Morten Hjorth-Jensen
469 lines
30 KiB
HTML
469 lines
30 KiB
HTML
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{'highest level': 2,
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('Using Automatic differentiation',
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2,
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('Back propagation and automatic differentiation',
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('The function to solve for',
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('The trial solution', 2, None, 'the-trial-solution'),
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('Reformulating the problem',
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('Back propagation', 2, None, 'back-propagation'),
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('Gradient descent', 2, None, 'gradient-descent'),
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('The code for solving the ODE',
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('The network with one input layer, specified number of hidden '
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'the-network-with-one-input-layer-specified-number-of-hidden-layers-and-one-output-layer'),
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('Example: Population growth',
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('Setting up the problem', 2, None, 'setting-up-the-problem'),
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('The trial solution', 2, None, 'the-trial-solution'),
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('The program using Autograd',
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('Using forward Euler to solve the ODE',
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2,
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('Example: Solving the one dimensional Poisson equation',
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('The specific equation to solve for',
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('Solving the equation using Autograd',
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('Comparing with a numerical scheme',
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('Setting up the code', 2, None, 'setting-up-the-code'),
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('Partial Differential Equations',
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2,
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'partial-differential-equations'),
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('Type of problem', 2, None, 'type-of-problem'),
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('Network requirements', 2, None, 'network-requirements'),
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('More details', 2, None, 'more-details'),
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('Example: The diffusion equation',
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2,
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'example-the-diffusion-equation'),
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('Defining the problem', 2, None, 'defining-the-problem'),
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('Setting up the network using Autograd',
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('Setting up the network using Autograd; The trial solution',
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('Why the jacobian?', 2, None, 'why-the-jacobian'),
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('Setting up the network using Autograd; The full program',
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2,
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('Example: Solving the wave equation with Neural Networks',
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('The problem to solve for', 2, None, 'the-problem-to-solve-for'),
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('The trial solution', 2, None, 'the-trial-solution'),
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('The analytical solution', 2, None, 'the-analytical-solution'),
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('Solving the wave equation - the full program using Autograd',
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2,
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('Resources on differential equations and deep learning',
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('Convolutional Neural Networks (recognizing images)',
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('Neural Networks vs CNNs', 2, None, 'neural-networks-vs-cnns'),
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('Why CNNS for images, sound files, medical images from CT scans '
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2,
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('3D volumes of neurons', 2, None, '3d-volumes-of-neurons'),
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('Layers used to build CNNs',
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('Transforming images', 2, None, 'transforming-images'),
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('CNNs in brief', 2, None, 'cnns-in-brief'),
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('Key Idea', 2, None, 'key-idea'),
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('Mathematics of CNNs', 2, None, 'mathematics-of-cnns'),
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('Convolution Examples: Polynomial multiplication',
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2,
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'convolution-examples-polynomial-multiplication'),
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('Efficient Polynomial Multiplication',
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'efficient-polynomial-multiplication'),
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('A more efficient way of coding the above Convolution',
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2,
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None,
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'a-more-efficient-way-of-coding-the-above-convolution'),
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('Convolution Examples: Principle of Superposition and Periodic '
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'Forces (Fourier Transforms)',
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2,
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None,
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'convolution-examples-principle-of-superposition-and-periodic-forces-fourier-transforms'),
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('Principle of Superposition',
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2,
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'principle-of-superposition'),
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('Simple Code Example', 2, None, 'simple-code-example'),
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('Wrapping up Fourier transforms',
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2,
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None,
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'wrapping-up-fourier-transforms'),
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('Finding the Coefficients', 2, None, 'finding-the-coefficients'),
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('Final words on Fourier Transforms',
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2,
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None,
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'final-words-on-fourier-transforms'),
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('Two-dimensional Objects', 2, None, 'two-dimensional-objects'),
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('Cross-Correlation', 2, None, 'cross-correlation'),
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('More on Dimensionalities', 2, None, 'more-on-dimensionalities'),
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('Further Dimensionality Remarks',
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2,
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None,
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'further-dimensionality-remarks'),
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('CNNs in more detail, Lecture from IN5400',
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2,
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None,
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'cnns-in-more-detail-lecture-from-in5400'),
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('CNNs in more detail, building convolutional neural networks in '
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'Tensorflow and Keras',
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2,
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None,
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'cnns-in-more-detail-building-convolutional-neural-networks-in-tensorflow-and-keras'),
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('Setting it up', 2, None, 'setting-it-up'),
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('The MNIST dataset again', 2, None, 'the-mnist-dataset-again'),
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('Strong correlations', 2, None, 'strong-correlations'),
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('Layers of a CNN', 2, None, 'layers-of-a-cnn'),
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('Systematic reduction', 2, None, 'systematic-reduction'),
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('Prerequisites: Collect and pre-process data',
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2,
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None,
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'prerequisites-collect-and-pre-process-data'),
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('Importing Keras and Tensorflow',
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2,
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None,
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'importing-keras-and-tensorflow'),
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('Running with Keras', 2, None, 'running-with-keras'),
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('Final part', 2, None, 'final-part'),
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('Final visualization', 2, None, 'final-visualization'),
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('The CIFAR01 data set', 2, None, 'the-cifar01-data-set'),
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('Verifying the data set', 2, None, 'verifying-the-data-set'),
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('Set up the model', 2, None, 'set-up-the-model'),
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('Add Dense layers on top', 2, None, 'add-dense-layers-on-top'),
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('Compile and train the model',
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2,
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None,
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'compile-and-train-the-model'),
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('Finally, evaluate the model',
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2,
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<a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
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<ul class="dropdown-menu">
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<!-- navigation toc: --> <li><a href="._week42-bs001.html#plan-for-week-42" style="font-size: 80%;">Plan for week 42</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs002.html#using-automatic-differentiation" style="font-size: 80%;">Using Automatic differentiation</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs003.html#back-propagation-and-automatic-differentiation" style="font-size: 80%;">Back propagation and automatic differentiation</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs004.html#solving-odes-with-deep-learning" style="font-size: 80%;">Solving ODEs with Deep Learning</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs005.html#ordinary-differential-equations" style="font-size: 80%;">Ordinary Differential Equations</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs048.html#the-trial-solution" style="font-size: 80%;">The trial solution</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs007.html#minimization-process" style="font-size: 80%;">Minimization process</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs008.html#minimizing-the-cost-function-using-gradient-descent-and-automatic-differentiation" 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="._week42-bs009.html#example-exponential-decay" style="font-size: 80%;">Example: Exponential decay</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs010.html#the-function-to-solve-for" style="font-size: 80%;">The function to solve for</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs048.html#the-trial-solution" style="font-size: 80%;">The trial solution</a></li>
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||
<!-- navigation toc: --> <li><a href="._week42-bs012.html#setup-of-network" style="font-size: 80%;">Setup of Network</a></li>
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||
<!-- navigation toc: --> <li><a href="._week42-bs013.html#reformulating-the-problem" style="font-size: 80%;">Reformulating the problem</a></li>
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||
<!-- navigation toc: --> <li><a href="._week42-bs014.html#more-technicalities" style="font-size: 80%;">More technicalities</a></li>
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||
<!-- navigation toc: --> <li><a href="._week42-bs039.html#more-details" style="font-size: 80%;">More details</a></li>
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||
<!-- navigation toc: --> <li><a href="._week42-bs016.html#a-possible-implementation-of-a-neural-network" style="font-size: 80%;">A possible implementation of a neural network</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs017.html#technicalities" style="font-size: 80%;">Technicalities</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs018.html#final-technicalities-i" style="font-size: 80%;">Final technicalities I</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs019.html#final-technicalities-ii" style="font-size: 80%;">Final technicalities II</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs020.html#final-technicalities-iii" style="font-size: 80%;">Final technicalities III</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs021.html#final-technicalities-iv" style="font-size: 80%;">Final technicalities IV</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs022.html#back-propagation" style="font-size: 80%;">Back propagation</a></li>
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||
<!-- navigation toc: --> <li><a href="._week42-bs023.html#gradient-descent" style="font-size: 80%;">Gradient descent</a></li>
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||
<!-- navigation toc: --> <li><a href="._week42-bs024.html#the-code-for-solving-the-ode" style="font-size: 80%;">The code for solving the ODE</a></li>
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||
<!-- navigation toc: --> <li><a href="._week42-bs025.html#the-network-with-one-input-layer-specified-number-of-hidden-layers-and-one-output-layer" style="font-size: 80%;">The network with one input layer, specified number of hidden layers, and one output layer</a></li>
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||
<!-- navigation toc: --> <li><a href="._week42-bs026.html#example-population-growth" style="font-size: 80%;">Example: Population growth</a></li>
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||
<!-- navigation toc: --> <li><a href="._week42-bs027.html#setting-up-the-problem" style="font-size: 80%;">Setting up the problem</a></li>
|
||
<!-- navigation toc: --> <li><a href="._week42-bs048.html#the-trial-solution" style="font-size: 80%;">The trial solution</a></li>
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||
<!-- navigation toc: --> <li><a href="._week42-bs029.html#the-program-using-autograd" style="font-size: 80%;">The program using Autograd</a></li>
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||
<!-- navigation toc: --> <li><a href="._week42-bs030.html#using-forward-euler-to-solve-the-ode" style="font-size: 80%;">Using forward Euler to solve the ODE</a></li>
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||
<!-- navigation toc: --> <li><a href="._week42-bs031.html#example-solving-the-one-dimensional-poisson-equation" style="font-size: 80%;">Example: Solving the one dimensional Poisson equation</a></li>
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||
<!-- navigation toc: --> <li><a href="._week42-bs032.html#the-specific-equation-to-solve-for" style="font-size: 80%;">The specific equation to solve for</a></li>
|
||
<!-- navigation toc: --> <li><a href="._week42-bs033.html#solving-the-equation-using-autograd" style="font-size: 80%;">Solving the equation using Autograd</a></li>
|
||
<!-- navigation toc: --> <li><a href="._week42-bs034.html#comparing-with-a-numerical-scheme" style="font-size: 80%;">Comparing with a numerical scheme</a></li>
|
||
<!-- navigation toc: --> <li><a href="._week42-bs035.html#setting-up-the-code" style="font-size: 80%;">Setting up the code</a></li>
|
||
<!-- navigation toc: --> <li><a href="._week42-bs036.html#partial-differential-equations" style="font-size: 80%;">Partial Differential Equations</a></li>
|
||
<!-- navigation toc: --> <li><a href="._week42-bs037.html#type-of-problem" style="font-size: 80%;">Type of problem</a></li>
|
||
<!-- navigation toc: --> <li><a href="._week42-bs038.html#network-requirements" style="font-size: 80%;">Network requirements</a></li>
|
||
<!-- navigation toc: --> <li><a href="._week42-bs039.html#more-details" style="font-size: 80%;">More details</a></li>
|
||
<!-- navigation toc: --> <li><a href="._week42-bs040.html#example-the-diffusion-equation" style="font-size: 80%;">Example: The diffusion equation</a></li>
|
||
<!-- navigation toc: --> <li><a href="._week42-bs041.html#defining-the-problem" style="font-size: 80%;">Defining the problem</a></li>
|
||
<!-- navigation toc: --> <li><a href="._week42-bs042.html#setting-up-the-network-using-autograd" style="font-size: 80%;">Setting up the network using Autograd</a></li>
|
||
<!-- navigation toc: --> <li><a href="._week42-bs043.html#setting-up-the-network-using-autograd-the-trial-solution" style="font-size: 80%;">Setting up the network using Autograd; The trial solution</a></li>
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||
<!-- navigation toc: --> <li><a href="._week42-bs044.html#why-the-jacobian" style="font-size: 80%;">Why the jacobian?</a></li>
|
||
<!-- navigation toc: --> <li><a href="._week42-bs045.html#setting-up-the-network-using-autograd-the-full-program" style="font-size: 80%;">Setting up the network using Autograd; The full program</a></li>
|
||
<!-- navigation toc: --> <li><a href="._week42-bs046.html#example-solving-the-wave-equation-with-neural-networks" style="font-size: 80%;">Example: Solving the wave equation with Neural Networks</a></li>
|
||
<!-- navigation toc: --> <li><a href="._week42-bs047.html#the-problem-to-solve-for" style="font-size: 80%;">The problem to solve for</a></li>
|
||
<!-- navigation toc: --> <li><a href="._week42-bs048.html#the-trial-solution" style="font-size: 80%;">The trial solution</a></li>
|
||
<!-- navigation toc: --> <li><a href="._week42-bs049.html#the-analytical-solution" style="font-size: 80%;">The analytical solution</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs050.html#solving-the-wave-equation-the-full-program-using-autograd" style="font-size: 80%;">Solving the wave equation - the full program using Autograd</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs051.html#resources-on-differential-equations-and-deep-learning" style="font-size: 80%;">Resources on differential equations and deep learning</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs052.html#convolutional-neural-networks-recognizing-images" style="font-size: 80%;">Convolutional Neural Networks (recognizing images)</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs053.html#what-is-the-difference" style="font-size: 80%;">What is the Difference</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs054.html#neural-networks-vs-cnns" style="font-size: 80%;">Neural Networks vs CNNs</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs055.html#why-cnns-for-images-sound-files-medical-images-from-ct-scans-etc" style="font-size: 80%;">Why CNNS for images, sound files, medical images from CT scans etc?</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs056.html#regular-nns-don-t-scale-well-to-full-images" style="font-size: 80%;">Regular NNs don’t scale well to full images</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs057.html#3d-volumes-of-neurons" style="font-size: 80%;">3D volumes of neurons</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs058.html#layers-used-to-build-cnns" style="font-size: 80%;">Layers used to build CNNs</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs059.html#transforming-images" style="font-size: 80%;">Transforming images</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs060.html#cnns-in-brief" style="font-size: 80%;">CNNs in brief</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs061.html#key-idea" style="font-size: 80%;">Key Idea</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs062.html#mathematics-of-cnns" style="font-size: 80%;">Mathematics of CNNs</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs063.html#convolution-examples-polynomial-multiplication" style="font-size: 80%;">Convolution Examples: Polynomial multiplication</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs064.html#efficient-polynomial-multiplication" style="font-size: 80%;">Efficient Polynomial Multiplication</a></li>
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||
<!-- navigation toc: --> <li><a href="._week42-bs065.html#a-more-efficient-way-of-coding-the-above-convolution" style="font-size: 80%;">A more efficient way of coding the above Convolution</a></li>
|
||
<!-- navigation toc: --> <li><a href="._week42-bs066.html#convolution-examples-principle-of-superposition-and-periodic-forces-fourier-transforms" style="font-size: 80%;">Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs067.html#principle-of-superposition" style="font-size: 80%;">Principle of Superposition</a></li>
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||
<!-- navigation toc: --> <li><a href="._week42-bs068.html#simple-code-example" style="font-size: 80%;">Simple Code Example</a></li>
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||
<!-- navigation toc: --> <li><a href="._week42-bs069.html#wrapping-up-fourier-transforms" style="font-size: 80%;">Wrapping up Fourier transforms</a></li>
|
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<!-- navigation toc: --> <li><a href="._week42-bs070.html#finding-the-coefficients" style="font-size: 80%;">Finding the Coefficients</a></li>
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||
<!-- navigation toc: --> <li><a href="._week42-bs071.html#final-words-on-fourier-transforms" style="font-size: 80%;">Final words on Fourier Transforms</a></li>
|
||
<!-- navigation toc: --> <li><a href="._week42-bs072.html#two-dimensional-objects" style="font-size: 80%;">Two-dimensional Objects</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs073.html#cross-correlation" style="font-size: 80%;">Cross-Correlation</a></li>
|
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<!-- navigation toc: --> <li><a href="._week42-bs074.html#more-on-dimensionalities" style="font-size: 80%;">More on Dimensionalities</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs075.html#further-dimensionality-remarks" style="font-size: 80%;">Further Dimensionality Remarks</a></li>
|
||
<!-- navigation toc: --> <li><a href="._week42-bs076.html#cnns-in-more-detail-lecture-from-in5400" style="font-size: 80%;">CNNs in more detail, Lecture from IN5400</a></li>
|
||
<!-- navigation toc: --> <li><a href="._week42-bs077.html#cnns-in-more-detail-building-convolutional-neural-networks-in-tensorflow-and-keras" style="font-size: 80%;">CNNs in more detail, building convolutional neural networks in Tensorflow and Keras</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs078.html#setting-it-up" style="font-size: 80%;">Setting it up</a></li>
|
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<!-- navigation toc: --> <li><a href="._week42-bs079.html#the-mnist-dataset-again" style="font-size: 80%;">The MNIST dataset again</a></li>
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<!-- navigation toc: --> <li><a href="._week42-bs080.html#strong-correlations" style="font-size: 80%;">Strong correlations</a></li>
|
||
<!-- navigation toc: --> <li><a href="._week42-bs081.html#layers-of-a-cnn" style="font-size: 80%;">Layers of a CNN</a></li>
|
||
<!-- navigation toc: --> <li><a href="._week42-bs082.html#systematic-reduction" style="font-size: 80%;">Systematic reduction</a></li>
|
||
<!-- navigation toc: --> <li><a href="._week42-bs083.html#prerequisites-collect-and-pre-process-data" style="font-size: 80%;">Prerequisites: Collect and pre-process data</a></li>
|
||
<!-- navigation toc: --> <li><a href="._week42-bs084.html#importing-keras-and-tensorflow" style="font-size: 80%;">Importing Keras and Tensorflow</a></li>
|
||
<!-- navigation toc: --> <li><a href="._week42-bs085.html#running-with-keras" style="font-size: 80%;">Running with Keras</a></li>
|
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<!-- navigation toc: --> <li><a href="._week42-bs086.html#final-part" style="font-size: 80%;">Final part</a></li>
|
||
<!-- navigation toc: --> <li><a href="._week42-bs087.html#final-visualization" style="font-size: 80%;">Final visualization</a></li>
|
||
<!-- navigation toc: --> <li><a href="._week42-bs088.html#the-cifar01-data-set" style="font-size: 80%;">The CIFAR01 data set</a></li>
|
||
<!-- navigation toc: --> <li><a href="._week42-bs089.html#verifying-the-data-set" style="font-size: 80%;">Verifying the data set</a></li>
|
||
<!-- navigation toc: --> <li><a href="._week42-bs090.html#set-up-the-model" style="font-size: 80%;">Set up the model</a></li>
|
||
<!-- navigation toc: --> <li><a href="._week42-bs091.html#add-dense-layers-on-top" style="font-size: 80%;">Add Dense layers on top</a></li>
|
||
<!-- navigation toc: --> <li><a href="._week42-bs092.html#compile-and-train-the-model" style="font-size: 80%;">Compile and train the model</a></li>
|
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<!-- navigation toc: --> <li><a href="._week42-bs093.html#finally-evaluate-the-model" style="font-size: 80%;">Finally, evaluate the model</a></li>
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|
||
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<h1>Week 42 Solving differential equations and Convolutional (CNN)</h1>
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<!-- author(s): Morten Hjorth-Jensen -->
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<b>Morten Hjorth-Jensen</b> [1, 2]
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[1] <b>Department of Physics, University of Oslo</b>
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[2] <b>Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University</b>
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<br>
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<h4>Oct 21, 2022</h4>
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