AdaPW: An adaptive point-weighting method for training physics-informed neural networks

Physics-informed neural networks (PINNs) provide a promising framework for solving partial differential equations (PDEs) through deep neural networks. One main challenge in the training of PINNs is how to quickly and effectively learn the collocation points causing big errors. In this paper, we propose an adaptive point-weighting (AdaPW) method to update the distribution of collocation-point weights not only based on their PDE residual errors but also considering the training status of the current network. In this manner, AdaPW is able to balance the attention on all collocation points so as to effectively improve the training performance. Different from the existing relevant point-weighting methods, the AdaPW method does not contain any trainable parameter, and thus has a high applicability. The theoretical analysis and numerical experiments validate the effectiveness and the superiority of the AdaPW method.