Adaptive deep density approximation for Fokker-Planck equations

TL;DR

This paper introduces an adaptive deep density approximation method using KRnet for efficiently solving high-dimensional steady-state Fokker-Planck equations on unbounded domains, overcoming limitations of traditional grid-based methods.

Contribution

The paper proposes ADDA-KR, a novel adaptive deep density approximation framework utilizing KRnet for high-dimensional Fokker-Planck equations, with an adaptive sampling strategy for improved accuracy.

Findings

Validated the accuracy of ADDA-KR through numerical experiments.

Demonstrated the efficiency of the method in high-dimensional settings.

Showed weaker dependence on dimensionality compared to traditional methods.

Abstract

In this paper we present an adaptive deep density approximation strategy based on KRnet (ADDA-KR) for solving the steady-state Fokker-Planck (F-P) equations. F-P equations are usually high-dimensional and defined on an unbounded domain, which limits the application of traditional grid based numerical methods. With the Knothe-Rosenblatt rearrangement, our newly proposed flow-based generative model, called KRnet, provides a family of probability density functions to serve as effective solution candidates for the Fokker-Planck equations, which has a weaker dependence on dimensionality than traditional computational approaches and can efficiently estimate general high-dimensional density functions. To obtain effective stochastic collocation points for the approximation of the F-P equation, we develop an adaptive sampling procedure, where samples are generated iteratively using the approximate density function at each iteration. We present a general framework of ADDA-KR, validate its accuracy and demonstrate its efficiency with numerical experiments.