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We discuss the possibility of learning a data-driven explicit model correction for inverse problems and whether such a model correction can be used within a variational framework to obtain regularized reconstructions. This paper discusses the conceptual difficulty of learning such a forward model correction and proceeds to present a possible solution as a forward-adjoint correction that explicitly corrects in both data and solution spaces. We then derive conditions under which solutions to the variational problem with a learned correction converge to solutions obtained with the correct operator. The proposed approach is evaluated on an application to limited view photoacoustic tomography and compared to the established framework of the Bayesian approximation error method.

Models of the glacial‐isostatic adjustment (GIA) to past ice‐mass changes exhibit large differences in north‐east Greenland owing to insufficient knowledge about glacial history and Earth rheology. The GIA uncertainties feed back to uncertainties in present‐day mass‐balance estimates from satellite gravimetry. Geodetic Global Navigation Satellite System (GNSS) measurements allow to directly observe displacement of bedrock. We present results from repeated and continuous GNSS measurements conducted within five measurement campaigns between 2008 and 2017. We used the observed uplift rates to validate different GIA models in conjunction with estimates of the elastic response of the solid Earth to present‐day ice‐mass changes. To determine present‐day ice‐mass changes and the associated elastic deformations, we combined satellite altimetry data from CryoSat‐2 with satellite gravimetry data from the Gravity Recovery and Climate Experiment (GRACE) for the entire Greenland Ice Sheet (GrIS) and included peripheral glaciers and ice caps. The different GIA models were consistently used in all processing steps. The GNSS measurements in north‐east Greenland revealed uplift rates in the range of 2.8 mm yr−1 to 8.9 mm yr−1. The comparison of the total displacement predicted by GIA and elastic modeling with the GNSS‐based displacement clearly favors GIA models that show low rates (0.7 mm yr−1 to 4.4 mm yr−1 at the GNSS sites) against GIA models with higher rates of up to 8.3 mm yr−1. The correction due to the favored GIA model in north‐east Greenland results in an ice‐mass loss of 233 ± 43 Gt yr−1 for the GrIS including peripheral glaciers over the period July 2010 to June 2017.

Inverse Problems in medical imaging and computer vision are traditionally solved using purely model-based methods. Among those variational regularization models are one of the most popular approaches. We propose a new framework for applying data-driven approaches to inverse problems, using a neural network as a regularization functional. The network learns to discriminate between the distribution of ground truth images and the distribution of unregularized reconstructions. Once trained, the network is applied to the inverse problem by solving the corresponding variational problem. Unlike other data-based approaches for inverse problems, the algorithm can be applied even if only unsupervised training data is available. Experiments demonstrate the potential of the framework for denoising on the BSDS dataset and for computed tomography reconstruction on the LIDC dataset., QC20190610