A STAGE-ADAPTIVE RESAMPLING PHYSICS-INFORMED NEURAL NETWORK

In recent years, physics-informed neural networks (PINNs) have attracted considerable attention as a novel approach for solving partial differential equations (PDEs). Although PINNs offer numerous advantages over traditional numerical methods, effectively ensuring model convergence and accuracy remains a core challenge that demands urgent resolution. To address this, this paper proposes a stage-adaptive resampling physics-informed neural network (stage-Adaptive resampling physics-informed neural networks, STAR-PINNs) for solving evolutionary equations. The method first discretizes the solution time domain into multiple consecutive stages. Within each stage, a sampling probability density function is constructed based on the loss values of the current residual points, and a subset of new sample points is resampled according to this function to replace the original residual points. This resampling and update process is performed repeatedly at fixed training intervals. By incorporating this adaptive resampling strategy, the spatial distribution of residual points is dynamically adjusted, enabling the sample points to adaptively focus on the stiff regions of the equation solution and thereby substantially accelerating the network convergence process. Recognizing that the prediction accuracy of early stages directly impacts the solution results of subsequent stages, STAR-PINNs introduces a causality weighting algorithm and proposes a novel adaptive update strategy for the causality strength coefficient, which enables dynamic adjustment of the weighting intensity during training. This design effectively suppresses the accumulation effect of errors evolving over time, significantly enhancing the stability and accuracy of long-term predictions. To validate its effectiveness, this paper adopts the Allen-Cahn equation—a challenging case for PINNs—as a test case for solution and further compares it with causal training. The results demonstrate that STAR-PINNs significantly reduces training costs while improving accuracy by approximately one order of magnitude, achieving a minimum relative L₂ error of 3.11 × 10⁻⁵. Further solutions to the reaction equation, reaction-diffusion equation, and wave equation show that the predicted solutions of STAR-PINNs are highly consistent with the reference solutions of the equations.