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.
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