This repository contains all files related to my capstone project for the Machine Learning Nanodegree by Udacity. I followed two approaches to construct approximate solutions for Initial-Boundary-Value (IBV) problems:

• Supervised learning: the numerical solution can be seen as set of feature/label pairs. The loss function is defined such that the difference between the MLP evaluated at the discrete mesh locations, and the numerical solution, calculated beforehand, becomes minimal.
• Partial derivatives of the MLP: because the MLP is always a continuous, smooth function, its gradient with respect to the input features can be computed. The loss function is then defined such that the residual of the approximated partial derivatives forming the differential equation becomes minimal.

A more detailed description of the basic ideas can be found in the capstone_report.pdf document. The maximum error of the final model is about one percent.

• pdeUtils.py: implementation of the neural network, custom functions, error norms, and complementary functions
• visuals.py: contains a function to visualize the approximate solution and its deviation from a given reference as surface plots
• fullRawData: contains the training data including boundary points
• internalRawData: contains the training data without boundary points
• capstone_baseline.ipynb: notebook with tests performed on the baseline model
• network_arch_test.ipynb: notebook to test different network architectures
• pureNetwork*, trialFunction*, customFunction*: folders which contain final weights and biases for different Python models
• keras: folder which contains the weights and biases of all keras models
• the initial capstone proposal and the final report

• Keras with Tensorflow backend: For the standard supervised learning based on feature-label pairs I used simple Keras models.
• Autograd: For the networks I implemented from scratch, Autograd is used to compute higher order derivative with respect to the inputs, and also to compute the derivative of the loss function with respect to the network weights.

MIT License

Copyright (c) 2018 Andre Weiner

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