Artificial Neural Network Solver for Time-Dependent Fokker-Planck Equations

TL;DR

This paper develops a neural network-based method to solve time-dependent Fokker-Planck equations, extending previous stationary solutions to dynamic cases with novel training algorithms and sampling strategies, demonstrated through 1D and 2D examples.

Contribution

It introduces a new neural network approach for time-dependent Fokker-Planck equations, including multi-scale loss training and a novel collocation sampling method.

Findings

Effective neural network solutions for 1D and 2D time-dependent Fokker-Planck equations

Improved training algorithms for multi-scale loss functions

Proposed new sampling strategy for collocation points

Abstract

Stochastic differential equations play an important role in various applications when modeling systems that have either random perturbations or chaotic dynamics at faster time scales. The time evolution of the probability distribution of a stochastic differential equation is described by the Fokker-Planck equation, which is a second order parabolic partial differential equation. Previous work combined artificial neural network and Monte Carlo data to solve stationary Fokker-Planck equations. This paper extends this approach to time dependent Fokker-Planck equations. The focus is on the investigation of algorithms for training a neural network that has multi-scale loss functions. Additionally, a new approach for collocation point sampling is proposed. A few 1D and 2D numerical examples are demonstrated.