Fractional physics-informed neural networks for time-fractional phase field models

Abstract

In this paper, a new fractional physics-informed neural networks (fPINNs) is proposed, which combines f-PINNs with spectral collocation method to solve the timefractional phase field models. Compared to fPINNs, it has large representation capacity due to the property of spectral collocation method, which reduces the number of approximate points of discrete fractional operators, improves the training efficiency and has higher error accuracy. Unlike traditional numerical method, it directly optimizes the spectral collocation coefficient, saves the time of matrix calculation, is easy to deal with the high-dimensional model, and also has higher error accuracy. First, fPINNs based on a spectral collocation method is used to obtain the numerical solutions of the models under consideration. The spectral collocation method is used to discretize the space direction, and the fractional backward difference formula is used to approximate the time-fractional derivative. The error accuracy in different cases is discussed, and it is observed that the point-wise error is 10 −5 to 10 −7 . Next, fPINNs is employed to solve several inverse problems in time-fractional phase field models to identify the order of fractional derivative, mobility constant, and other coefficients. The results of numerical experiments are presented to prove the effectiveness of fPINNs in solving time-fractional phase field models and their inverse problems.

Solving inverse problems with sparse noisy data, operator splitting and physics-constraints machine learning

“…Gao et al [36] proposed physicsinformed graph-neural Galerkin networks to solve PDEgoverned forward and inverse problems. For more references about physics-constrained learning, please refer to [37][38][39][40][41][42][43][44][45][46][47][48][49].…”