HRR-PINN: A deep learning method for solving complex data-driven solutions

Physics-informed neural networks (PINNs) have emerged as a powerful tool for the data-driven solution of the partial differential equations. However, when solving complex solutions which have sharp gradient region or waveform mutation region, the traditional PINNs method often have significantly higher prediction errors in those critical regions than in other areas due to 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 adopts a two-stage paradigm. The pre-training focuses on global modeling to obtain the residual of network training, which is helpful to achieve more precise sampling optimization. The secondary training concentrates computational resources on critical regions to specialize in optimizing high-residual regions based on the global results of pre-training, 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 apply it to solve single-periodic solutions, rogue wave solutions on single-periodic backgrounds, and double-periodic solutions of the second-type derivative nonlinear Schrödinger equation. The 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.