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 toward 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. In particular, 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 diffeomorphic 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 into ResNet architectures.

中文翻译：

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残差网络作为不同态流

本文介绍了残差网络（ResNet）的理解和特征，残差网络是针对各种监督学习问题的最新深度学习体系结构之一。我们专注于ResNets的映射组件，该组件将嵌入空间映射到一个新的未知空间，在该空间中可以根据线性准则来进行预测或分类。我们表明，该映射分量可以看作是由常微分方程控制的亚纯连续流的数值实现。特别是，具有共享权重的ResNet被完全表征为指数微分算子的数值近似。我们在理论上和数字上都强调了实施微晶性质的相关性以及数值问题对使连续公式和离散化ResNet实现一致的重要性。我们将进一步讨论由此产生的对ResNet架构的理论和计算见解。