{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/deep-neural-networks-motivated-by-partial","title":"Deep Neural Networks Motivated by Partial Differential Equations","arxiv_id":"1804.04272","date":"2018-04-12","proceeding":null,"authors":["Lars Ruthotto","Eldad Haber"],"abstract":"Partial differential equations (PDEs) are indispensable for modeling many\nphysical phenomena and also commonly used for solving image processing tasks.\nIn the latter area, PDE-based approaches interpret image data as\ndiscretizations of multivariate functions and the output of image processing\nalgorithms as solutions to certain PDEs. Posing image processing problems in\nthe infinite dimensional setting provides powerful tools for their analysis and\nsolution. Over the last few decades, the reinterpretation of classical image\nprocessing problems through the PDE lens has been creating multiple celebrated\napproaches that benefit a vast area of tasks including image segmentation,\ndenoising, registration, and reconstruction.\n  In this paper, we establish a new PDE-interpretation of a class of deep\nconvolutional neural networks (CNN) that are commonly used to learn from\nspeech, image, and video data. Our interpretation includes convolution residual\nneural networks (ResNet), which are among the most promising approaches for\ntasks such as image classification having improved the state-of-the-art\nperformance in prestigious benchmark challenges. Despite their recent\nsuccesses, deep ResNets still face some critical challenges associated with\ntheir design, immense computational costs and memory requirements, and lack of\nunderstanding of their reasoning.\n  Guided by well-established PDE theory, we derive three new ResNet\narchitectures that fall into two new classes: parabolic and hyperbolic CNNs. We\ndemonstrate how PDE theory can provide new insights and algorithms for deep\nlearning and demonstrate the competitiveness of three new CNN architectures\nusing numerical experiments.","url_abs":"http://arxiv.org/abs/1804.04272v2","url_pdf":"http://arxiv.org/pdf/1804.04272v2.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"abstracts"},"code_links":[{"paper_slug":"deep-neural-networks-motivated-by-partial","repo_url":"https://github.com/EmoryMLIP/DynamicBlocks","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"pytorch","reach":{"status":"ok","spdx":"MIT"}}],"tasks":[{"task_slug":"denoising","task_name":"Denoising"},{"task_slug":"image-classification","task_name":"Image Classification"},{"task_slug":"image-segmentation","task_name":"Image Segmentation"},{"task_slug":"semantic-segmentation","task_name":"Semantic Segmentation"},{"task_slug":"image-classification","task_name":"image-classification"}],"methods":[{"method_slug":"convolution","method_name":"Convolution"}],"datasets_introduced":[],"methods_introduced":[],"results":[{"leaderboard":"/sota/image-classification-on-stl-10","task":"Image Classification","dataset":"STL-10","model":"Hamiltonian","rank_in_archive_order":68,"of":117,"metrics":{"Percentage correct":"78.3"},"uses_additional_data":false},{"leaderboard":"/sota/image-classification-on-stl-10","task":"Image Classification","dataset":"STL-10","model":"Parabolic","rank_in_archive_order":71,"of":117,"metrics":{"Percentage correct":"77.0"},"uses_additional_data":false},{"leaderboard":"/sota/image-classification-on-stl-10","task":"Image Classification","dataset":"STL-10","model":"Second-order","rank_in_archive_order":80,"of":117,"metrics":{"Percentage correct":"74.3"},"uses_additional_data":false}],"syntology":{"atlas_url":"https://app.syntology.ai/?focus=1804.04272","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}