Structure-preserving deep learning

Over the past few years, deep learning has risen to foreground as a topic of massive interest, mainly result successes obtained in solving large-scale image processing tasks. There are multiple challenging mathematical problems involved applying learning: most methods require solution hard optimisation problems, and good understanding trade-off between computational effort, amount data model complexity is required successfully design approach for given problem.. A large progress made been based on heuristic explorations, but there growing effort mathematically understand structure existing systematically new preserve certain types learning. In this article, we review number these directions: some neural networks can be understood discretisations dynamical systems, designed have desirable properties such invertibility or group equivariance algorithmic frameworks conformal Hamiltonian systems Riemannian manifolds solve proposed. We conclude our each topics by discussing open that consider interesting directions future research.