  We address the solution of large-scale Bayesian optimal experimental design
(OED) problems governed by partial differential equations (PDEs) with
infinite-dimensional parameter fields. The OED problem seeks to find sensor
locations that maximize the expected information gain (EIG) in the solution of
the underlying Bayesian inverse problem. Computation of the EIG is usually
prohibitive for PDE-based OED problems. To make the evaluation of the EIG
tractable, we approximate the (PDE-based) parameter-to-observable map with a
derivative-informed projected neural network (DIPNet) surrogate, which exploits
the geometry, smoothness, and intrinsic low-dimensionality of the map using a
small and dimension-independent number of PDE solves. The surrogate is then
deployed within a greedy algorithm-based solution of the OED problem such that
no further PDE solves are required. We analyze the EIG approximation error in
terms of the generalization error of the DIPNet and show they are of the same
order. Finally, the efficiency and accuracy of the method are demonstrated via
numerical experiments on OED problems governed by inverse scattering and
inverse reactive transport with up to 16,641 uncertain parameters and 100
experimental design variables, where we observe up to three orders of magnitude
speedup relative to a reference double loop Monte Carlo method.
