A STAGE-ADAPTIVE RESAMPLING PHYSICS-INFORMED NEURAL NETWORK

In recent years, Physics-Informed Neural Networks (PINNs) have gained significant attention as a novel approach for solving partial differential equations. Despite their advantages over traditional numerical methods, effectively ensuring model convergence and accuracy remains a critical challenge. To address this, we propose Stage-Adaptive Resampling Physics-Informed Neural Networks (STAR-PINNs) for solving evolution equations. This method decomposes the temporal domain into multiple stages and employs an adaptive resampling strategy within each stage to dynamically adjust the distribution of collocation points, concentrating them on stiff regions difficult to fit, thereby accelerating network convergence. Given that prediction accuracy in early stages directly impacts later results, STAR-PINNs incorporates a causality weighting strategy into the stage-wise loss functions, ensuring strict adherence to physical causality during training and reducing error accumulation over time. To validate efficacy, the notoriously challenging Allen-Cahn equation—known for its difficulty in conventional PINN solutions—serves as a test case. Comparative results with temporally causal algorithms demonstrate that STAR-PINNs significantly reduce training costs while improving accuracy by approximately an order of magnitude, achieving a minimal relative L2 error of 3.11×10-5. Further validation on reaction equations, reaction-diffusion equations, and wave equations confirms that STAR-PINNs' predictions maintain high consistency with reference solutions across all cases.