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（PDEs）是科学计算中的核心任务。近年来，神经网络对PDE的近似因其灵活的无网格离散化及在高维问题中的潜力而受到广泛关注。一个基本的数值困难在于，训练集中的随机样本会将统计误差引入损失函数泛函的离散化，该误差可能成为最终近似中的主导误差，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，以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中的随机样本。关键思想是利用深度生成模型调整训练集中的随机样本，使得近似PDE解所诱导的残差在最小化过程中能够保持平滑轮廓。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实现，该距离随后与残差共同被最小化。近乎均匀的残差轮廓意味着，对于任何归一化权重函数，其方差较小，从而在给定样本量下显著降低损失泛函的蒙特卡罗近似误差。本文提出的对抗自适应采样（AAS）方法是首次尝试将最小化残差和寻求最优训练集这两个核心组件统一为一个用于PDE神经网络近似的极小极大目标泛函。