mhjensen
360 lines
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360 lines
20 KiB
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<!-- navigation toc: --> <li><a href="._odenn-bs001.html#___sec0" style="font-size: 80%;">Differential equations</a></li>
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<!-- navigation toc: --> <li><a href="._odenn-bs002.html#___sec1" style="font-size: 80%;">Description of the equation to solve for</a></li>
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<!-- navigation toc: --> <li><a href="._odenn-bs003.html#___sec2" style="font-size: 80%;">Ordinary Differential Equations</a></li>
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<!-- navigation toc: --> <li><a href="._odenn-bs004.html#___sec3" style="font-size: 80%;">The trial solution</a></li>
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<!-- navigation toc: --> <li><a href="._odenn-bs005.html#___sec4" style="font-size: 80%;">Minimizing the cost function using gradient descent and automatic differentiation</a></li>
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<!-- navigation toc: --> <li><a href="._odenn-bs006.html#___sec5" style="font-size: 80%;">Example: Exponential decay and setting up the network using Autograd</a></li>
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<!-- navigation toc: --> <li><a href="._odenn-bs007.html#___sec6" style="font-size: 80%;">The function to solve for</a></li>
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<!-- navigation toc: --> <li><a href="._odenn-bs008.html#___sec7" style="font-size: 80%;">The trial solution</a></li>
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<!-- navigation toc: --> <li><a href="._odenn-bs009.html#___sec8" style="font-size: 80%;">Reformulating the problem</a></li>
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<!-- navigation toc: --> <li><a href="._odenn-bs010.html#___sec9" style="font-size: 80%;">A possible implementation of a neural network using Autograd</a></li>
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<!-- navigation toc: --> <li><a href="._odenn-bs011.html#___sec10" style="font-size: 80%;">Backpropagation using Autograd</a></li>
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<!-- navigation toc: --> <li><a href="._odenn-bs012.html#___sec11" style="font-size: 80%;">Gradient descent</a></li>
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<!-- navigation toc: --> <li><a href="._odenn-bs013.html#___sec12" style="font-size: 80%;">The network with one input, hidden, and output layer</a></li>
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<!-- navigation toc: --> <li><a href="._odenn-bs014.html#___sec13" style="font-size: 80%;">The network with one input layer, specified number of hidden layers, and one output layer output layer</a></li>
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<!-- navigation toc: --> <li><a href="._odenn-bs015.html#___sec14" style="font-size: 80%;">Example: Population growth, comparing Autograd, TensorFlow, and Euler's scheme</a></li>
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<!-- navigation toc: --> <li><a href="._odenn-bs016.html#___sec15" style="font-size: 80%;">Setting up the problem</a></li>
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<!-- navigation toc: --> <li><a href="._odenn-bs017.html#___sec16" style="font-size: 80%;">The trial solution</a></li>
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<!-- navigation toc: --> <li><a href="._odenn-bs018.html#___sec17" style="font-size: 80%;">The program using Autograd</a></li>
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<!-- navigation toc: --> <li><a href="._odenn-bs019.html#___sec18" style="font-size: 80%;">Using forward Euler to solve the ODE</a></li>
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<!-- navigation toc: --> <li><a href="._odenn-bs020.html#___sec19" style="font-size: 80%;">Using TensorFlow to model logistic population growth</a></li>
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<!-- navigation toc: --> <li><a href="._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>
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|
<!-- navigation toc: --> <li><a href="._odenn-bs022.html#___sec24" style="font-size: 80%;">Example: Solving the one dimensional Poisson equation using Autograd and TensorFlow</a></li>
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<!-- navigation toc: --> <li><a href="._odenn-bs023.html#___sec25" style="font-size: 80%;">The specific equation to solve for</a></li>
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<!-- navigation toc: --> <li><a href="._odenn-bs024.html#___sec26" style="font-size: 80%;">Solving the equation using Autograd</a></li>
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<!-- navigation toc: --> <li><a href="._odenn-bs025.html#___sec27" style="font-size: 80%;">Comparing with a numerical scheme</a></li>
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<!-- navigation toc: --> <li><a href="._odenn-bs026.html#___sec28" style="font-size: 80%;">Using gradient descent in TensorFlow to solve Poisson equation</a></li>
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<!-- navigation toc: --> <li><a href="._odenn-bs027.html#___sec29" style="font-size: 80%;">Using a different optimization algorithm implemented in TensorFlow to solve Poisson equation</a></li>
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<!-- navigation toc: --> <li><a href="._odenn-bs028.html#___sec30" style="font-size: 80%;">Partial Differential Equations</a></li>
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<!-- navigation toc: --> <li><a href="._odenn-bs029.html#___sec31" style="font-size: 80%;">Example: The diffusion equation</a></li>
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<!-- navigation toc: --> <li><a href="._odenn-bs030.html#___sec32" style="font-size: 80%;">Defining the problem</a></li>
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<!-- navigation toc: --> <li><a href="._odenn-bs031.html#___sec33" style="font-size: 80%;">Setting up the network using Autograd</a></li>
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<!-- navigation toc: --> <li><a href="#___sec34" style="font-size: 80%;">Setting up the network using Autograd; The trial solution</a></li>
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<!-- navigation toc: --> <li><a href="._odenn-bs033.html#___sec35" style="font-size: 80%;">Setting up the network using Autograd; The full program</a></li>
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<!-- navigation toc: --> <li><a href="._odenn-bs034.html#___sec36" style="font-size: 80%;">Example: Solving the wave equation using Autograd and TensorFlow</a></li>
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<!-- navigation toc: --> <li><a href="._odenn-bs035.html#___sec37" style="font-size: 80%;">The problem to solve for</a></li>
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<!-- navigation toc: --> <li><a href="._odenn-bs036.html#___sec38" style="font-size: 80%;">The trial solution</a></li>
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<!-- navigation toc: --> <li><a href="._odenn-bs037.html#___sec39" style="font-size: 80%;">The analytical solution</a></li>
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<!-- navigation toc: --> <li><a href="._odenn-bs038.html#___sec40" style="font-size: 80%;">Solving the wave equation - the full program using Autograd</a></li>
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<!-- navigation toc: --> <li><a href="._odenn-bs039.html#___sec41" style="font-size: 80%;">Solving the wave equation - the full program using TensorFlow</a></li>
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<!-- navigation toc: --> <li><a href="._odenn-bs040.html#___sec42" style="font-size: 80%;">Resources</a></li>
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<a name="part0032"></a>
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<!-- !split -->
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<h2 id="___sec34" class="anchor">Setting up the network using Autograd; The trial solution </h2>
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The cost function must then iterate through the given arrays containing values for \( x \) and \( t \), defines a point \( (x,t) \) the deep neural network and the trial solution is evaluated at, and then finds the Jacobian of the trial solution.
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<p>
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A possible trial solution for this PDE is
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$$
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g_t(x,t) = h_1(x,t) + x(1-x)tN(x,t,P)
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$$
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<p>
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with \( A(x,t) \) being a function ensuring that \( g_t(x,t) \) satisfies our given conditions, and \( N(x,t,P) \) being the output from the deep neural network using weights and biases for each layer from \( P \).
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<p>
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To fulfill the conditions, \( A(x,t) \) could be:
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$$
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h_1(x,t) = (1-t)\Big(u(x) - \big((1-x)u(0) + x u(1)\big)\Big) = (1-t)u(x) = (1-t)\sin(\pi x)
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$$
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since \( (0) = u(1) = 0 \) and \( u(x) = \sin(\pi x) \).
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<p>
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The Jacobian is used because the program must find the derivative of the trial solution with respect to \( x \) and \( t \).
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<p>
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This gives the necessity of computing the Jacobian matrix, as we want to evaluate the gradient with respect to \( x \) and \( t \) (note that the Jacobian of a scalar-valued multivariate function is simply its gradient).
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<p>
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In Autograd, the differentiation is by default done with respect to the first input argument of your Python function. Since the points is an array representing \( x \) and \( t \), the Jacobian is calculated using the values of \( x \) and \( t \).
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<p>
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To find the second derivative with respect to \( x \) and \( t \), the Jacobian can be found for the second time. The result is a Hessian matrix, which is the matrix containing all the possible second order mixed derivatives of \( g(x,t) \).
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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"># Set up the trial function:</span>
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<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">u</span>(x):
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<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)
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<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">g_trial</span>(point,P):
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x,t <span style="color: #666666">=</span> point
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<span style="color: #008000; font-weight: bold">return</span> (<span style="color: #666666">1-</span>t)<span style="color: #666666">*</span>u(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">*</span>deep_neural_network(P,point)
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<span style="color: #408080; font-style: italic"># The right side of the ODE:</span>
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<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">f</span>(point):
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<span style="color: #008000; font-weight: bold">return</span> <span style="color: #666666">0.</span>
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<span style="color: #408080; font-style: italic"># The cost function:</span>
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<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">cost_function</span>(P, x, t):
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cost_sum <span style="color: #666666">=</span> <span style="color: #666666">0</span>
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g_t_jacobian_func <span style="color: #666666">=</span> jacobian(g_trial)
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g_t_hessian_func <span style="color: #666666">=</span> hessian(g_trial)
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<span style="color: #008000; font-weight: bold">for</span> x_ <span style="color: #AA22FF; font-weight: bold">in</span> x:
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<span style="color: #008000; font-weight: bold">for</span> t_ <span style="color: #AA22FF; font-weight: bold">in</span> t:
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point <span style="color: #666666">=</span> np<span style="color: #666666">.</span>array([x_,t_])
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g_t <span style="color: #666666">=</span> g_trial(point,P)
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g_t_jacobian <span style="color: #666666">=</span> g_t_jacobian_func(point,P)
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g_t_hessian <span style="color: #666666">=</span> g_t_hessian_func(point,P)
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g_t_dt <span style="color: #666666">=</span> g_t_jacobian[<span style="color: #666666">1</span>]
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g_t_d2x <span style="color: #666666">=</span> g_t_hessian[<span style="color: #666666">0</span>][<span style="color: #666666">0</span>]
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func <span style="color: #666666">=</span> f(point)
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err_sqr <span style="color: #666666">=</span> ( (g_t_dt <span style="color: #666666">-</span> g_t_d2x) <span style="color: #666666">-</span> func)<span style="color: #666666">**2</span>
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cost_sum <span style="color: #666666">+=</span> err_sqr
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<span style="color: #008000; font-weight: bold">return</span> cost_sum
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</pre></div>
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<p>
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<p>
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