  With the recent study of deep learning in scientific computation, the
Physics-Informed Neural Networks (PINNs) method has drawn widespread attention
for solving Partial Differential Equations (PDEs). Compared to traditional
methods, PINNs can efficiently handle high-dimensional problems, but the
accuracy is relatively low, especially for highly irregular problems. Inspired
by the idea of adaptive finite element methods and incremental learning, we
propose GAS, a Gaussian mixture distribution-based adaptive sampling method for
PINNs. During the training procedure, GAS uses the current residual information
to generate a Gaussian mixture distribution for the sampling of additional
points, which are then trained together with historical data to speed up the
convergence of the loss and achieve higher accuracy. Several numerical
simulations on 2D and 10D problems show that GAS is a promising method that
achieves state-of-the-art accuracy among deep solvers, while being comparable
with traditional numerical solvers.
