{"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/geometric-deep-learning-grids-groups-graphs","title":"Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges","arxiv_id":"2104.13478","date":"2021-04-27","proceeding":null,"authors":["Michael M. Bronstein","Joan Bruna","Taco Cohen","Petar Veličković"],"abstract":"The last decade has witnessed an experimental revolution in data science and machine learning, epitomised by deep learning methods. Indeed, many high-dimensional learning tasks previously thought to be beyond reach -- such as computer vision, playing Go, or protein folding -- are in fact feasible with appropriate computational scale. Remarkably, the essence of deep learning is built from two simple algorithmic principles: first, the notion of representation or feature learning, whereby adapted, often hierarchical, features capture the appropriate notion of regularity for each task, and second, learning by local gradient-descent type methods, typically implemented as backpropagation. While learning generic functions in high dimensions is a cursed estimation problem, most tasks of interest are not generic, and come with essential pre-defined regularities arising from the underlying low-dimensionality and structure of the physical world. This text is concerned with exposing these regularities through unified geometric principles that can be applied throughout a wide spectrum of applications. Such a 'geometric unification' endeavour, in the spirit of Felix Klein's Erlangen Program, serves a dual purpose: on one hand, it provides a common mathematical framework to study the most successful neural network architectures, such as CNNs, RNNs, GNNs, and Transformers. On the other hand, it gives a constructive procedure to incorporate prior physical knowledge into neural architectures and provide principled way to build future architectures yet to be invented.","url_abs":"https://arxiv.org/abs/2104.13478v2","url_pdf":"https://arxiv.org/pdf/2104.13478v2.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":"geometric-deep-learning-grids-groups-graphs","repo_url":"https://github.com/BojanFaletic/IQ_net","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"pytorch","reach":{"status":"ok"}},{"paper_slug":"geometric-deep-learning-grids-groups-graphs","repo_url":"https://github.com/BrianYang2013/JHU_AI_Journey","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"pytorch","reach":{"status":"ok","spdx":"GPL-2.0"}},{"paper_slug":"geometric-deep-learning-grids-groups-graphs","repo_url":"https://github.com/GuangfuWang/geometric-deep-learning-chinese-translation-scripts","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":{"status":"ok","spdx":"MIT"}},{"paper_slug":"geometric-deep-learning-grids-groups-graphs","repo_url":"https://github.com/elias-sundqvist/obsidian-annotator","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":{"status":"ok","spdx":"AGPL-3.0"}},{"paper_slug":"geometric-deep-learning-grids-groups-graphs","repo_url":"https://github.com/subhobrata/Courses_ML_DL4","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":{"status":"ok"}},{"paper_slug":"geometric-deep-learning-grids-groups-graphs","repo_url":"https://github.com/AIMPer73/Geometric-Deep-Learning-Grids-Groups-Graphs-Geodesics-and-Gauges","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":0,"framework":"none","reach":{"status":"ok"}}],"tasks":[{"task_slug":"deep-learning","task_name":"Deep Learning"},{"task_slug":"protein-folding","task_name":"Protein Folding"}],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":"https://app.syntology.ai/?focus=2104.13478","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}