  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.
