Solving Non-local Fokker-Planck Equations by Deep Learning

TL;DR

This paper introduces trapz-PiNNs, a physics-informed neural network method enhanced with a modified trapezoidal rule, to accurately solve space-fractional Fokker-Planck equations in multiple dimensions, demonstrating high accuracy and potential for generalization.

Contribution

The paper develops a novel trapz-PiNN approach incorporating a modified trapezoidal rule for fractional Laplacian evaluation, enabling accurate, mesh-independent solutions of space-fractional PDEs in higher dimensions.

Findings

High accuracy with low $\\mathcal{L}^2$ error in numerical examples

Effective local error analysis and improvement methods

Ability to solve PDEs with fractional Laplacian for any $\\alpha \in (0,2)$

Abstract

Physics-informed neural networks (PiNNs) recently emerged as a powerful solver for a large class of partial differential equations under various initial and boundary conditions. In this paper, we propose trapz-PiNNs, physics-informed neural networks incorporated with a modified trapezoidal rule recently developed for accurately evaluating fractional laplacian and solve the space-fractional Fokker-Planck equations in 2D and 3D. We describe the modified trapezoidal rule in detail and verify the second-order accuracy. We demonstrate trapz-PiNNs have high expressive power through predicting solution with low $\mathcal{L}^2$ relative error on a variety of numerical examples. We also use local metrics such as pointwise absolute and relative errors to analyze where could be further improved. We present an effective method for improving performance of trapz-PiNN on local metrics, provided that physical observations of high-fidelity simulation of the true solution are available. Besides the usual advantages of the deep learning solvers such as adaptivity and mesh-independence, the trapz-PiNN is able to solve PDEs with fractional laplacian with arbitrary $\alpha\in (0,2)$ and specializes on rectangular domain. It also has potential to be generalized into higher dimensions.