Fast tempered fractional physics-informed neural network for nonlinear tempered time-fractional subdiffusion equations: Forward and inverse problems

We develop FT-fPINNs, a fast physics-informed learning framework for nonlinear tempered time-fractional subdiffusion equations (tTFSEs) that couples a tempered fast L 2-1 θ discretization of the tempered Caputo derivative with a neural surrogate trained in an end-to-end manner. The time-fractional operator is enforced through its time-discrete fast representation embedded directly into the physics-informed residual, which reduces the memory and computational cost associated with long-history evaluation while retaining the higher temporal accuracy of the L 2-1 θ scheme. To enhance optimization efficiency, we employ a stagewise training strategy that progressively expands the effective time horizon, thereby stabilizing the learning dynamics and concentrating updates on informative residual structures. The resulting approach is mesh-free in space, uses automatic differentiation for spatial derivatives, and is applicable to forward simulation and inverse parameter identification in tempered fractional-order dynamics. We validate FT-fPINNs on one-dimensional and two-dimensional benchmark problems for nonlinear tTFSEs in both forward and inverse settings. The numerical results demonstrate accurate reconstruction of the solution field and reliable recovery of the unknown parameter set from terminal observations. Robustness is further examined by contaminating only terminal measurements with additive Gaussian noise at representative levels, under which reconstruction errors remain controlled and the inferred parameters remain close to their reference values. These results indicate that FT-fPINNs provides an effective and computationally efficient physics-informed learning framework for nonlinear tTFSEs with tempered memory.