Deep Operator Learning Lessens the Curse of Dimensionality for PDEs

Deep neural networks (DNNs) have seen tremendous success in many fields and their developments in PDE-related problems are rapidly growing. 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. We apply our results to various PDEs, including elliptic equations, parabolic equations, and Burgers equations.