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.

翻译：求解偏微分方程是科学计算中的核心任务。近年来，神经网络因其灵活的无网格离散化特性及处理高维问题的潜力，在偏微分方程逼近领域受到越来越多的关注。其中一个基本数值困难在于，训练集中的随机样本会在损失泛函的离散化过程中引入统计误差，这种误差可能成为最终逼近中的主导误差，从而掩盖神经网络的建模能力。本文提出一种新的极小极大形式，用于同时优化由神经网络模型给出的近似解和由深度生成模型提供的训练集随机样本。核心思想是使用深度生成模型调整训练集中的随机样本，使得在最小化近似偏微分方程解产生的残差时，该残差能保持平滑分布。这一思想通过将残差诱导分布与均匀分布之间的Wasserstein距离隐式嵌入损失函数中实现，并与残差联合最小化。近乎均匀的残差分布意味着对于任何归一化的权重函数，其方差都很小，从而在给定样本量下显著降低损失泛函的蒙特卡洛近似误差。本文提出的对抗自适应采样方法是首次尝试将最小化残差和寻找最优训练集这两个核心要素整合为单一极小极大目标泛函，用于神经网络逼近偏微分方程问题。