Adaptive trajectories sampling for solving PDEs with deep learning methods

In this paper, we propose a novel adaptive technique, named adaptive trajectories sampling (ATS), to select training points for learning the solution of partial differential equations (PDEs). By the ATS, the training points are selected adaptively according to an empirical-value-type instead of residual-type error indicator from trajectories which are generated by a PDE-related stochastic process. We incorporate the ATS into three known deep learning solvers for PDEs, namely, the adaptive physics-informed neural network method (ATS-PINN), the adaptive derivative-free-loss method (ATS-DFLM), and the adaptive temporal-difference method for forward-backward stochastic differential equations (ATS-FBSTD). Our numerical experiments show that the ATS remarkably improves the computational accuracy and efficiency of the original deep solvers. In particular, for a high-dimensional peak problem, the relative errors by the ATS-PINN can achieve the order of $O\left({10}^{-3}\right)$, even when the vanilla PINN fails.