  We derive and analyse a new variant of the iteratively regularized Landweber
iteration, for solving linear and nonlinear ill-posed inverse problems. The
method takes into account training data, which are used to estimate the
interior of a black box, which is used to define the iteration process. We
prove convergence and stability for the scheme in infinite dimensional Hilbert
spaces. These theoretical results are complemented by several numerical
experiments for solving linear inverse problems for the Radon transform and a
nonlinear inverse problem for Schlieren tomography.
