关键词: partial differential equation, physics-informed neural networks, residual-based adaptive sampling, temporal causality

Abstract: Physics-informed neural networks (PINNs) have become an attractive machine learning framework for obtaining solutions to partial differential equations (PDEs). PINNs embed initial, boundary, and PDE constraints into the loss function. The performance of PINNs is generally affected by both training and sampling. Specifically, training methods focus on how to overcome the training difficulties caused by the special PDE residual loss of PINNs, and sampling methods are concerned with the location and distribution of the sampling points upon which evaluations of PDE residual loss are accomplished. However, a common problem among these original PINNs is that they omit special temporal information utilization during the training or sampling stages when dealing with an important PDE category, namely, time-dependent PDEs, where temporal information plays a key role in the algorithms used. There is one method, called Causal PINN, that considers temporal causality at the training level but not special temporal utilization at the sampling level. Incorporating temporal knowledge into sampling remains to be studied. To fill this gap, we propose a novel temporal causality-based adaptive sampling method that dynamically determines the sampling ratio according to both PDE residual and temporal causality. By designing a sampling ratio determined by both residual loss and temporal causality to control the number and location of sampled points in each temporal sub-domain, we provide a practical solution by incorporating temporal information into sampling. Numerical experiments of several nonlinear time-dependent PDEs, including the Cahn-Hilliard, Korteweg-de Vries, Allen-Cahn and wave equations, show that our proposed sampling method can improve the performance. We demonstrate that using such a relatively simple sampling method can improve prediction performance by up to two orders of magnitude compared with the results from other methods, especially when points are limited.

Key words: partial differential equation, physics-informed neural networks, residual-based adaptive sampling, temporal causality