  Solving partial differential equations (PDEs) is a central task in scientific
computing. Recently, neural network approximation of PDEs has received
increasing attention due to its flexible meshless discretization and its
potential for high-dimensional problems. One fundamental numerical difficulty
is that random samples in the training set introduce statistical errors into
the discretization of loss functional which may become the dominant error in
the final approximation, and therefore overshadow the modeling capability of
the neural network. In this work, we propose a new minmax formulation to
optimize simultaneously the approximate solution, given by a neural network
model, and the random samples in the training set, provided by a deep
generative model. The key idea is to use a deep generative model to adjust
random samples in the training set such that the residual induced by the
approximate PDE solution can maintain a smooth profile when it is being
minimized. Such an idea is achieved by implicitly embedding the Wasserstein
distance between the residual-induced distribution and the uniform distribution
into the loss, which is then minimized together with the residual. A nearly
uniform residual profile means that its variance is small for any normalized
weight function such that the Monte Carlo approximation error of the loss
functional is reduced significantly for a certain sample size. The adversarial
adaptive sampling (AAS) approach proposed in this work is the first attempt to
formulate two essential components, minimizing the residual and seeking the
optimal training set, into one minmax objective functional for the neural
network approximation of PDEs.
