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Deep Neural Networks Motivated by Partial Differential Equations

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Abstract

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.

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Acknowledgements

L.R. is supported by the U.S. National Science Foundation (NSF) through awards DMS 1522599 and DMS 1751636 and by the NVIDIA Corporation’s GPU grant program. We thank Martin Burger for outlining how to show stability using monotone operator theory and Eran Treister and other contributors of the Meganet package. We also thank the Isaac Newton Institute (INI) for Mathematical Sciences for support and hospitality during the program on Generative Models, Parameter Learning and Sparsity (VMVW02) when work on this paper was undertaken. INI was supported by EPSRC Grant Number: LNAG/036, RG91310.

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Ruthotto, L., Haber, E. Deep Neural Networks Motivated by Partial Differential Equations. J Math Imaging Vis 62, 352–364 (2020). https://doi.org/10.1007/s10851-019-00903-1

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