Deep Neural Networks Motivated by Partial Differential Equations

• Published in: Journal of mathematical imaging and vision Vol. 62; no. 3; pp. 352 - 364
• Main Authors: Ruthotto, Lars, Haber, Eldad
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.04.2020; Springer Nature B.V
• Subjects: Algorithms; Applications of Mathematics; Artificial neural networks; Computer Science; Convolution; Image classification; Image processing; Image Processing and Computer Vision; Image reconstruction; Image segmentation; Machine learning; Mathematical Methods in Physics; Multivariate analysis; Neural networks; Noise reduction; Partial differential equations; Signal,Image and Speech Processing; Video data; Deep neural networks; Partial differential equations; PDE-constrained optimization; Machine learning; Image classification
• ISSN: 0924-9907; 1573-7683
• DOI: 10.1007/s10851-019-00903-1

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.