{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,10]],"date-time":"2026-02-10T18:49:33Z","timestamp":1770749373705,"version":"3.50.0"},"reference-count":22,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2020,12,23]],"date-time":"2020-12-23T00:00:00Z","timestamp":1608681600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>We provide a rigorous mathematical formulation of Deep Learning (DL) methodologies through an in-depth analysis of the learning procedures characterizing Neural Network (NN) models within the theoretical frameworks of Stochastic Optimal Control (SOC) and Mean-Field Games (MFGs). In particular, we show how the supervised learning approach can be translated in terms of a (stochastic) mean-field optimal control problem by applying the Hamilton\u2013Jacobi\u2013Bellman (HJB) approach and the mean-field Pontryagin maximum principle. Our contribution sheds new light on a possible theoretical connection between mean-field problems and DL, melting heterogeneous approaches and reporting the state-of-the-art within such fields to show how the latter different perspectives can be indeed fruitfully unified.<\/jats:p>","DOI":"10.3390\/sym13010014","type":"journal-article","created":{"date-parts":[[2020,12,23]],"date-time":"2020-12-23T12:19:51Z","timestamp":1608725991000},"page":"14","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["Deep Learning and Mean-Field Games: A Stochastic Optimal Control Perspective"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0317-4351","authenticated-orcid":false,"given":"Luca Di","family":"Persio","sequence":"first","affiliation":[{"name":"Department of Computer Science, University of Verona, 37134 Verona, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1642-0846","authenticated-orcid":false,"given":"Matteo","family":"Garbelli","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Trento, 38123 Trento, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,12,23]]},"reference":[{"key":"ref_1","unstructured":"Cardaliguet, P. 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The Theory of Differential Equations: Classical and Qualitative, Springer Science and Business Media.","DOI":"10.1007\/978-1-4419-5783-2"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/1\/14\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T10:49:03Z","timestamp":1760179743000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/1\/14"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,12,23]]},"references-count":22,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2021,1]]}},"alternative-id":["sym13010014"],"URL":"https:\/\/doi.org\/10.3390\/sym13010014","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,12,23]]}}}