  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-invariance in operator learning.
