Alikhanov-XfPINNs: Adaptive Physics-Informed Learning for Nonlinear Fractional PDEs on Nonuniform Meshes

TL;DR

This paper introduces Alikhanov-XfPINNs, an adaptive physics-informed neural network framework that efficiently solves nonlinear fractional PDEs on nonuniform meshes, addressing singularities and reducing computational costs.

Contribution

It develops an accelerated Alikhanov discretization within PINNs, incorporating adaptive activation functions and an auxiliary time-marching scheme for improved accuracy and efficiency in solving fractional PDEs.

Findings

Demonstrates robustness on problems with known and unknown solutions.

Achieves significant CPU time savings compared to traditional methods.

Effectively handles initial singularities and nonlocal memory effects.

Abstract

To address the initial singularity inherent in solutions to fractional partial differential equations (fPDEs), we propose an accelerated Alikhanov discretization formulation implemented on nonuniform time grids. Based on the physics-informed neural networks (PINNs) framework, we introduce an Alikhanov-extended fractional PINNs (XfPINNs) architecture that combines high-order temporal discretization and deep learning. The nonlocal memory term in fPDEs leads to high computational cost, while the weak singularity near $t\to 0^+$ can deteriorate accuracy on uniform meshes. To separate temporal discretization effects from optimization and sampling errors, we further develop an auxiliary time-marching configuration that enables auditable temporal-convergence studies under controlled training tolerances. This architecture can solve general nonlinear fPDEs. The XfPINNs approach is designed for forward and inverse problems, allowing for data-driven solution reconstruction and parameter estimation. First, the neural network approximates the solution of nonlinear fPDEs; then, an adaptive activation function accelerates convergence and enhances training efficiency. The optimization framework embeds a variational loss function constructed from the Alikhanov scheme, where the initial and boundary conditions are imposed using a combination of hard and soft constraints. Numerical experiments, including cases with known and unknown exact solutions which demonstrate the robustness, computational efficiency, and significant CPU time savings of the Alikhanov-XfPINNs method.