  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. Over 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.
