mhjensen
366 lines
18 KiB
HTML
366 lines
18 KiB
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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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<!-- 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="#___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>
|
|
<!-- 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="._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="._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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<p> </p><p> </p><p> </p> <!-- add vertical space -->
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<a name="part0009"></a>
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<!-- !split -->
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<h2 id="___sec8" class="anchor">Reformulating the problem </h2>
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We wish that our neural network manages to minimize a given cost function.
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<p>
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A reformulation of out equation, <a href="._odenn-bs007.html#mjx-eqn-6">(6)</a>, must therefore be done,
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such that it describes the problem a neural network can solve for.
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<p>
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The neural network must find the set of weights and biases \( P \) such that the trial solution in <a href="._odenn-bs008.html#mjx-eqn-7">(7)</a> satisfies <a href="._odenn-bs007.html#mjx-eqn-6">(6)</a>.
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<p>
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The trial solution
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$$
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g_t(x, P) = g_0 + x \cdot N(x, P)
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$$
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<p>
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has been chosen such that it already solves the condition \( g(0) = g_0 \). What remains, is to find \( P \) such that
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$$
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\begin{equation} \tag{8}
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g_t'(x, P) = - \gamma g_t(x, P)
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\end{equation}
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$$
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<p>
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is fulfilled as <em>best as possible</em>.
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<p>
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The left hand side and right hand side of <a href="#mjx-eqn-8">(8)</a> must be computed separately, and then the neural network must choose weights and biases, contained in \( P \), such that the sides are equal as best as possible.
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This means that the absolute or squared difference between the sides must be as close to zero, ideally equal to zero.
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In this case, the difference squared shows to be an appropriate measurement of how erroneous the trial solution is with respect to \( P \) of the neural network.
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<p>
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This gives the following cost function our neural network must solve for:
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$$
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\min_{P}\Big\{ \big(g_t'(x, P) - ( -\gamma g_t(x, P) \big)^2 \Big\}
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$$
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<p>
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(the notation \( \min_{P}\{ f(x, P) \} \) means that we desire to find \( P \) that yields the minimum of \( f(x, P) \))
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<p>
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or, in terms of weights and biases for the hidden and output layer in our network:
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$$
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\min_{P_{\text{hidden} }, \ P_{\text{output} }}\Big\{ \big(g_t'(x, \{ P_{\text{hidden} }, P_{\text{output} }\}) - ( -\gamma g_t(x, \{ P_{\text{hidden} }, P_{\text{output} }\}) \big)^2 \Big\}
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$$
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<p>
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for an input value \( x \).
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<p>
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If the neural network evaluates \( g_t(x, P) \) at more values for \( x \), say \( N \) values \( x_i \) for \( i = 1, \dots, N \), then the <em>total</em> error to minimize becomes
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$$
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\begin{equation} \tag{9}
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\min_{P}\Big\{\frac{1}{N} \sum_{i=1}^N \big(g_t'(x_i, P) - ( -\gamma g_t(x_i, P) \big)^2 \Big\}
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\end{equation}
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$$
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<p>
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Letting \( \vec x \) be a vector with elements \( x_i \) and \( c(\vec x, P) = \frac{1}{N} \sum_i \big(g_t'(x_i, P) - ( -\gamma g_t(x_i, P) \big)^2 \) denote the cost function, the minimization problem that our network must solve, becomes
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$$
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\min_{P} c(\vec x, P)
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$$
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<p>
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In terms of \( P_{\text{hidden} } \) and \( P_{\text{output} } \), this could also be expressed as
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$$
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\min_{P_{\text{hidden} }, \ P_{\text{output} }} c(\vec x, \{P_{\text{hidden} }, P_{\text{output} }\})
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$$
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<p>
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<p>
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