In this thesis, we investigate properties of deep neural networks and their application to inverse problems.

A successful classical approach to inverse problems is variational regularisation, combining knowledge and modelling of the imaging modality at hand with a regularisation functional that incorporates prior knowledge about solutions to the inverse problem. With the success of deep neural networks in many imaging tasks such as image classification or semantic segmentation, recently algorithms that leverage the power of neural networks have been explored to solve inverse problems.

In this thesis, we discuss various approaches to incorporate deep learning into reconstruction algorithms for inverse problems and in particular into variational approaches. We propose and discuss an algorithm to train a neural network as regularisation functional. This is achieved by training the network to tell apart an unregularised pseudo-inverse from ground truth images. The resulting regulariser decreases the Wasserstein distance between reconstructions and ground truth images at an optimal rate. We present computational results for computed tomography (CT) and magnetic resonance imaging (MRI) reconstruction and investigate generalisation properties of the learned regularisation functional.

In another line of research, we turn our attention to making use of neural networks to correct for errors in the forward operator. While an approximate model of the forward operator is available in many applications, this model can exhibit artefacts compared to the true behaviour of the imaging modality. We train a neural network to learn how to correct for these shortcomings by learning a correction from data. The aim is to obtain a corrected operator that can be employed within a variational framework for reconstruction. We investigate key challenges of this approach and propose a recursive forward-adjoint algorithm to efficiently train an operator correction for photo-acoustic tomography reconstruction.