Multiprecision computing for multistage fractional physics-informed neural networks

Fractional physics-informed neural networks (fPINNs) have been successfully introduced in Pang et al. [SIAM J. Sci. Comput. 41, A2603-A2626 (2019)], which observe relative errors of 10-3∼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 [Y. Wang and C.-Y. Lai, J. Comput. Phys. 504, 112865 (2024)]. Numerical experiments show that the relative errors improve to 10-7∼10-8 for the subdiffusion equations on uniform or nonuniform meshes.