An Eigenvalue PDE Solver by Neural Networks
This Python package trains artificial neural networks to solve the leading (in ascending order) eigenvalues/eigenfunctions of the eigenvalue PDE problem
, where
is the infiniesimal generator of certain type of diffusion processes whose invariant measure is
.
A trajectory data (possibly biased with biasing weights) is needed as training data. Such data can be generated either
- using a numerical scheme (e.g., Euler-Maruyama scheme), for diffusion processes whose stochastic differential equation (SDE) is given explicitly; or
- using a molecular simulation package (e.g., NAMD), for molecular systems.
The code in this repository was used to produce the numerical results in the paper [1]. See instructions for the two concrete examples below. Based on this repository, a Python package called ColVars-Finder with more detailed documentation was developed for finding collective variables of molecular systems.
-
MDAnalysis, used to load MD data.
-
NAMD, molecular simulation code. It is needed to run the second example below.
-
slepc4py and petsc4py, needed for solving 2D eigenvalue PDE problems using finite volume method. In some examples, the solution given by finite volume method can be used to compare with the neural network solution. These two packages are not needed if one just wants to solve a PDE problem by training neural networks.
git clone https://github.com/zwpku/EigenPDE-NN.git
The directory ./examples contains necessary files of two numerical examples studied in the paper [1]. The first example is A 50-dimensional system, while the second example is the simple MD system alanine dipeptide. Please refer to the README files in ./examples/test-ex1-50d and ./examples/test-ex2 for detailed instructions to run the simulations.
[1] Solving eigenvalue PDEs of metastable diffusion processes using artificial neural networks, W. Zhang, T. Li and Ch. Schütte, 2021, https://arxiv.org/abs/2110.14523