  Neural Operators offer a powerful, data-driven tool for solving parametric
PDEs as they can represent maps between infinite-dimensional function spaces.
In this work, we employ physics-informed Neural Operators in the context of
high-dimensional, Bayesian inverse problems. Traditional solution strategies
necessitate an enormous, and frequently infeasible, number of forward model
solves, as well as the computation of parametric derivatives. In order to
enable efficient solutions, we extend Deep Operator Networks (DeepONets) by
employing a RealNVP architecture which yields an invertible and differentiable
map between the parametric input and the branch-net output. This allows us to
construct accurate approximations of the full posterior, irrespective of the
number of observations and the magnitude of the observation noise, without any
need for additional forward solves nor for cumbersome, iterative sampling
procedures. We demonstrate the efficacy and accuracy of the proposed
methodology in the context of inverse problems for three benchmarks: an
anti-derivative equation, reaction-diffusion dynamics and flow through porous
media.
