  We develop a general framework for data-driven approximation of input-output
maps between infinite-dimensional spaces. The proposed approach is motivated by
the recent successes of neural networks and deep learning, in combination with
ideas from model reduction. This combination results in a neural network
approximation which, in principle, is defined on infinite-dimensional spaces
and, in practice, is robust to the dimension of finite-dimensional
approximations of these spaces required for computation. For a class of
input-output maps, and suitably chosen probability measures on the inputs, we
prove convergence of the proposed approximation methodology. We also include
numerical experiments which demonstrate the effectiveness of the method,
showing convergence and robustness of the approximation scheme with respect to
the size of the discretization, and compare it with existing algorithms from
the literature; our examples include the mapping from coefficient to solution
in a divergence form elliptic partial differential equation (PDE) problem, and
the solution operator for viscous Burgers' equation.
