Solving Fokker-Planck equation using deep learning

TL;DR

This paper introduces a deep learning-based method to solve the Fokker-Planck equation for stochastic differential equations, avoiding traditional numerical methods and demonstrating effectiveness through various examples.

Contribution

The paper presents a novel deep neural network approach with penalty factors and normalization conditions to solve the Fokker-Planck equation efficiently and accurately.

Findings

Deep learning method effectively solves FP equations in 1D and 2D.

Penalty factors improve convergence and solution quality.

Neural network performance depends on network structure and optimization choices.

Abstract

The probability density function of stochastic differential equations is governed by the Fokker-Planck (FP) equation. A novel machine learning method is developed to solve the general FP equations based on deep neural networks. The proposed algorithm does not require any interpolation and coordinate transformation, which is different from the traditional numercial methods. The main novelty of this paper is that penalty factors are introduced to overcome the local optimization for the deep learning approach, and the corresponding setting rules are given. Meanwhile, we consider a normalization condition as a supervision condition to effectively avoid that the trial solution is zero. Several numerical examples are presented to illustrate performances of the proposed algorithm, including one- and two-dimensional systems. All the results suggest that the deep learning is quite feasible and effective to calculate the FP equation. Further, influences of the number of hidden layers, the penalty factors, and the optimization algorithm are discussed in detail. These results indicate that the performances of the machine learning technique can be improved through constructing the neural networks appropriately.