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Structure-preserving deep learning

Published online by Cambridge University Press:  27 May 2021

E. CELLEDONI
Affiliation:
Department of Mathematical Sciences, NTNU, N-7491 Trondheim, Norway emails: elena.celledoni@ntnu.no; brynjulf.owren@ntnu.no
M. J. EHRHARDT
Affiliation:
Institute for Mathematical Innovation, University of Bath, Bath BA2 7JU, UK email: m.ehrhardt@bath.ac.uk
C. ETMANN
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK emails: cetmann@damtp.cam.ac.uk; cbs31@cam.ac.uk; fs436@cam.ac.uk
R. I. MCLACHLAN
Affiliation:
School of Fundamental Sciences, Massey University, Private Bag 11-222, Palmerston North, New Zealand email: r.mclachlan@massey.ac.nz
B. OWREN
Affiliation:
Department of Mathematical Sciences, NTNU, N-7491 Trondheim, Norway emails: elena.celledoni@ntnu.no; brynjulf.owren@ntnu.no
C.-B. SCHONLIEB
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK emails: cetmann@damtp.cam.ac.uk; cbs31@cam.ac.uk; fs436@cam.ac.uk
F. SHERRY
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK emails: cetmann@damtp.cam.ac.uk; cbs31@cam.ac.uk; fs436@cam.ac.uk
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Abstract

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

Type
Papers
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2021. Published by Cambridge University Press

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