  Deep learning based reconstruction methods deliver outstanding results for
solving inverse problems and are therefore becoming increasingly important. A
recently invented class of learning-based reconstruction methods is the
so-called NETT (for Network Tikhonov Regularization), which contains a trained
neural network as regularizer in generalized Tikhonov regularization. The
existing analysis of NETT considers fixed operator and fixed regularizer and
analyzes the convergence as the noise level in the data approaches zero. In
this paper, we extend the frameworks and analysis considerably to reflect
various practical aspects and take into account discretization of the data
space, the solution space, the forward operator and the neural network defining
the regularizer. We show the asymptotic convergence of the discretized NETT
approach for decreasing noise levels and discretization errors. Additionally,
we derive convergence rates and present numerical results for a limited data
problem in photoacoustic tomography.
