{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,28]],"date-time":"2026-02-28T06:46:15Z","timestamp":1772261175782,"version":"3.50.1"},"reference-count":21,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2025,11,20]],"date-time":"2025-11-20T00:00:00Z","timestamp":1763596800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100004901","name":"FAPEMIG","doi-asserted-by":"crossref","award":["FAPEMIG-APQ-06611-24"],"award-info":[{"award-number":["FAPEMIG-APQ-06611-24"]}],"id":[{"id":"10.13039\/501100004901","id-type":"DOI","asserted-by":"crossref"}]},{"name":"CNPQ","award":["302182\/2022-5"],"award-info":[{"award-number":["302182\/2022-5"]}]},{"DOI":"10.13039\/501100004911","name":"FAPEPI","doi-asserted-by":"crossref","award":["00110.000202\/2022-28"],"award-info":[{"award-number":["00110.000202\/2022-28"]}],"id":[{"id":"10.13039\/501100004911","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>We investigate the critical properties of kinetic continuous opinion dynamics using deep learning techniques. The system consists of N continuous spin variables in the interval [\u22121,1]. Dense neural networks are trained on spin configuration data generated via kinetic Monte Carlo simulations, accurately identifying the critical point on both square and triangular lattices. Classical unsupervised learning with principal component analysis reproduces the magnetization and allows estimation of critical exponents. Additionally, variational autoencoders are implemented to study the phase transition through the loss function, which behaves as an order parameter. A correlation function between real and reconstructed data is defined and found to be universal at the critical point.<\/jats:p>","DOI":"10.3390\/e27111173","type":"journal-article","created":{"date-parts":[[2025,11,20]],"date-time":"2025-11-20T09:44:53Z","timestamp":1763631893000},"page":"1173","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Deep Learning of the Biswas\u2013Chatterjee\u2013Sen Model"],"prefix":"10.3390","volume":"27","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5738-9895","authenticated-orcid":false,"given":"Jos\u00e9 F. S.","family":"Neto","sequence":"first","affiliation":[{"name":"Departamento de Matem\u00e1tica e F\u00edsica, Universidade Estadual do Maranh\u00e3o, Caxias 65604-380, MA, Brazil"}]},{"given":"David S. M.","family":"Alencar","sequence":"additional","affiliation":[{"name":"Departamento de Matem\u00e1tica e F\u00edsica, Universidade Estadual do Maranh\u00e3o, Caxias 65604-380, MA, Brazil"}]},{"ORCID":"https:\/\/orcid.org\/0009-0006-3890-0495","authenticated-orcid":false,"given":"Lenilson T.","family":"Brito","sequence":"additional","affiliation":[{"name":"Departamento de F\u00edsica, Universidade Estadual do Piau\u00ed, Teresina 64002-150, PI, Brazil"}]},{"given":"Gladstone A.","family":"Alves","sequence":"additional","affiliation":[{"name":"Departamento de F\u00edsica, Universidade Estadual do Piau\u00ed, Teresina 64002-150, PI, Brazil"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0895-4186","authenticated-orcid":false,"given":"Francisco Welington S.","family":"Lima","sequence":"additional","affiliation":[{"name":"Departamento de F\u00edsica, Universidade Federal do Piau\u00ed, Teresina 57072-970, PI, Brazil"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9419-5595","authenticated-orcid":false,"given":"Ant\u00f4nio M.","family":"Filho","sequence":"additional","affiliation":[{"name":"Departamento de F\u00edsica, Universidade Estadual do Piau\u00ed, Teresina 64002-150, PI, Brazil"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7623-1361","authenticated-orcid":false,"given":"Ronan S.","family":"Ferreira","sequence":"additional","affiliation":[{"name":"Departamento de Ci\u00eancias Exatas e Aplicadas, Universidade Federal de Ouro Preto, Jo\u00e3o Monlevade 35931-008, MG, Brazil"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9937-7980","authenticated-orcid":false,"given":"Tayroni F. A.","family":"Alves","sequence":"additional","affiliation":[{"name":"Departamento de F\u00edsica, Universidade Federal do Piau\u00ed, Teresina 57072-970, PI, Brazil"}]}],"member":"1968","published-online":{"date-parts":[[2025,11,20]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"3257","DOI":"10.1016\/j.physa.2012.01.046","article-title":"Disorder Induced Phase Transition in Kinetic Models of Opinion Dynamics","volume":"391","author":"Biswas","year":"2012","journal-title":"Phys. A"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"062317","DOI":"10.1103\/PhysRevE.94.062317","article-title":"Disorder-Induced Phase Transition in an Opinion Dynamics Model: Results in Two and Three Dimensions","volume":"94","author":"Mukherjee","year":"2016","journal-title":"Phys. Rev. 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