Finite difference/deep learning scheme for tempered fractional generalized Burgers’ equations with fast evaluation of Caputo derivative

Considering the initial singularity in the solution, we develop an accelerated tempered Alikhanov formula with a tempering parameter λ on nonuniform time grids. We propose numerical schemes based on nonuniform tempered Alikhanov formula leveraging the sum-of-exponents (SOE) based algorithm to efficiently approximate the Caputo tempered fractional derivative for the generalized Burgers’ problem. Spatial derivatives are approximated with standard second-order central differences, ensuring overall second-order accuracy in space. By exploiting the convolutional structure of the consistency error, we establish refined maximum-norm error estimates that faithfully reflect and rigorously account for the solution´s temporal regularity. The resulting finite-difference solver achieves a computational cost of O(MKtlogKt) and a memory requirement of O(MlogKt), where M and Kt denote the spatial and temporal grid sizes. Finally, we embed this accelerated tempered Alikhanov scheme into an adaptive extended-fractional physics-informed neural network (XfPINN) on non-uniform meshes, leveraging adaptive activation functions to accelerate training convergence. Numerical results confirm the accuracy of analysis and the efficiency of the proposed algorithm.