  Learning operators between infinitely dimensional spaces is an important
learning task arising in wide applications in machine learning, imaging
science, mathematical modeling and simulations, etc. This paper studies the
nonparametric estimation of Lipschitz operators using deep neural networks.
Non-asymptotic upper bounds are derived for the generalization error of the
empirical risk minimizer over a properly chosen network class. Under the
assumption that the target operator exhibits a low dimensional structure, our
error bounds decay as the training sample size increases, with an attractive
fast rate depending on the intrinsic dimension in our estimation. Our
assumptions cover most scenarios in real applications and our results give rise
to fast rates by exploiting low dimensional structures of data in operator
estimation. We also investigate the influence of network structures (e.g.,
network width, depth, and sparsity) on the generalization error of the neural
network estimator and propose a general suggestion on the choice of network
structures to maximize the learning efficiency quantitatively.
