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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-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>
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<!-- 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="._week43-bs049.html#___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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<center><h1>Week 43: Solving Differential Equations with Deep Learning and Dimensionality Reduction methods</h1></center> <!-- document title -->
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<!-- author(s): Morten Hjorth-Jensen -->
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<b>Morten Hjorth-Jensen</b> [1, 2]
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<center>[1] <b>Department of Physics, University of Oslo</b></center>
<center>[2] <b>Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University</b></center>
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<center><h4>Oct 26, 2020</h4></center> <!-- date -->
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