  We discuss the possibility to learn a data-driven explicit model correction
for inverse problems and whether such a model correction can be used within a
variational framework to obtain regularised reconstructions. This paper
discusses the conceptual difficulty to learn such a forward model correction
and proceeds to present a possible solution as 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 Bayesian approximation error method.
