Multiprecision computing for multistage fractional physics-informed neural networks

TL;DR

This paper tackles the accuracy bottleneck of fractional physics-informed neural networks (fPINNs) for subdiffusion problems governed by Caputo derivatives. It introduces a multistage, multi-scale fractional PINN framework that cascades neural correction stages and employs MscaleDNN-2 to capture multi-frequency residuals arising from nonlocal fractional operators. By deriving amplitude–frequency parameter relations from residual spectra and using equal data/physics loss weights, the method achieves relative $L^2$ errors as small as $10^{-7}$–$10^{-8}$ on uniform and nonuniform grids with far fewer collocation points than traditional fPINNs. The results demonstrate substantial accuracy gains, data efficiency, and robustness across discretization schemes, signaling a practical pathway to high-precision FPDE solutions in scientific computing.

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.

Figures (8)

• Figure 1: PINN architecture, see Fig. 2 in cuomo2022scientific
• Figure 1: Architectures of MscaleDNN in liu2020multi
• Figure 1: Loss and L2 relative error (Stage 1)
• Figure 2: Schematic architecture of fPINNs in pang2019fpinns
• Figure 3: Equation residual (Stage 1)