Input gradient annealing neural network for solving low-temperature Fokker-Planck equations

TL;DR

This paper introduces the IGANN method, a neural network approach that effectively solves high-dimensional, low-temperature Fokker-Planck equations by overcoming numerical difficulties caused by small eigenvalues.

Contribution

The paper proposes the input gradient annealing neural network (IGANN), a novel meshless deep learning method to address low-temperature Fokker-Planck equations with multiple metastable states.

Findings

Successfully solves high-dimensional Fokker-Planck equations.

Overcomes numerical difficulties at low temperatures.

Demonstrates effectiveness through numerical experiments.

Abstract

We present a novel yet simple deep learning approach, called input gradient annealing neural network (IGANN), for solving stationary Fokker-Planck equations. Traditional methods, such as finite difference and finite elements, suffer from the curse of dimensionality. Neural network based algorithms are meshless methods, which can avoid the curse of dimensionality. However, at low temperature, when directly solving a stationary Fokker-Planck equation with more than two metastable states in the generalized potential landscape, the small eigenvalue introduces numerical difficulties due to a large condition number. To overcome these problems, we introduce the IGANN method, which uses a penalty of negative input gradient annealing during the training. We demonstrate that the IGANN method can effectively solve high-dimensional and low-temperature Fokker-Planck equations through our numerical experiments.