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Solving partial differential equations (PDEs) is a central task in scientificcomputing. Recently, neural network approximation of PDEs has receivedincreasing attention due to its flexible meshless discretization and itspotential for high-dimensional problems. One fundamental numerical difficultyis that random samples in the training set introduce statistical errors intothe discretization of loss functional which may become the dominant error inthe final approximation, and therefore overshadow the modeling capability ofthe neural network. In this work, we propose a new minmax formulation tooptimize simultaneously the approximate solution, given by a neural networkmodel, and the random samples in the training set, provided by a deepgenerative model. The key idea is to use a deep generative model to adjustrandom samples in the training set such that the residual induced by theapproximate PDE solution can maintain a smooth profile when it is beingminimized. Such an idea is achieved by implicitly embedding the Wassersteindistance between the residual-induced distribution and the uniform distributioninto the loss, which is then minimized together with the residual. A nearlyuniform residual profile means that its variance is small for any normalizedweight function such that the Monte Carlo approximation error of the lossfunctional is reduced significantly for a certain sample size. The adversarialadaptive sampling (AAS) approach proposed in this work is the first attempt toformulate two essential components, minimizing the residual and seeking theoptimal training set, into one minmax objective functional for the neuralnetwork approximation of PDEs.

Adversarial Adaptive Sampling: Unify PINN and Optimal Transport for the Approximation of PDEs