  We consider the solution of nonlinear inverse problems where the forward
problem is a discretization of a partial differential equation. Such problems
are notoriously difficult to solve in practice and require minimizing a
combination of a data-fit term and a regularization term. The main
computational bottleneck of typical algorithms is the direct estimation of the
data misfit. Therefore, likelihood-free approaches have become appealing
alternatives. Nonetheless, difficulties in generalization and limitations in
accuracy have hindered their broader utility and applicability. In this work,
we use a paired autoencoder framework as a likelihood-free estimator for
inverse problems. We show that the use of such an architecture allows us to
construct a solution efficiently and to overcome some known open problems when
using likelihood-free estimators. In particular, our framework can assess the
quality of the solution and improve on it if needed. We demonstrate the
viability of our approach using examples from full waveform inversion and
inverse electromagnetic imaging.
