  We develop a theoretical analysis for special neural network architectures,
termed operator recurrent neural networks, for approximating nonlinear
functions whose inputs are linear operators. Such functions commonly arise in
solution algorithms for inverse boundary value problems. Traditional neural
networks treat input data as vectors, and thus they do not effectively capture
the multiplicative structure associated with the linear operators that
correspond to the data in such inverse problems. We therefore introduce a new
family that resembles a standard neural network architecture, but where the
input data acts multiplicatively on vectors. Motivated by compact operators
appearing in boundary control and the analysis of inverse boundary value
problems for the wave equation, we promote structure and sparsity in selected
weight matrices in the network. After describing this architecture, we study
its representation properties as well as its approximation properties. We
furthermore show that an explicit regularization can be introduced that can be
derived from the mathematical analysis of the mentioned inverse problems, and
which leads to certain guarantees on the generalization properties. We observe
that the sparsity of the weight matrices improves the generalization estimates.
Lastly, we discuss how operator recurrent networks can be viewed as a deep
learning analogue to deterministic algorithms such as boundary control for
reconstructing the unknown wavespeed in the acoustic wave equation from
boundary measurements.
