  We explore using neural operators, or neural network representations of
nonlinear maps between function spaces, to accelerate infinite-dimensional
Bayesian inverse problems (BIPs) with models governed by nonlinear parametric
partial differential equations (PDEs). Neural operators have gained significant
attention in recent years for their ability to approximate the
parameter-to-solution maps defined by PDEs using as training data solutions of
PDEs at a limited number of parameter samples. The computational cost of BIPs
can be drastically reduced if the large number of PDE solves required for
posterior characterization are replaced with evaluations of trained neural
operators. However, reducing error in the resulting BIP solutions via reducing
the approximation error of the neural operators in training can be challenging
and unreliable. We provide an a priori error bound result that implies certain
BIPs can be ill-conditioned to the approximation error of neural operators,
thus leading to inaccessible accuracy requirements in training. To reliably
deploy neural operators in BIPs, we consider a strategy for enhancing the
performance of neural operators, which is to correct the prediction of a
trained neural operator by solving a linear variational problem based on the
PDE residual. We show that a trained neural operator with error correction can
achieve a quadratic reduction of its approximation error, all while retaining
substantial computational speedups of posterior sampling when models are
governed by highly nonlinear PDEs. The strategy is applied to two numerical
examples of BIPs based on a nonlinear reaction--diffusion problem and
deformation of hyperelastic materials. We demonstrate that posterior
representations of the two BIPs produced using trained neural operators are
greatly and consistently enhanced by error correction.
