  In this work we propose a deep adaptive sampling (DAS) method for solving
partial differential equations (PDEs), where deep neural networks are utilized
to approximate the solutions of PDEs and deep generative models are employed to
generate new collocation points that refine the training set. The overall
procedure of DAS consists of two components: solving the PDEs by minimizing the
residual loss on the collocation points in the training set and generating a
new training set to further improve the accuracy of current approximate
solution. In particular, we treat the residual as a probability density
function and approximate it with a deep generative model, called KRnet. The new
samples from KRnet are consistent with the distribution induced by the
residual, i.e., more samples are located in the region of large residual and
less samples are located in the region of small residual. Analogous to
classical adaptive methods such as the adaptive finite element, KRnet acts as
an error indicator that guides the refinement of the training set. Compared to
the neural network approximation obtained with uniformly distributed
collocation points, the developed algorithms can significantly improve the
accuracy, especially for low regularity and high-dimensional problems. We
demonstrate the effectiveness of the proposed DAS method with numerical
experiments.
