Communications in Mathematical Sciences

Volume 20 (2022)

Number 3

An efficient data-driven solver for Fokker–Planck equations: algorithm and analysis

Pages: 803 – 827

DOI: https://dx.doi.org/10.4310/CMS.2022.v20.n3.a8

Authors

Matthew Dobson (Department of Mathematics and Statistics, University of Massachusetts, Amherst, Mass., U.S.A.)

Yao Li (Department of Mathematics and Statistics, University of Massachusetts, Amherst, Mass., U.S.A.)

Jiayu Zhai (Department of Mathematics and Statistics, University of Massachusetts, Amherst, Mass., U.S.A.)

Abstract

Computing the invariant probability measure of a randomly perturbed dynamical system usually means solving the stationary Fokker–Planck equation. This paper studies several key properties of a novel data-driven solver for low-dimensional Fokker–Planck equations proposed in [Y. Li, Commun. Math. Sci., 17(4):1045–1059, 2019]. Based on these results, we propose a new “block solver” for the stationary Fokker–Planck equation, which significantly improves the performance of the original algorithm. Some possible ways of reducing numerical artifacts caused by the block solver are discussed and tested with examples.

Keywords

Fokker–Planck equation, Monte Carlo simulation, data-driven method

2010 Mathematics Subject Classification

37M25, 65C05, 65N99

Received 30 January 2021

Received revised 9 July 2021

Accepted 13 September 2021

Published 21 March 2022