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<a class="navbar-brand" href="week43-bs.html">Week 43: Solving Differential Equations with Deep Learning and Dimensionality Reduction methods</a>
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<!-- navigation toc: --> <li><a href="._week43-bs001.html#___sec0" style="font-size: 80%;"><b>Plans for week 43</b></a></li>
<!-- navigation toc: --> <li><a href="._week43-bs002.html#___sec1" style="font-size: 80%;"><b>Recurrent Neural Networks</b></a></li>
<!-- navigation toc: --> <li><a href="._week43-bs003.html#___sec2" style="font-size: 80%;"><b>Solving ODEs with Deep Learning</b></a></li>
<!-- navigation toc: --> <li><a href="._week43-bs004.html#___sec3" style="font-size: 80%;"><b>Ordinary Differential Equations</b></a></li>
<!-- navigation toc: --> <li><a href="._week43-bs005.html#___sec4" style="font-size: 80%;"><b>The trial solution</b></a></li>
<!-- navigation toc: --> <li><a href="._week43-bs006.html#___sec5" style="font-size: 80%;"><b>Minimization process</b></a></li>
<!-- navigation toc: --> <li><a href="._week43-bs007.html#___sec6" style="font-size: 80%;"><b>Minimizing the cost function using gradient descent and automatic differentiation</b></a></li>
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<!-- navigation toc: --> <li><a href="._week43-bs009.html#___sec8" style="font-size: 80%;"><b>The function to solve for</b></a></li>
<!-- navigation toc: --> <li><a href="._week43-bs010.html#___sec9" style="font-size: 80%;"><b>The trial solution</b></a></li>
<!-- navigation toc: --> <li><a href="._week43-bs011.html#___sec10" style="font-size: 80%;"><b>Setup of Network</b></a></li>
<!-- navigation toc: --> <li><a href="._week43-bs012.html#___sec11" style="font-size: 80%;"><b>Reformulating the problem</b></a></li>
<!-- navigation toc: --> <li><a href="._week43-bs013.html#___sec12" style="font-size: 80%;"><b>More technicalities</b></a></li>
<!-- navigation toc: --> <li><a href="._week43-bs014.html#___sec13" style="font-size: 80%;"><b>More details</b></a></li>
<!-- navigation toc: --> <li><a href="._week43-bs015.html#___sec14" style="font-size: 80%;"><b>A possible implementation of a neural network</b></a></li>
<!-- navigation toc: --> <li><a href="._week43-bs016.html#___sec15" style="font-size: 80%;"><b>Technicalities</b></a></li>
<!-- navigation toc: --> <li><a href="._week43-bs017.html#___sec16" style="font-size: 80%;"><b>Final technicalities I</b></a></li>
<!-- navigation toc: --> <li><a href="._week43-bs018.html#___sec17" style="font-size: 80%;"><b>Final technicalities II</b></a></li>
<!-- navigation toc: --> <li><a href="._week43-bs019.html#___sec18" style="font-size: 80%;"><b>Final technicalities III</b></a></li>
<!-- navigation toc: --> <li><a href="._week43-bs020.html#___sec19" style="font-size: 80%;"><b>Final technicalities IV</b></a></li>
<!-- navigation toc: --> <li><a href="._week43-bs021.html#___sec20" style="font-size: 80%;"><b>Back propagation</b></a></li>
<!-- navigation toc: --> <li><a href="._week43-bs022.html#___sec21" style="font-size: 80%;"><b>Gradient descent</b></a></li>
<!-- navigation toc: --> <li><a href="._week43-bs023.html#___sec22" style="font-size: 80%;"><b>The code for solving the ODE</b></a></li>
<!-- navigation toc: --> <li><a href="._week43-bs024.html#___sec23" style="font-size: 80%;"><b>The network with one input layer, specified number of hidden layers, and one output layer</b></a></li>
<!-- navigation toc: --> <li><a href="._week43-bs025.html#___sec24" style="font-size: 80%;"><b>Example: Population growth</b></a></li>
<!-- navigation toc: --> <li><a href="._week43-bs026.html#___sec25" style="font-size: 80%;"><b>Setting up the problem</b></a></li>
<!-- navigation toc: --> <li><a href="._week43-bs027.html#___sec26" style="font-size: 80%;"><b>The trial solution</b></a></li>
<!-- navigation toc: --> <li><a href="._week43-bs028.html#___sec27" style="font-size: 80%;"><b>The program using Autograd</b></a></li>
<!-- navigation toc: --> <li><a href="._week43-bs029.html#___sec28" style="font-size: 80%;"><b>Using forward Euler to solve the ODE</b></a></li>
<!-- navigation toc: --> <li><a href="._week43-bs030.html#___sec29" style="font-size: 80%;"><b>Example: Solving the one dimensional Poisson equation</b></a></li>
<!-- navigation toc: --> <li><a href="._week43-bs031.html#___sec30" style="font-size: 80%;"><b>The specific equation to solve for</b></a></li>
<!-- navigation toc: --> <li><a href="._week43-bs032.html#___sec31" style="font-size: 80%;"><b>Solving the equation using Autograd</b></a></li>
<!-- navigation toc: --> <li><a href="._week43-bs033.html#___sec32" style="font-size: 80%;"><b>Comparing with a numerical scheme</b></a></li>
<!-- navigation toc: --> <li><a href="._week43-bs034.html#___sec33" style="font-size: 80%;"><b>Setting up the code</b></a></li>
<!-- navigation toc: --> <li><a href="._week43-bs035.html#___sec34" style="font-size: 80%;"><b>Partial Differential Equations</b></a></li>
<!-- navigation toc: --> <li><a href="._week43-bs036.html#___sec35" style="font-size: 80%;"><b>Type of problem</b></a></li>
<!-- navigation toc: --> <li><a href="._week43-bs037.html#___sec36" style="font-size: 80%;"><b>Network requirements</b></a></li>
<!-- navigation toc: --> <li><a href="._week43-bs038.html#___sec37" style="font-size: 80%;"><b>More details</b></a></li>
<!-- navigation toc: --> <li><a href="._week43-bs039.html#___sec38" style="font-size: 80%;"><b>Example: The diffusion equation</b></a></li>
<!-- navigation toc: --> <li><a href="._week43-bs040.html#___sec39" style="font-size: 80%;"><b>Defining the problem</b></a></li>
<!-- navigation toc: --> <li><a href="._week43-bs041.html#___sec40" style="font-size: 80%;"><b>Setting up the network using Autograd</b></a></li>
<!-- navigation toc: --> <li><a href="._week43-bs042.html#___sec41" style="font-size: 80%;"><b>Setting up the network using Autograd; The trial solution</b></a></li>
<!-- navigation toc: --> <li><a href="._week43-bs043.html#___sec42" style="font-size: 80%;"><b>Why the jacobian?</b></a></li>
<!-- navigation toc: --> <li><a href="._week43-bs044.html#___sec43" style="font-size: 80%;"><b>Setting up the network using Autograd; The full program</b></a></li>
<!-- navigation toc: --> <li><a href="._week43-bs045.html#___sec44" style="font-size: 80%;"><b>Example: Solving the wave equation with Neural Networks</b></a></li>
<!-- navigation toc: --> <li><a href="._week43-bs046.html#___sec45" style="font-size: 80%;"><b>The problem to solve for</b></a></li>
<!-- navigation toc: --> <li><a href="._week43-bs047.html#___sec46" style="font-size: 80%;"><b>The trial solution</b></a></li>
<!-- navigation toc: --> <li><a href="._week43-bs048.html#___sec47" style="font-size: 80%;"><b>The analytical solution</b></a></li>
<!-- navigation toc: --> <li><a href="#___sec48" style="font-size: 80%;"><b>Solving the wave equation - the full program using Autograd</b></a></li>
<!-- navigation toc: --> <li><a href="._week43-bs050.html#___sec49" style="font-size: 80%;"><b>Resources on differential equations and deep learning</b></a></li>
<!-- navigation toc: --> <li><a href="._week43-bs051.html#___sec50" style="font-size: 80%;"><b>Friday, Principal Component Analysis</b></a></li>
<!-- navigation toc: --> <li><a href="._week43-bs052.html#___sec51" style="font-size: 80%;"><b>Basic ideas of the Principal Component Analysis (PCA)</b></a></li>
<!-- navigation toc: --> <li><a href="._week43-bs053.html#___sec52" style="font-size: 80%;"><b>Introducing the Covariance and Correlation functions</b></a></li>
<!-- navigation toc: --> <li><a href="._week43-bs054.html#___sec53" style="font-size: 80%;"><b>Correlation Function and Design/Feature Matrix</b></a></li>
<!-- navigation toc: --> <li><a href="._week43-bs055.html#___sec54" style="font-size: 80%;"><b>Covariance Matrix Examples</b></a></li>
<!-- navigation toc: --> <li><a href="._week43-bs056.html#___sec55" style="font-size: 80%;"><b>Correlation Matrix</b></a></li>
<!-- navigation toc: --> <li><a href="._week43-bs057.html#___sec56" style="font-size: 80%;"><b>Correlation Matrix with Pandas</b></a></li>
<!-- navigation toc: --> <li><a href="._week43-bs058.html#___sec57" style="font-size: 80%;"><b>Correlation Matrix with Pandas and the Franke function</b></a></li>
<!-- navigation toc: --> <li><a href="._week43-bs059.html#___sec58" style="font-size: 80%;"><b>Rewriting the Covariance and/or Correlation Matrix</b></a></li>
<!-- navigation toc: --> <li><a href="._week43-bs060.html#___sec59" style="font-size: 80%;"><b>Towards the PCA theorem</b></a></li>
<!-- navigation toc: --> <li><a href="._week43-bs061.html#___sec60" style="font-size: 80%;"><b>The Algorithm before the Theorem</b></a></li>
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec61" style="font-size: 80%;"><b>Writing our own PCA code</b></a></li>
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec62" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Compute the sample mean and center the data</a></li>
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec63" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Compute the sample covariance</a></li>
<!-- navigation toc: --> <li><a href="._week43-bs062.html#___sec64" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Diagonalize the sample covariance matrix to obtain the principal components</a></li>
<!-- navigation toc: --> <li><a href="._week43-bs063.html#___sec65" style="font-size: 80%;"><b>Classical PCA Theorem</b></a></li>
<!-- navigation toc: --> <li><a href="._week43-bs064.html#___sec66" style="font-size: 80%;"><b>Proof of the PCA Theorem</b></a></li>
<!-- navigation toc: --> <li><a href="._week43-bs065.html#___sec67" style="font-size: 80%;"><b>PCA Proof continued</b></a></li>
<!-- navigation toc: --> <li><a href="._week43-bs066.html#___sec68" style="font-size: 80%;"><b>The final step</b></a></li>
<!-- navigation toc: --> <li><a href="._week43-bs067.html#___sec69" style="font-size: 80%;"><b>Geometric Interpretation and link with Singular Value Decomposition</b></a></li>
<!-- navigation toc: --> <li><a href="._week43-bs068.html#___sec70" style="font-size: 80%;"><b>Principal Component Analysis</b></a></li>
<!-- navigation toc: --> <li><a href="._week43-bs069.html#___sec71" style="font-size: 80%;"><b>PCA and scikit-learn</b></a></li>
<!-- navigation toc: --> <li><a href="._week43-bs070.html#___sec72" style="font-size: 80%;"><b>Back to the Cancer Data</b></a></li>
<!-- navigation toc: --> <li><a href="._week43-bs071.html#___sec73" style="font-size: 80%;"><b>More on the PCA</b></a></li>
<!-- navigation toc: --> <li><a href="._week43-bs072.html#___sec74" style="font-size: 80%;"><b>Incremental PCA</b></a></li>
<!-- navigation toc: --> <li><a href="._week43-bs073.html#___sec75" style="font-size: 80%;"><b>Randomized PCA</b></a></li>
<!-- navigation toc: --> <li><a href="._week43-bs074.html#___sec76" style="font-size: 80%;"><b>Kernel PCA</b></a></li>
<!-- navigation toc: --> <li><a href="._week43-bs075.html#___sec77" style="font-size: 80%;"><b>LLE</b></a></li>
<!-- navigation toc: --> <li><a href="._week43-bs076.html#___sec78" style="font-size: 80%;"><b>Other techniques</b></a></li>
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<h2 id="___sec48" class="anchor">Solving the wave equation - the full program using Autograd </h2>
<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">autograd.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">autograd</span> <span style="color: #008000; font-weight: bold">import</span> hessian,grad
<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>
<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
<span style="color: #408080; font-style: italic">## Set up the trial function:</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> np<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>np<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">h1</span>(point):
x,t <span style="color: #666666">=</span> point
<span style="color: #008000; font-weight: bold">return</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: #008000; font-weight: bold">def</span> <span style="color: #0000FF">g_trial</span>(point,P):
x,t <span style="color: #666666">=</span> point
<span style="color: #008000; font-weight: bold">return</span> h1(point) <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>deep_neural_network(P,point)
<span style="color: #408080; font-style: italic">## Define the cost function</span>
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">cost_function</span>(P, x, t):
cost_sum <span style="color: #666666">=</span> <span style="color: #666666">0</span>
g_t_hessian_func <span style="color: #666666">=</span> hessian(g_trial)
<span style="color: #008000; font-weight: bold">for</span> x_ <span style="color: #AA22FF; font-weight: bold">in</span> x:
<span style="color: #008000; font-weight: bold">for</span> t_ <span style="color: #AA22FF; font-weight: bold">in</span> t:
point <span style="color: #666666">=</span> np<span style="color: #666666">.</span>array([x_,t_])
g_t_hessian <span style="color: #666666">=</span> g_t_hessian_func(point,P)
g_t_d2x <span style="color: #666666">=</span> g_t_hessian[<span style="color: #666666">0</span>][<span style="color: #666666">0</span>]
g_t_d2t <span style="color: #666666">=</span> g_t_hessian[<span style="color: #666666">1</span>][<span style="color: #666666">1</span>]
err_sqr <span style="color: #666666">=</span> ( (g_t_d2t <span style="color: #666666">-</span> g_t_d2x) )<span style="color: #666666">**2</span>
cost_sum <span style="color: #666666">+=</span> err_sqr
<span style="color: #008000; font-weight: bold">return</span> cost_sum <span style="color: #666666">/</span> (np<span style="color: #666666">.</span>size(t) <span style="color: #666666">*</span> np<span style="color: #666666">.</span>size(x))
<span style="color: #408080; font-style: italic">## The neural network</span>
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">sigmoid</span>(z):
<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))
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">deep_neural_network</span>(deep_params, x):
<span style="color: #408080; font-style: italic"># x is now a point and a 1D numpy array; make it a column vector</span>
num_coordinates <span style="color: #666666">=</span> np<span style="color: #666666">.</span>size(x,<span style="color: #666666">0</span>)
x <span style="color: #666666">=</span> x<span style="color: #666666">.</span>reshape(num_coordinates,<span style="color: #666666">-1</span>)
num_points <span style="color: #666666">=</span> np<span style="color: #666666">.</span>size(x,<span style="color: #666666">1</span>)
<span style="color: #408080; font-style: italic"># N_hidden is the number of hidden layers</span>
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>
<span style="color: #408080; font-style: italic"># Assume that the input layer does nothing to the input x</span>
x_input <span style="color: #666666">=</span> x
x_prev <span style="color: #666666">=</span> x_input
<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>(N_hidden):
<span style="color: #408080; font-style: italic"># From the list of parameters P; find the correct weigths and bias for this layer</span>
w_hidden <span style="color: #666666">=</span> deep_params[l]
<span style="color: #408080; font-style: italic"># Add a row of ones to include bias</span>
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_points)), x_prev ), axis <span style="color: #666666">=</span> <span style="color: #666666">0</span>)
z_hidden <span style="color: #666666">=</span> np<span style="color: #666666">.</span>matmul(w_hidden, x_prev)
x_hidden <span style="color: #666666">=</span> sigmoid(z_hidden)
<span style="color: #408080; font-style: italic"># Update x_prev such that next layer can use the output from this layer</span>
x_prev <span style="color: #666666">=</span> x_hidden
<span style="color: #408080; font-style: italic">## Output layer:</span>
<span style="color: #408080; font-style: italic"># Get the weights and bias for this layer</span>
w_output <span style="color: #666666">=</span> deep_params[<span style="color: #666666">-1</span>]
<span style="color: #408080; font-style: italic"># Include bias:</span>
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_points)), x_prev), axis <span style="color: #666666">=</span> <span style="color: #666666">0</span>)
z_output <span style="color: #666666">=</span> np<span style="color: #666666">.</span>matmul(w_output, x_prev)
x_output <span style="color: #666666">=</span> z_output
<span style="color: #008000; font-weight: bold">return</span> x_output[<span style="color: #666666">0</span>][<span style="color: #666666">0</span>]
<span style="color: #408080; font-style: italic">## The analytical solution</span>
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">g_analytic</span>(point):
x,t <span style="color: #666666">=</span> point
<span style="color: #008000; font-weight: bold">return</span> np<span style="color: #666666">.</span>sin(np<span style="color: #666666">.</span>pi<span style="color: #666666">*</span>x)<span style="color: #666666">*</span>np<span style="color: #666666">.</span>cos(np<span style="color: #666666">.</span>pi<span style="color: #666666">*</span>t) <span style="color: #666666">-</span> np<span style="color: #666666">.</span>sin(np<span style="color: #666666">.</span>pi<span style="color: #666666">*</span>x)<span style="color: #666666">*</span>np<span style="color: #666666">.</span>sin(np<span style="color: #666666">.</span>pi<span style="color: #666666">*</span>t)
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">solve_pde_deep_neural_network</span>(x,t, num_neurons, num_iter, lmb):
<span style="color: #408080; font-style: italic">## Set up initial weigths and biases</span>
N_hidden <span style="color: #666666">=</span> np<span style="color: #666666">.</span>size(num_neurons)
<span style="color: #408080; font-style: italic">## Set up initial weigths and biases</span>
<span style="color: #408080; font-style: italic"># Initialize the list of parameters:</span>
P <span style="color: #666666">=</span> [<span style="color: #008000; font-weight: bold">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>
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> <span style="color: #666666">+</span> <span style="color: #666666">1</span> ) <span style="color: #408080; font-style: italic"># 2 since we have two points, +1 to include bias</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>(<span style="color: #666666">1</span>,N_hidden):
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>
<span style="color: #408080; font-style: italic"># For the output layer</span>
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>
<span style="color: #008000">print</span>(<span style="color: #BA2121">&#39;Initial cost: &#39;</span>,cost_function(P, x, t))
cost_function_grad <span style="color: #666666">=</span> grad(cost_function,<span style="color: #666666">0</span>)
<span style="color: #408080; font-style: italic"># Let the update be done num_iter times</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):
cost_grad <span style="color: #666666">=</span> cost_function_grad(P, x , t)
<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>):
P[l] <span style="color: #666666">=</span> P[l] <span style="color: #666666">-</span> lmb <span style="color: #666666">*</span> cost_grad[l]
<span style="color: #008000">print</span>(<span style="color: #BA2121">&#39;Final cost: &#39;</span>,cost_function(P, x, t))
<span style="color: #008000; font-weight: bold">return</span> P
<span style="color: #008000; font-weight: bold">if</span> <span style="color: #19177C">__name__</span> <span style="color: #666666">==</span> <span style="color: #BA2121">&#39;__main__&#39;</span>:
<span style="color: #408080; font-style: italic">### Use the neural network:</span>
npr<span style="color: #666666">.</span>seed(<span style="color: #666666">15</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>; Nt <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)
t <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)
<span style="color: #408080; font-style: italic">## Set up the parameters for the network</span>
num_hidden_neurons <span style="color: #666666">=</span> [<span style="color: #666666">50</span>,<span style="color: #666666">20</span>]
num_iter <span style="color: #666666">=</span> <span style="color: #666666">1000</span>
lmb <span style="color: #666666">=</span> <span style="color: #666666">0.01</span>
P <span style="color: #666666">=</span> solve_pde_deep_neural_network(x,t, num_hidden_neurons, num_iter, lmb)
<span style="color: #408080; font-style: italic">## Store the results</span>
res <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros((Nx, Nt))
res_analytical <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros((Nx, Nt))
<span style="color: #008000; font-weight: bold">for</span> i,x_ <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">enumerate</span>(x):
<span style="color: #008000; font-weight: bold">for</span> j, t_ <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">enumerate</span>(t):
point <span style="color: #666666">=</span> np<span style="color: #666666">.</span>array([x_, t_])
res[i,j] <span style="color: #666666">=</span> g_trial(point,P)
res_analytical[i,j] <span style="color: #666666">=</span> g_analytic(point)
diff <span style="color: #666666">=</span> np<span style="color: #666666">.</span>abs(res <span style="color: #666666">-</span> res_analytical)
<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;Max difference between analytical and solution from nn: </span><span style="color: #BB6688; font-weight: bold">%g</span><span style="color: #BA2121">&quot;</span><span style="color: #666666">%</span>np<span style="color: #666666">.</span>max(diff))
<span style="color: #408080; font-style: italic">## Plot the solutions in two dimensions, that being in position and time</span>
T,X <span style="color: #666666">=</span> np<span style="color: #666666">.</span>meshgrid(t,x)
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(T,X,res,linewidth<span style="color: #666666">=0</span>,antialiased<span style="color: #666666">=</span><span style="color: #008000; font-weight: bold">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(T,X,res_analytical,linewidth<span style="color: #666666">=0</span>,antialiased<span style="color: #666666">=</span><span style="color: #008000; font-weight: bold">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(T,X,diff,linewidth<span style="color: #666666">=0</span>,antialiased<span style="color: #666666">=</span><span style="color: #008000; font-weight: bold">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 slices of the 3D plots just to see the solutions at particular times</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[indx1]
t2 <span style="color: #666666">=</span> t[indx2]
t3 <span style="color: #666666">=</span> t[indx3]
<span style="color: #408080; font-style: italic"># Slice the results from the DNN</span>
res1 <span style="color: #666666">=</span> res[:,indx1]
res2 <span style="color: #666666">=</span> res[:,indx2]
res3 <span style="color: #666666">=</span> res[:,indx3]
<span style="color: #408080; font-style: italic"># Slice the analytical results</span>
res_analytical1 <span style="color: #666666">=</span> res_analytical[:,indx1]
res_analytical2 <span style="color: #666666">=</span> res_analytical[:,indx2]
res_analytical3 <span style="color: #666666">=</span> res_analytical[:,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, res1)
plt<span style="color: #666666">.</span>plot(x,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, res2)
plt<span style="color: #666666">.</span>plot(x,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, res3)
plt<span style="color: #666666">.</span>plot(x,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()
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