  We propose a non-stationary iterated network Tikhonov (iNETT) method for the
solution of ill-posed inverse problems. The iNETT employs deep neural networks
to build a data-driven regularizer, and it avoids the difficult task of
estimating the optimal regularization parameter. To achieve the theoretical
convergence of iNETT, we introduce uniformly convex neural networks to build
the data-driven regularizer. Rigorous theories and detailed algorithms are
proposed for the construction of convex and uniformly convex neural networks.
In particular, given a general neural network architecture, we prescribe
sufficient conditions to achieve a trained neural network which is
component-wise convex or uniformly convex; moreover, we provide concrete
examples of realizing convexity and uniform convexity in the modern U-net
architecture. With the tools of convex and uniformly convex neural networks,
the iNETT algorithm is developed and a rigorous convergence analysis is
provided. Lastly, we show applications of the iNETT algorithm in 2D
computerized tomography, where numerical examples illustrate the efficacy of
the proposed algorithm.
