  The solution of inverse problems is central to a wide range of applications
including medicine, biology, and engineering. These problems require finding a
desired solution in the presence of noisy observations. A key feature of
inverse problems is their ill-posedness, which leads to unstable behavior under
noise when standard solution methods are used. For this reason, regularization
methods have been developed that compromise between data fitting and prior
structure. Recently, data-driven variational regularization methods have been
introduced, where the prior in the form of a regularizer is derived from
provided ground truth data. However, these methods have mainly been analyzed
for Tikhonov regularization, referred to as Network Tikhonov Regularization
(NETT). In this paper, we propose and analyze Morozov regularization in
combination with a learned regularizer. The regularizers, which can be adapted
to the training data, are defined by neural networks and are therefore
non-convex. We give a convergence analysis in the non-convex setting allowing
noise-dependent regularizers, and propose a possible training strategy. We
present numerical results for attenuation correction in the context of
photoacoustic tomography.
