Alikhanov-XfPINNs: Adaptive Physics-Informed Learning for Nonlinear Fractional PDEs on Nonuniform Meshes
For researchers solving fractional PDEs, this method offers improved accuracy and efficiency over standard PINNs, but it is an incremental combination of existing techniques.
The paper proposes Alikhanov-XfPINNs, a physics-informed neural network framework that uses high-order Alikhanov temporal discretization on nonuniform meshes to solve nonlinear fractional PDEs, addressing initial singularities and reducing computational cost. Numerical experiments show significant CPU time savings and robustness.
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.