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Abstract 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 (LFE) 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 LFEs. 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. 

Two segmentation methods, one atlas-based and one neural-network-based, were compared to see how well they can each automatically segment the brain stem and cerebellum in Displacement Encoding with Stimulated Echoes Magnetic Resonance Imaging (DENSE-MRI) data. The segmentation is a pre-requisite for estimating the average displacements in these regions, which have recently been proposed as biomarkers in the diagnosis of Chiari Malformation type I (CMI). In numerical experiments, the segmentations of both methods were similar to manual segmentations provided by trained experts. It was found that, overall, the neural-network-based method alone produced more accurate segmentations than the atlas-based method did alone, but that a combination of the two methods -- in which the atlas-based method is used for the segmentation of the brain stem and the neural-network is used for the segmentation of the cerebellum -- may be the most successful.