A Gaussian mixture distribution-based adaptive sampling method for physics-informed neural networks

In this article, we present a novel sampling method to improve the training accuracy of Physics-Informed Neural Networks (PINNs). Inspired by the idea of incremental learning in artificial intelligence, we propose a risk min-max framework to do the adaptive sampling. Within this framework, we develop a simple yet effective strategy known as Gaussian mixture distribution-based adaptive sampling (GAS), which enables us to achieve a lower error in the solution of PINNs with even fewer training epochs and samples. In practical training procedure, GAS uses the current residual information to generate a Gaussian mixture distribution for the sampling of additional points, which can be used to speed up the convergence of the loss and achieve higher accuracy. In our experiments with a two-peak Poisson problem, GAS achieves a mean square error of 1.5E-05, surpassing other sampling methods by two orders of magnitude. Other numerical examples on 2-dimensional and 10-dimensional problems further demonstrate that GAS consistently outperforms several existing adaptive methods in terms of accuracy and efficiency. In general, the proposed method can also be applied to various types of partial differential equations (PDEs), including the elliptic equations, wave equations and Burgers equations.