mlhiphy

Machine Learning for Hidden Physics and Partial Differential Equations

Understanding Gaussian Processes

• A simple Gaussian Process
• Using pyGPs
• Using GPy

Using Gaussian processes to estimate parameters in Linear operators

• Using pyGPs (has limitations)
• With custom code for kernels - 1D example
• 2D example

Parameter estimation for the Heat equation

• Using pyGPs (has limitations)
• with the Backward Euler scheme
• Non-homogeneous problem
• Using 2D kernel

Installation of the Python package

The Python package mlhiphy can be installed in the traditional way

• Standard mode::

    python3 -m pip install .

• Development mode::

    python3 -m pip install --user -e .

Automatic computation of Kernels

1. 1d example
2. 2d example
3. 3d example

Version 2!:

In https://github.com/Slowpuncher24/mlhiphy_v2 you can find a considerably improved version of mlhiphy. It features:

• A much more efficient and stable implementation of the negative log-likelihood. This vastly improves the algorithm, as the optimization of the negative log-likelihood is at its center. This was done by utilizing the block matrix structure of the covariance matrix and by using the Cholesky decomposition.
• The inference of up to four hidden parameters in three dimensions, as opposed to mainly one hidden parameter in two dimensions in mlhiphy (respectively counting the temporal dimension as one).
• An alternative implementation of the negative log-likelihood for the noise-free case, where we can optimize over one hyperparameter less (the signal variance can be written in terms of other values).
• The implementation and tests of using the Matérn-5/2-kernel, which is the most promising alternative to the SE kernel.
• The implementation and tests of a viable alternative to the Nelder-Mead optimization algorithm, namely a variant of the nonlinear conjugate gradient method (it is scipy's implementation up to a minor tweak to the line search algorithm).