  In recent years, deep learning techniques have shown great success in various
tasks related to inverse problems, where a target quantity of interest can only
be observed through indirect measurements by a forward operator. Common
approaches apply deep neural networks in a post-processing step to the
reconstructions obtained by classical reconstruction methods. However, the
latter methods can be computationally expensive and introduce artifacts that
are not present in the measured data and, in turn, can deteriorate the
performance on the given task. To overcome these limitations, we propose a
class of equivariant neural networks that can be directly applied to the
measurements to solve the desired task. To this end, we build appropriate
network structures by developing layers that are equivariant with respect to
data transformations induced by well-known symmetries in the domain of the
forward operator. We rigorously analyze the relation between the measurement
operator and the resulting group representations and prove a representer
theorem that characterizes the class of linear operators that translate between
a given pair of group actions. Based on this theory, we extend the existing
concepts of Lie group equivariant deep learning to inverse problems and
introduce new representations that result from the involved measurement
operations. This allows us to efficiently solve classification, regression or
even reconstruction tasks based on indirect measurements also for very sparse
data problems, where a classical reconstruction-based approach may be hard or
even impossible. We illustrate the effectiveness of our approach in numerical
experiments and compare with existing methods.
