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.

中文翻译：

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结构保持深度学习

在过去的几年里，深度学习作为一个引起广泛关注的话题而备受关注，这主要是由于在解决大规模图像处理任务方面取得的成功。应用深度学习涉及多个具有挑战性的数学问题：大多数深度学习方法需要解决难以优化的问题，并且需要很好地理解计算工作量、数据量和模型复杂度之间的权衡，以成功设计一个深度给定问题的学习方法。深度学习取得的大量进展是基于启发式探索的，但是在数学上理解现有深度学习方法的结构并系统地设计新的深度学习方法以保留深度学习中的某些类型的结构。在本文中，我们回顾了其中的一些方向：一些深度神经网络可以理解为动态系统的离散化，神经网络可以设计为具有理想的属性，例如可逆性或群等方差，以及基于保形哈密顿系统的新算法框架和已经提出了解决优化问题的黎曼流形。我们通过讨论一些我们认为是未来研究的有趣方向的开放性问题来结束对这些主题的回顾。神经网络可以设计为具有理想的属性，例如可逆性或群等方差，并且已经提出了基于保形哈密顿系统和黎曼流形的新算法框架来解决优化问题。我们通过讨论一些我们认为是未来研究的有趣方向的开放性问题来结束对这些主题的回顾。神经网络可以设计为具有理想的属性，例如可逆性或群等方差，并且已经提出了基于保形哈密顿系统和黎曼流形的新算法框架来解决优化问题。我们通过讨论一些我们认为是未来研究的有趣方向的开放性问题来结束对这些主题的回顾。