Physics-Informed Neural Networks with Generalized Residual-Based Adaptive Sampling

Physics-informed neural networks (PINNs) are the powerful tools in solving partial differential equations (PDEs). In general, the performance of PINNs heavily relies on the sampling distribution of residual points. However, existing sampling methods still suffer from the following problems: 1) poor performance due to ignoring the locations with small PDE residuals; and 2) limited generalizability, i.e., the need to manually tune hyperparameters for every specific PDE. To address these issues, we propose a Generalized Residual-based Adaptive Sampling (G-RAS) method for PINNs. G-RAS incorporates a novel probability density function, which can concern locations with small PDE residuals. In addition, the hyperparameter setting is much less than others, i.e., various PDEs only need to set hyperparameter ranges rather than tuning for each one. Experiments on six widely used benchmarks demonstrate that G-RAS can improve prediction accuracy and convergence speed compared to 10 SOTA methods. The supplementary materials and source code are available at https://github.com/songxt3/G-RAS.