eval()
<span style="color: #408080; font-style: italic">## Compare with the analutical solution</span>
diff <span style="color: #666666">=</span> np<span style="color: #666666">.</span>abs(g_analytic <span style="color: #666666">-</span> g_dnn)
<span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">&#39;Max absolute difference between analytical solution and TensorFlow DNN = &#39;</span>,np<span style="color: #666666">.</span>max(diff))
G_analytic <span style="color: #666666">=</span> g_analytic<span style="color: #666666">.</span>reshape((Nt,Nx))
G_dnn <span style="color: #666666">=</span> g_dnn<span style="color: #666666">.</span>reshape((Nt,Nx))
diff <span style="color: #666666">=</span> np<span style="color: #666666">.</span>abs(G_analytic <span style="color: #666666">-</span> G_dnn)
<span style="color: #408080; font-style: italic"># Plot the results</span>
X,T <span style="color: #666666">=</span> np<span style="color: #666666">.</span>meshgrid(x_np, t_np)
fig <span style="color: #666666">=</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>))
ax <span style="color: #666666">=</span> fig<span style="color: #666666">.</span>gca(projection<span style="color: #666666">=</span><span style="color: #BA2121">&#39;3d&#39;</span>)
ax<span style="color: #666666">.</span>set_title(<span style="color: #BA2121">&#39;Solution from the deep neural network w/ </span><span style="color: #BB6688; font-weight: bold">%d</span><span style="color: #BA2121"> layer&#39;</span><span style="color: #666666">%</span><span style="color: #008000">len</span>(num_hidden_neurons))
s <span style="color: #666666">=</span> ax<span style="color: #666666">.</span>plot_surface(X,T,G_dnn,linewidth<span style="color: #666666">=0</span>,antialiased<span style="color: #666666">=</span><span style="color: #008000">False</span>,cmap<span style="color: #666666">=</span>cm<span style="color: #666666">.</span>viridis)
ax<span style="color: #666666">.</span>set_xlabel(<span style="color: #BA2121">&#39;Time $t$&#39;</span>)
ax<span style="color: #666666">.</span>set_ylabel(<span style="color: #BA2121">&#39;Position $x$&#39;</span>);
fig <span style="color: #666666">=</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>))
ax <span style="color: #666666">=</span> fig<span style="color: #666666">.</span>gca(projection<span style="color: #666666">=</span><span style="color: #BA2121">&#39;3d&#39;</span>)
ax<span style="color: #666666">.</span>set_title(<span style="color: #BA2121">&#39;Analytical solution&#39;</span>)
s <span style="color: #666666">=</span> ax<span style="color: #666666">.</span>plot_surface(X,T,G_analytic,linewidth<span style="color: #666666">=0</span>,antialiased<span style="color: #666666">=</span><span style="color: #008000">False</span>,cmap<span style="color: #666666">=</span>cm<span style="color: #666666">.</span>viridis)
ax<span style="color: #666666">.</span>set_xlabel(<span style="color: #BA2121">&#39;Time $t$&#39;</span>)
ax<span style="color: #666666">.</span>set_ylabel(<span style="color: #BA2121">&#39;Position $x$&#39;</span>);
fig <span style="color: #666666">=</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>))
ax <span style="color: #666666">=</span> fig<span style="color: #666666">.</span>gca(projection<span style="color: #666666">=</span><span style="color: #BA2121">&#39;3d&#39;</span>)
ax<span style="color: #666666">.</span>set_title(<span style="color: #BA2121">&#39;Difference&#39;</span>)
s <span style="color: #666666">=</span> ax<span style="color: #666666">.</span>plot_surface(X,T,diff,linewidth<span style="color: #666666">=0</span>,antialiased<span style="color: #666666">=</span><span style="color: #008000">False</span>,cmap<span style="color: #666666">=</span>cm<span style="color: #666666">.</span>viridis)
ax<span style="color: #666666">.</span>set_xlabel(<span style="color: #BA2121">&#39;Time $t$&#39;</span>)
ax<span style="color: #666666">.</span>set_ylabel(<span style="color: #BA2121">&#39;Position $x$&#39;</span>);
<span style="color: #408080; font-style: italic">## Take some 3D slices</span>
indx1 <span style="color: #666666">=</span> <span style="color: #666666">0</span>
indx2 <span style="color: #666666">=</span> <span style="color: #008000">int</span>(Nt<span style="color: #666666">/2</span>)
indx3 <span style="color: #666666">=</span> Nt<span style="color: #666666">-1</span>
t1 <span style="color: #666666">=</span> t_np[indx1]
t2 <span style="color: #666666">=</span> t_np[indx2]
t3 <span style="color: #666666">=</span> t_np[indx3]
<span style="color: #408080; font-style: italic"># Slice the results from the DNN</span>
res1 <span style="color: #666666">=</span> G_dnn[indx1,:]
res2 <span style="color: #666666">=</span> G_dnn[indx2,:]
res3 <span style="color: #666666">=</span> G_dnn[indx3,:]
<span style="color: #408080; font-style: italic"># Slice the analytical results</span>
res_analytical1 <span style="color: #666666">=</span> G_analytic[indx1,:]
res_analytical2 <span style="color: #666666">=</span> G_analytic[indx2,:]
res_analytical3 <span style="color: #666666">=</span> G_analytic[indx3,:]
<span style="color: #408080; font-style: italic"># Plot the slices</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">&quot;Computed solutions at time = </span><span style="color: #BB6688; font-weight: bold">%g</span><span style="color: #BA2121">&quot;</span><span style="color: #666666">%</span>t1)
plt<span style="color: #666666">.</span>plot(x_np, res1)
plt<span style="color: #666666">.</span>plot(x_np,res_analytical1)
plt<span style="color: #666666">.</span>legend([<span style="color: #BA2121">&#39;dnn&#39;</span>,<span style="color: #BA2121">&#39;analytical&#39;</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">&quot;Computed solutions at time = </span><span style="color: #BB6688; font-weight: bold">%g</span><span style="color: #BA2121">&quot;</span><span style="color: #666666">%</span>t2)
plt<span style="color: #666666">.</span>plot(x_np, res2)
plt<span style="color: #666666">.</span>plot(x_np,res_analytical2)
plt<span style="color: #666666">.</span>legend([<span style="color: #BA2121">&#39;dnn&#39;</span>,<span style="color: #BA2121">&#39;analytical&#39;</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">&quot;Computed solutions at time = </span><span style="color: #BB6688; font-weight: bold">%g</span><span style="color: #BA2121">&quot;</span><span style="color: #666666">%</span>t3)
plt<span style="color: #666666">.</span>plot(x_np, res3)
plt<span style="color: #666666">.</span>plot(x_np,res_analytical3)
plt<span style="color: #666666">.</span>legend([<span style="color: #BA2121">&#39;dnn&#39;</span>,<span style="color: #BA2121">&#39;analytical&#39;</span>])
plt<span style="color: #666666">.</span>show()
</pre></div>
<p>
The program manages to find a solution having max absolute difference to the analytical
at approximately 0.0059, by just using some minutes!
It was found, by some testing, that one hidden layer with 90 neurons actually performed well.
<p>
<p>
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