Solving eigenvalue PDEs of metastable diffusion processes using artificial neural networks

TL;DR

This paper introduces a neural network-based numerical method to efficiently solve high-dimensional eigenvalue PDEs related to metastable diffusion processes, aiding understanding of their long-term dynamics.

Contribution

The paper presents a novel neural network algorithm capable of computing multiple eigenpairs of high-dimensional eigenvalue PDEs in a single training process.

Findings

Successfully applied to high-dimensional model problems

Effectively captures multiple eigenpairs simultaneously

Provides insights into metastable process dynamics

Abstract

In this paper, we consider the eigenvalue PDE problem of the infinitesimal generators of metastable diffusion processes. We propose a numerical algorithm based on training artificial neural networks for solving the leading eigenvalues and eigenfunctions of such high-dimensional eigenvalue problem. The algorithm is able to find multiple leading eigenpairs by solving a single training task. It is useful in understanding the dynamical behaviors of metastable processes on large timescales. We demonstrate the capability of our algorithm on a high-dimensional model problem, and on the simple molecular system alanine dipeptide.