  This paper addresses the understanding and characterization of residual
networks (ResNet), which are among the state-of-the-art deep learning
architectures for a variety of supervised learning problems. We focus on the
mapping component of ResNets, which map the embedding space towards a new
unknown space where the prediction or classification can be stated according to
linear criteria. We show that this mapping component can be regarded as the
numerical implementation of continuous flows of diffeomorphisms governed by
ordinary differential equations. Especially, ResNets with shared weights are
fully characterized as numerical approximation of exponential diffeomorphic
operators. We stress both theoretically and numerically the relevance of the
enforcement of diffeormorphic properties and the importance of numerical issues
to make consistent the continuous formulation and the discretized ResNet
implementation. We further discuss the resulting theoretical and computational
insights on ResNet architectures.
