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

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 temperatures, 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 Fokker-Planck equations with low temperature through our numerical experiments.