HRR-PINN: A Deep Learning Method for Solving Complex Data-Driven Solutions

Physics-informed neural networks (PINNs) have emerged as powerful tools for data-driven solutions of partial differential equations. However, when solving complex solutions with a sharp gradient or waveform mutation region, the traditional PINNs method frequently has significantly higher prediction errors in critical regions than in other areas because of randomly or uniformly distributed sampling points. To overcome the limitations of PINNs in solving complex solutions, we propose a high-residual region resampling PINN (HRR-PINN) method. The HRR-PINN method uses a two-stage paradigm. Pre-training focuses on global modeling to obtain the residual of the network training, which is helpful for achieving a more precise sampling optimization. Secondary training focuses on computational resources of critical regions to specialize in optimizing high-residual regions based on the global results of pretraining, that is, adding new points in high-residual regions and removing low-residual points from the original set. To illustrate the effectiveness of the HRR-PINN method, we applied it to single-periodic solutions, rogue wave solutions on single-periodic backgrounds, and double-periodic solutions of the second-type derivative nonlinear Schrödinger equation. Numerical experiments show that the HRR-PINN method significantly optimizes the distribution of sampling points and reduces prediction errors. This confirms the effectiveness for solving complex solutions with abrupt waveform changes or sharp gradients of the HRR-PINN method.