Solving the non-local Fokker–Planck equations by deep learning

• Published in: Chaos (Woodbury, N.Y.) N.Y.), 2023-04, Vol.33 (4)
• Main Authors: Jiang, Senbao, Li, Xiaofan
• Format: Article
• Language: English
• Subjects: Boundary conditions; Domains; Fokker-Planck equation; Mathematical analysis; Neural networks; Partial differential equations
• ISSN: 1054-1500; 1089-7682
• DOI: 10.1063/5.0128935

Physics-informed neural networks (PiNNs) recently emerged as a powerful solver for a large class of partial differential equations (PDEs) 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 that trapz-PiNNs have high expressive power through predicting the solution with low L 2 relative error by a variety of numerical examples. We also use local metrics, such as point-wise absolute and relative errors, to analyze where it could be further improved. We present an effective method for improving the performance of trapz-PiNN on local metrics, provided that physical observations or high-fidelity simulation of the true solution are available. The trapz-PiNN is able to solve PDEs with fractional Laplacian with arbitrary α ∈ ( 0 , 2 ) and on rectangular domains. It also has the potential to be generalized into higher dimensions or other bounded domains.