Partial differential equations (PDEs) are indispensable for modeling many physical phenomena and also commonly used for solving image processing tasks. In the latter area, PDE-based approaches interpret image data as discretizations of multivariate functions and the output of image processing algorithms as solutions to certain PDEs. Posing image processing problems in the infinite-dimensional setting provides powerful tools for their analysis and solution. For the last few decades, the reinterpretation of classical image processing problems through the PDE lens has been creating multiple celebrated approaches that benefit a vast area of tasks including image segmentation, denoising, registration, and reconstruction. In this paper, we establish a new PDE interpretation of a class of deep convolutional neural networks (CNN) that are commonly used to learn from speech, image, and video data. Our interpretation includes convolution residual neural networks (ResNet), which are among the most promising approaches for tasks such as image classification having improved the state-of-the-art performance in prestigious benchmark challenges. Despite their recent successes, deep ResNets still face some critical challenges associated with their design, immense computational costs and memory requirements, and lack of understanding of their reasoning. Guided by well-established PDE theory, we derive three new ResNet architectures that fall into two new classes: parabolic and hyperbolic CNNs. We demonstrate how PDE theory can provide new insights and algorithms for deep learning and demonstrate the competitiveness of three new CNN architectures using numerical experiments.

中文翻译：

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偏微分方程激发的深层神经网络

偏微分方程（PDE）对于建模许多物理现象必不可少，并且通常用于求解图像处理任务。在后一领域，基于PDE的方法将图像数据解释为多元函数的离散化，而将图像处理算法的输出解释为某些PDE的解决方案。将图像处理问题摆在无穷大的设置中提供了强大的工具来进行分析和解决。在过去的几十年中，通过PDE镜头对经典图像处理问题的重新解释已经创建了多种著名的方法，这些方法使很多领域受益，包括图像分割，去噪，配准和重建。在本文中，我们为一类深度卷积神经网络（CNN）建立了新的PDE解释，该网络通常用于从语音，图像和视频数据中学习。我们的解释包括卷积残差神经网络（ResNet），这是用于诸如图像分类等任务的最有前途的方法之一，这些方法改善了在著名基准挑战中的最新性能。尽管最近获得了成功，但深层ResNet仍然面临与其设计相关的一些关键挑战，巨大的计算成本和内存需求以及对推理的理解不足。在行之有效的PDE理论的指导下，我们得出了三个新的ResNet架构，它们分为两个新类别：抛物线和双曲线CNN。