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425 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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<a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></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="#___sec18" style="font-size: 80%;">Using forward Euler to solve the ODE</a></li>
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<!-- navigation toc: --> <li><a href="._odenn-bs020.html#___sec19" style="font-size: 80%;">Using TensorFlow to model logistic population growth</a></li>
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<!-- navigation toc: --> <li><a href="._odenn-bs021.html#___sec20" style="font-size: 80%;">The general program flow in TensorFlow</a></li>
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<!-- navigation toc: --> <li><a href="._odenn-bs021.html#___sec21" style="font-size: 80%;">Program flow in TensorFlow - Construction phase</a></li>
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<!-- navigation toc: --> <li><a href="._odenn-bs021.html#___sec22" style="font-size: 80%;">Program flow in TensorFlow - Execution phase</a></li>
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<!-- navigation toc: --> <li><a href="._odenn-bs021.html#___sec23" style="font-size: 80%;">The full program modeling logistic population growth using TensorFlow</a></li>
|
|
<!-- 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>
|
|
<!-- 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>
|
|
<!-- navigation toc: --> <li><a href="._odenn-bs033.html#___sec35" style="font-size: 80%;">Setting up the network using Autograd; The full program</a></li>
|
|
<!-- navigation toc: --> <li><a href="._odenn-bs034.html#___sec36" style="font-size: 80%;">Example: Solving the wave equation using Autograd and TensorFlow</a></li>
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<!-- navigation toc: --> <li><a href="._odenn-bs035.html#___sec37" style="font-size: 80%;">The problem to solve for</a></li>
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<!-- navigation toc: --> <li><a href="._odenn-bs036.html#___sec38" style="font-size: 80%;">The trial solution</a></li>
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<!-- navigation toc: --> <li><a href="._odenn-bs037.html#___sec39" style="font-size: 80%;">The analytical solution</a></li>
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|
<!-- navigation toc: --> <li><a href="._odenn-bs038.html#___sec40" style="font-size: 80%;">Solving the wave equation - the full program using Autograd</a></li>
|
|
<!-- 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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</li>
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</ul>
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</div>
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</div>
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</div> <!-- end of navigation bar -->
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<div class="container">
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<p> </p><p> </p><p> </p> <!-- add vertical space -->
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<a name="part0019"></a>
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<!-- !split -->
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<h2 id="___sec18" class="anchor">Using forward Euler to solve the ODE </h2>
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<p>
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A straight-forward way of solving an ODE numerically, is to use Euler's method.
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<p>
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Euler's method uses Taylor series to approximate the value at a function \( f \) at a step \( \Delta x \) from \( x \):
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$$
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f(x + \Delta x) \approx f(x) + \Delta x f'(x)
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$$
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<p>
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In our case, using Euler's method to approximate the value of \( g \) at a step \( \Delta t \) from \( t \) yields
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$$
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\begin{aligned}
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g(t + \Delta t) &\approx g(t) + \Delta t g'(t) \\
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&= g(t) + \Delta t \big(\alpha g(t)(A - g(t))\big)
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\end{aligned}
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$$
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along with the condition that \( g(0) = g_0 \).
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<p>
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Let \( t_i = i \cdot \Delta t \) where \( \Delta t = \frac{T}{N_t-1} \) where \( T \) is the final time our solver must solve for and \( N_t \) the number of values for \( t \in [0, T] \) for \( i = 0, \dots, N_t-1 \).
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<p>
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For \( i \geq 1 \), we have that
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$$
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\begin{aligned}
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t_i &= i\Delta t \\
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&= (i - 1)\Delta t + \Delta t \\
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&= t_{i-1} + \Delta t
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\end{aligned}
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$$
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<p>
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Now, if \( g_i = g(t_i) \) then
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$$
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\begin{equation}
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\begin{aligned}
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g_i &= g(t_i) \\
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&= g(t_{i-1} + \Delta t) \\
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&\approx g(t_{i-1}) + \Delta t \big(\alpha g(t_{i-1})(A - g(t_{i-1}))\big) \\
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&= g_{i-1} + \Delta t \big(\alpha g_{i-1}(A - g_{i-1})\big)
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\end{aligned}
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\end{equation} \tag{12}
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$$
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for \( i \geq 1 \) and \( g_0 = g(t_0) = g(0) = g_0 \).
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<p>
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Equation <a href="#mjx-eqn-12">(12)</a> could be implemented in the following way,
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extending the program that uses the network using Autograd:
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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: #408080; font-style: italic"># Assume that all function definitions from the example program using Autograd</span>
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<span style="color: #408080; font-style: italic"># are located here.</span>
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<span style="color: #008000; font-weight: bold">if</span> <span style="color: #19177C">__name__</span> <span style="color: #666666">==</span> <span style="color: #BA2121">'__main__'</span>:
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npr<span style="color: #666666">.</span>seed(<span style="color: #666666">4155</span>)
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<span style="color: #408080; font-style: italic">## Decide the vales of arguments to the function to solve</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">## Set up the initial parameters</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_iter <span style="color: #666666">=</span> <span style="color: #666666">1000</span>
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lmb <span style="color: #666666">=</span> <span style="color: #666666">1e-3</span>
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P <span style="color: #666666">=</span> solve_ode_deep_neural_network(t, num_hidden_neurons, num_iter, lmb)
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g_dnn_ag <span style="color: #666666">=</span> g_trial_deep(t,P)
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g_analytical <span style="color: #666666">=</span> g_analytic(t)
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<span style="color: #408080; font-style: italic"># Find the maximum absolute difference between the solutons:</span>
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diff_ag <span style="color: #666666">=</span> np<span style="color: #666666">.</span>max(np<span style="color: #666666">.</span>abs(g_dnn_ag <span style="color: #666666">-</span> g_analytical))
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<span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">"The max absolute difference between the solutions is: </span><span style="color: #BB6688; font-weight: bold">%g</span><span style="color: #BA2121">"</span><span style="color: #666666">%</span>diff_ag)
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plt<span style="color: #666666">.</span>figure(figsize<span style="color: #666666">=</span>(<span style="color: #666666">10</span>,<span style="color: #666666">10</span>))
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plt<span style="color: #666666">.</span>title(<span style="color: #BA2121">'Performance of neural network solving an ODE compared to the analytical solution'</span>)
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plt<span style="color: #666666">.</span>plot(t, g_analytical)
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plt<span style="color: #666666">.</span>plot(t, g_dnn_ag[<span style="color: #666666">0</span>,:])
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plt<span style="color: #666666">.</span>legend([<span style="color: #BA2121">'analytical'</span>,<span style="color: #BA2121">'nn'</span>])
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plt<span style="color: #666666">.</span>xlabel(<span style="color: #BA2121">'t'</span>)
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plt<span style="color: #666666">.</span>ylabel(<span style="color: #BA2121">'g(t)'</span>)
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<span style="color: #408080; font-style: italic">## Find an approximation to the funtion using forward Euler</span>
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alpha, A, g0 <span style="color: #666666">=</span> get_parameters()
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dt <span style="color: #666666">=</span> T<span style="color: #666666">/</span>(Nt <span style="color: #666666">-</span> <span style="color: #666666">1</span>)
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<span style="color: #408080; font-style: italic"># Perform forward Euler to solve the ODE</span>
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g_euler <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros(Nt)
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g_euler[<span style="color: #666666">0</span>] <span style="color: #666666">=</span> g0
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<span style="color: #008000; font-weight: bold">for</span> i <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(<span style="color: #666666">1</span>,Nt):
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g_euler[i] <span style="color: #666666">=</span> g_euler[i<span style="color: #666666">-1</span>] <span style="color: #666666">+</span> dt<span style="color: #666666">*</span>(alpha<span style="color: #666666">*</span>g_euler[i<span style="color: #666666">-1</span>]<span style="color: #666666">*</span>(A <span style="color: #666666">-</span> g_euler[i<span style="color: #666666">-1</span>]))
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<span style="color: #408080; font-style: italic"># Print the errors done by each method</span>
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diff1 <span style="color: #666666">=</span> np<span style="color: #666666">.</span>max(np<span style="color: #666666">.</span>abs(g_euler <span style="color: #666666">-</span> g_analytical))
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diff2 <span style="color: #666666">=</span> np<span style="color: #666666">.</span>max(np<span style="color: #666666">.</span>abs(g_dnn_ag[<span style="color: #666666">0</span>,:] <span style="color: #666666">-</span> g_analytical))
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<span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">'Max absolute difference between Euler method and analytical: </span><span style="color: #BB6688; font-weight: bold">%g</span><span style="color: #BA2121">'</span><span style="color: #666666">%</span>diff1)
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<span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">'Max absolute difference between deep neural network and analytical: </span><span style="color: #BB6688; font-weight: bold">%g</span><span style="color: #BA2121">'</span><span style="color: #666666">%</span>diff2)
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<span style="color: #408080; font-style: italic"># Plot results</span>
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plt<span style="color: #666666">.</span>figure(figsize<span style="color: #666666">=</span>(<span style="color: #666666">10</span>,<span style="color: #666666">10</span>))
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plt<span style="color: #666666">.</span>plot(t,g_euler)
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plt<span style="color: #666666">.</span>plot(t,g_analytical)
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plt<span style="color: #666666">.</span>plot(t,g_dnn_ag[<span style="color: #666666">0</span>,:])
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plt<span style="color: #666666">.</span>legend([<span style="color: #BA2121">'euler'</span>,<span style="color: #BA2121">'analytical'</span>,<span style="color: #BA2121">'dnn'</span>])
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plt<span style="color: #666666">.</span>xlabel(<span style="color: #BA2121">'Time t'</span>)
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plt<span style="color: #666666">.</span>ylabel(<span style="color: #BA2121">'g(t)'</span>)
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plt<span style="color: #666666">.</span>show()
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</pre></div>
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<p>
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Running the program gives
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<p>
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<!-- code=text typeset with pygments style "default" -->
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<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span>Max absolute difference between Euler method and analytical: 0.011225
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Max absolute difference between deep neural network and analytical: 0.00424909
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</pre></div>
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<p>
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<p>
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