  In this paper, we propose a new adaptive technique, named adaptive
trajectories sampling (ATS), which is used to select training points for the
numerical solution of partial differential equations (PDEs) with deep learning
methods. The key feature of the ATS is that all training points are adaptively
selected from trajectories that are generated according to a PDE-related
stochastic process. We incorporate the ATS into three known deep learning
solvers for PDEs, namely the adaptive derivative-free-loss method (ATS-DFLM),
the adaptive physics-informed neural network method (ATS-PINN), and the
adaptive temporal-difference method for forward-backward stochastic
differential equations (ATS-FBSTD). Our numerical experiments demonstrate that
the ATS remarkably improves the computational accuracy and efficiency of the
original deep learning solvers for the PDEs. In particular, for some specific
high-dimensional PDEs, the ATS can even improve the accuracy of the PINN by two
orders of magnitude.
