Multiprecision computing for multistage fractional physics-informed neural networks

TL;DR

This paper introduces multistage fractional physics-informed neural networks (fPINNs) with multiprecision computing, significantly enhancing solution accuracy for subdiffusion equations from 10^{-3}–10^{-4} to 10^{-7}–10^{-8}.

Contribution

It develops a multistage fPINNs approach combined with multiprecision computing to achieve high-precision solutions for subdiffusion equations, addressing limitations of previous methods.

Findings

Relative errors reduced to 10^{-7}–10^{-8} with the new method.

Effective on both uniform and non-uniform meshes.

Demonstrates improved accuracy over traditional fPINNs.

Abstract

Fractional physics-informed neural networks (fPINNs) have been successfully introduced in [Pang, Lu and Karniadakis, SIAM J. Sci. Comput. 41 (2019) A2603-A2626], which observe relative errors of $10^{-3} \, \sim \, 10^{-4}$ for the subdiffusion equations. However their high-precision (multiprecision) numerical solution remains challenging, due to the limited regularity of the subdiffusion model caused by the nonlocal operator. To fill in the gap, we present the multistage fPINNs based on traditional multistage PINNs [Wang and Lai, J. Comput. Phys. 504 (2024) 112865]. Numerical experiments show that the relative errors improve to $10^{-7} \, \sim \, 10^{-8}$ for the subdiffusion equations on uniform or nouniform meshes.