Deep neural networks (DNNs) have achieved remarkable success in numerous domains, and their application to PDE-related problems has been rapidly advancing. This paper provides an estimate for the generalization error of learning Lipschitz operators over Banach spaces using DNNs with applications to various PDE solution operators. The goal is to specify DNN width, depth, and the number of training samples needed to guarantee a certain testing error. Under mild assumptions on data distributions or operator structures, our analysis shows that deep operator learning can have a relaxed dependence on the discretization resolution of PDEs and, hence, lessen the curse of dimensionality in many PDE-related problems including elliptic equations, parabolic equations, and Burgers equations. Our results are also applied to give insights about discretization-invariant in operator learning.

翻译：深度神经网络(DNN)已在众多领域取得显著成功，其在偏微分方程(PDE)相关问题的应用也在迅速发展。本文给出了利用DNN学习巴拿赫空间上Lipschitz算子的泛化误差估计，并将其应用于各类PDE求解算子。研究目标在于明确为保证特定测试误差所需的DNN宽度、深度及训练样本数量。在关于数据分布或算子结构的温和假设下，我们的分析表明，深度算子学习可降低对PDE离散化分辨率的依赖程度，从而缓解椭圆方程、抛物方程和Burgers方程等众多PDE相关问题的维度灾难。本文结果还可用于揭示算子学习中的离散化不变性。