A Fractional-Memory Physics-Informed Neural Network with Fast History Compression for Tempered Fractional Coupled Phase-Field Systems

Abstract: Tempered time-fractional coupled phase-field (tTFCP) systems are used to model interfacial phenomena involving memory-dependent transport and relaxation mechanisms. Numerical solutions to these systems are challenging due to the simultaneous presence of nonlocal temporal operators, weak initial singularities, moving diffuse interfaces, and strongly coupled multiphysics dynamics. In this work, we introduce FM-tfPINN (fractional-memory physics-informed neural network), which is used for forward simulation and inverse parameter identification in tempered fractional coupled phase-field systems. Unlike conventional fractional PINNs, which enforce memory effects solely through residual constraints, our framework incorporates tempered fractional memory directly into the neural representation via latent memory-source functions and a tempered fractional integral operator. We develop a fast shifted residual formulation based on graded temporal meshes and sum-of-exponentials (SOE) history compression to efficiently evaluate the tempered fractional operators. This framework combines interface-aware and residual-adaptive collocation strategies, improving resolution near evolving diffuse interfaces. A unified, physics-informed loss formulation allows for the forward prediction and inverse recovery of unknown physical parameters from sparse observations. We assess the proposed method on a class of tempered fractional corrosion phase-field models, including one-dimensional corrosion-front propagation, activation- and diffusion-controlled regimes, two-dimensional pitting corrosion, and inverse mobility identification problems. The numerical results demonstrate the accurate recovery of coupled phase and concentration fields, the robust prediction of physically relevant interface diagnostics, and the reliable estimation of parameters from limited data.