Solving the Fractional Allen–Cahn Equation and the Fractional Cahn–Hilliard Equation with the Fractional Physics-Informed Neural Networks

In this paper, we present a computational framework based on fractional Physics-Informed Neural Networks combined with the L1 approximation of the Caputo fractional derivative to solve the fractional Allen–Cahn and fractional Cahn–Hilliard equations. Considering the limitation of the original fractional Physics-Informed Neural Networks in achieving high accuracy when applied to these highly stiff fractional equations, we propose three improved optimization strategies: adaptive non-uniform sampling, adaptive exponential moving average ratio loss weighting, and two–stage adaptive quasi optimization. By combining these strategies, three improved fPINNs algorithms are developed: f–A–PINNs, f–A–A–PINNs, and f–A–T–PINNs. Numerical experiments demonstrate that the f–A–T–PINNs algorithm achieves superior computational accuracy and improved parameter stability compared to the other algorithms.