  Data assisted reconstruction algorithms, incorporating trained neural
networks, are a novel paradigm for solving inverse problems. One approach is to
first apply a classical reconstruction method and then apply a neural network
to improve its solution. Empirical evidence shows that such two-step methods
provide high-quality reconstructions, but they lack a convergence analysis. In
this paper we formalize the use of such two-step approaches with classical
regularization theory. We propose data-consistent neural networks that we
combine with classical regularization methods. This yields a data-driven
regularization method for which we provide a full convergence analysis with
respect to noise. Numerical simulations show that compared to standard two-step
deep learning methods, our approach provides better stability with respect to
structural changes in the test set, while performing similarly on test data
similar to the training set. Our method provides a stable solution of inverse
problems that exploits both the known nonlinear forward model as well as the
desired solution manifold from data.
