ESR-PINNs: Physics-informed neural networks with expansion-shrinkage resampling selection strategies

Neural network methods have been widely used in many fields of scientific research with the rapid increase of computing power. The physics-informed neural networks (PINNs) have received much attention as a major breakthrough in solving partial differential equations using neural networks. In this paper, a resampling technique based on the expansion-shrinkage point (ESP) selection strategy is developed to dynamically modify the distribution of training points in accordance with the performance of the neural networks. In this new approach both training sites with slight changes in residual values and training points with large residuals are taken into account. In order to make the distribution of training points more uniform, the concept of continuity is further introduced and incorporated. This method successfully addresses the issue that the neural network becomes ill or even crashes due to the extensive alteration of training point distribution. The effectiveness of the improved physics-informed neural networks with expansion-shrinkage resampling is demonstrated through a series of numerical experiments.