Solving stationary nonlinear Fokker-Planck equations via sampling

TL;DR

This paper introduces a sampling-based method to solve stationary nonlinear Fokker-Planck equations, avoiding long-time simulations and propagation of chaos, with proven convergence and validated through numerical experiments.

Contribution

It proposes a novel sampling approach using Gibbs measures to efficiently solve nonlinear Fokker-Planck equations, bypassing traditional simulation challenges.

Findings

Convergence of Gibbs measure to the stationary solution is established.

Numerical experiments validate the method for Poisson-Boltzmann and neural network applications.

The approach works under bounded interaction kernels and moderate temperatures.

Abstract

Solving the stationary nonlinear Fokker-Planck equations is important in applications and examples include the Poisson-Boltzmann equation and the two layer neural networks. Making use of the connection between the interacting particle systems and the nonlinear Fokker-Planck equations, we propose to solve the stationary solution by sampling from the $N$-body Gibbs distribution. This avoids simulation of the $N$-body system for long time and more importantly such a method can avoid the requirement of uniform propagation of chaos from direct simulation of the particle systems. We establish the convergence of the Gibbs measure to the stationary solution when the interaction kernel is bounded (not necessarily continuous) and the temperature is not very small. Numerical experiments are performed for the Poisson-Boltzmann equations and the two-layer neural networks to validate the method and the theory.