  Physics-Informed Neural Networks (PINNs) have become a kind of attractive
machine learning method for obtaining solutions of partial differential
equations (PDEs). Training PINNs can be seen as a semi-supervised learning
task, in which only exact values of initial and boundary points can be obtained
in solving forward problems, and in the whole spatio-temporal domain
collocation points are sampled without exact labels, which brings training
difficulties. Thus the selection of collocation points and sampling methods are
quite crucial in training PINNs. Existing sampling methods include fixed and
dynamic types, and in the more popular latter one, sampling is usually
controlled by PDE residual loss. We point out that it is not sufficient to only
consider the residual loss in adaptive sampling and sampling should obey
temporal causality. We further introduce temporal causality into adaptive
sampling and propose a novel adaptive causal sampling method to improve the
performance and efficiency of PINNs. Numerical experiments of several PDEs with
high-order derivatives and strong nonlinearity, including Cahn Hilliard and KdV
equations, show that the proposed sampling method can improve the performance
of PINNs with few collocation points. We demonstrate that by utilizing such a
relatively simple sampling method, prediction performance can be improved up to
two orders of magnitude compared with state-of-the-art results with almost no
extra computation cost, especially when points are limited.
