{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,8]],"date-time":"2026-08-08T13:03:59Z","timestamp":1786194239404,"version":"build-2736575974"},"reference-count":40,"publisher":"American Physical Society (APS)","issue":"5","license":[{"start":{"date-parts":[[2024,2,1]],"date-time":"2024-02-01T00:00:00Z","timestamp":1706745600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100004063","name":"Knut och Alice Wallenbergs Stiftelse","doi-asserted-by":"publisher","award":["2019.0079"],"award-info":[{"award-number":["2019.0079"]}],"id":[{"id":"10.13039\/501100004063","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100004359","name":"Vetenskapsr\u00e5det","doi-asserted-by":"publisher","award":["2021-4452"],"award-info":[{"award-number":["2021-4452"]}],"id":[{"id":"10.13039\/501100004359","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100004359","name":"Vetenskapsr\u00e5det","doi-asserted-by":"publisher","award":["2021-4452"],"award-info":[{"award-number":["2021-4452"]}],"id":[{"id":"10.13039\/501100004359","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100004359","name":"Vetenskapsr\u00e5det","doi-asserted-by":"publisher","award":["2018-03974"],"award-info":[{"award-number":["2018-03974"]}],"id":[{"id":"10.13039\/501100004359","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100013371","name":"Universit\u00e9 C\u00f4te d\u2019Azur","doi-asserted-by":"publisher","award":["ANR-15-IDEX-01"],"award-info":[{"award-number":["ANR-15-IDEX-01"]}],"id":[{"id":"10.13039\/501100013371","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["journals.aps.org"],"crossmark-restriction":true},"short-container-title":["Phys. Rev. Lett."],"accepted":{"date-parts":[[2024,1,3]]},"abstract":"<jats:p>We compute how small input perturbations affect the output of deep neural networks, exploring an analogy between deep feed-forward networks and dynamical systems, where the growth or decay of local perturbations is characterized by finite-time Lyapunov exponents. We show that the maximal exponent forms geometrical structures in input space, akin to coherent structures in dynamical systems. Ridges of large positive exponents divide input space into different regions that the network associates with different classes. These ridges visualize the geometry that deep networks construct in input space, shedding light on the fundamental mechanisms underlying their learning capabilities.<\/jats:p>\n          <jats:sec>\n            <jats:title\/>\n            <jats:supplementary-material>\n              <jats:permissions>\n                <jats:copyright-statement>Published by the American Physical Society<\/jats:copyright-statement>\n                <jats:copyright-year>2024<\/jats:copyright-year>\n              <\/jats:permissions>\n            <\/jats:supplementary-material>\n          <\/jats:sec>","DOI":"10.1103\/physrevlett.132.057301","type":"journal-article","created":{"date-parts":[[2024,2,1]],"date-time":"2024-02-01T18:34:22Z","timestamp":1706812462000},"update-policy":"https:\/\/doi.org\/10.1103\/crossmark-policy","source":"Crossref","is-referenced-by-count":20,"title":["Finite-Time Lyapunov Exponents of Deep Neural Networks"],"prefix":"10.1103","volume":"132","author":[{"given":"L.","family":"Storm","sequence":"first","affiliation":[{"name":"Department of Physics, University of Gothenburg, 41296 Gothenburg, Sweden"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"H.","family":"Linander","sequence":"additional","affiliation":[{"name":"Department of Mathematical Sciences, Chalmers Technical University and University of Gothenburg, Gothenburg, Sweden"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"J.","family":"Bec","sequence":"additional","affiliation":[{"name":"MINES Paris, PSL Research University, CNRS, Cemef, Valbonne, F-06900, France"},{"name":"Universit\u00e9 C\u00f4te d\u2019Azur, Inria, CNRS, Cemef, Valbonne, F-06900, France"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"K.","family":"Gustavsson","sequence":"additional","affiliation":[{"name":"Department of Physics, University of Gothenburg, 41296 Gothenburg, Sweden"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3672-6538","authenticated-orcid":true,"given":"B.","family":"Mehlig","sequence":"additional","affiliation":[{"name":"Department of Physics, University of Gothenburg, 41296 Gothenburg, Sweden"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"16","published-online":{"date-parts":[[2024,2,1]]},"reference":[{"key":"PhysRevLett.132.057301Cc1R1","first-page":"3360","volume":"29","author":"B. Poole","year":"2016","journal-title":"Adv. Neural Inf. Process. Syst.","ISSN":"https:\/\/id.crossref.org\/issn\/1049-5258","issn-type":"print"},{"key":"PhysRevLett.132.057301Cc2R1","doi-asserted-by":"publisher","DOI":"10.1038\/s41586-021-03819-2"},{"key":"PhysRevLett.132.057301Cc3R1","volume-title":"International Conference on Machine Learning","author":"M. Tan","year":"2019"},{"key":"PhysRevLett.132.057301Cc4R1","first-page":"8580","volume":"31","author":"A. Jacot","year":"2018","journal-title":"Adv. Neural Inf. Process. Syst.","ISSN":"https:\/\/id.crossref.org\/issn\/1049-5258","issn-type":"print"},{"key":"PhysRevLett.132.057301Cc5R1","doi-asserted-by":"publisher","DOI":"10.1016\/j.physrep.2021.04.001"},{"key":"PhysRevLett.132.057301Cc7R1","doi-asserted-by":"publisher","DOI":"10.1038\/s41467-023-36361-y"},{"key":"PhysRevLett.132.057301Cc8R1","volume-title":"International Conference on Artificial Intelligence and Statistics","author":"A. Baratin","year":"2021"},{"key":"PhysRevLett.132.057301Cc9R1","doi-asserted-by":"publisher","DOI":"10.1088\/1742-5468\/abf1f3"},{"key":"PhysRevLett.132.057301Cc10R1","doi-asserted-by":"publisher","DOI":"10.1038\/s42256-023-00767-6"},{"key":"PhysRevLett.132.057301Cc11R1","doi-asserted-by":"crossref","DOI":"10.1017\/9781108860604","volume-title":"Machine Learning with Neural Networks: An Introduction for Scientists and Engineers","author":"B. Mehlig","year":"2021"},{"key":"PhysRevLett.132.057301Cc12R1","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9780511803260","volume-title":"Chaos in Dynamical Systems","author":"E. Ott","year":"2002"},{"key":"PhysRevLett.132.057301Cc13R1","volume-title":"Chaos: Classical and Quantum","author":"P. Cvitanovi\u0107","year":"2020"},{"key":"PhysRevLett.132.057301Cc14R1","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-642-84942-8","volume-title":"Products of Random Matrices in Statistical Physics","author":"A. Crisanti","year":"1993"},{"key":"PhysRevLett.132.057301Cc15R1","first-page":"4785","volume":"30","author":"J. Pennington","year":"2017","journal-title":"Adv. Neural Inf. Process. Syst.","ISSN":"https:\/\/id.crossref.org\/issn\/1049-5258","issn-type":"print"},{"key":"PhysRevLett.132.057301Cc16R1","volume-title":"Proceedings of the 30th International Conference on Machine Learning\u2014Volume 28","author":"I. Sutskever","year":"2013"},{"key":"PhysRevLett.132.057301Cc17R1","volume-title":"Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics","author":"X. Glorot","year":"2010"},{"key":"PhysRevLett.132.057301Cc18R1","first-page":"15847","volume":"33","author":"S. Chen","year":"2020","journal-title":"Adv. Neural Inf. Process. Syst.","ISSN":"https:\/\/id.crossref.org\/issn\/1049-5258","issn-type":"print"},{"key":"PhysRevLett.132.057301Cc19R1","first-page":"2933","volume":"32","author":"L. Chizat","year":"2019","journal-title":"Adv. Neural Inf. Process. Syst.","ISSN":"https:\/\/id.crossref.org\/issn\/1049-5258","issn-type":"print"},{"key":"PhysRevLett.132.057301Cc20R1","doi-asserted-by":"publisher","DOI":"10.1016\/S0167-2789(00)00142-1"},{"key":"PhysRevLett.132.057301Cc21R1","doi-asserted-by":"publisher","DOI":"10.1088\/1361-6544\/aa6b11"},{"key":"PhysRevLett.132.057301Cc22R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevResearch.2.033258"},{"key":"PhysRevLett.132.057301Cc25R1","volume-title":"The Non-Linear Field Theories of Mechanics","author":"C. Truesdell","year":"1965"},{"key":"PhysRevLett.132.057301Cc26R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.86.134303"},{"key":"PhysRevLett.132.057301Cc27R1","doi-asserted-by":"publisher","DOI":"10.1016\/j.measurement.2020.108855"},{"key":"PhysRevLett.132.057301Cc28R1","doi-asserted-by":"publisher","DOI":"10.1146\/annurev-fluid-010313-141322"},{"key":"PhysRevLett.132.057301Cc29R1","doi-asserted-by":"publisher","DOI":"10.1063\/1.4979042"},{"key":"PhysRevLett.132.057301Cc30R1","doi-asserted-by":"publisher","DOI":"10.1016\/j.patter.2021.100226"},{"key":"PhysRevLett.132.057301Cc31R1","doi-asserted-by":"publisher","DOI":"10.1016\/j.plrev.2019.09.004"},{"key":"PhysRevLett.132.057301Cc32R1","doi-asserted-by":"publisher","DOI":"10.1017\/jfm.2014.32"},{"key":"PhysRevLett.132.057301Cc33R1","doi-asserted-by":"publisher","DOI":"10.1209\/0295-5075\/89\/34003"},{"key":"PhysRevLett.132.057301Cc34R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevFluids.2.014605"},{"key":"PhysRevLett.132.057301Cc35R1","doi-asserted-by":"publisher","DOI":"10.1017\/S002211200800267X"},{"key":"PhysRevLett.132.057301Cc38R1","doi-asserted-by":"publisher","DOI":"10.1103\/RevModPhys.57.617"},{"key":"PhysRevLett.132.057301Cc39R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.91.254101"},{"key":"PhysRevLett.132.057301Cc41R1","doi-asserted-by":"publisher","DOI":"10.1109\/PGEC.1965.264137"},{"key":"PhysRevLett.132.057301Cc42R1","volume-title":"2012 IEEE Conference on Computer Vision and Pattern Recognition","author":"D. Ciregan","year":"2012"},{"key":"PhysRevLett.132.057301Cc43R1","doi-asserted-by":"publisher","DOI":"10.1016\/j.neucom.2015.08.104"},{"key":"PhysRevLett.132.057301Cc45R1","first-page":"6402","volume":"30","author":"B. Lakshminarayanan","year":"2017","journal-title":"Adv. Neural Inf. Process. Syst.","ISSN":"https:\/\/id.crossref.org\/issn\/1049-5258","issn-type":"print"},{"key":"PhysRevLett.132.057301Cc47R1","volume-title":"International Conference on Learning Representations","author":"A. Madry","year":"2018"},{"key":"PhysRevLett.132.057301Cc48R1","volume-title":"Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition","author":"K. He","year":"2016"}],"container-title":["Physical Review Letters"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.aps.org\/article\/10.1103\/PhysRevLett.132.057301","content-type":"unspecified","content-version":"vor","intended-application":"syndication"},{"URL":"http:\/\/harvest.aps.org\/v2\/journals\/articles\/10.1103\/PhysRevLett.132.057301\/fulltext","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,1,3]],"date-time":"2025-01-03T01:11:24Z","timestamp":1735866684000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.aps.org\/doi\/10.1103\/PhysRevLett.132.057301"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,2,1]]},"references-count":40,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2024,2]]}},"URL":"https:\/\/doi.org\/10.1103\/physrevlett.132.057301","relation":{},"ISSN":["0031-9007","1079-7114"],"issn-type":[{"value":"0031-9007","type":"print"},{"value":"1079-7114","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,2,1]]},"article-number":"057301"}}