{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,2]],"date-time":"2026-07-02T00:33:06Z","timestamp":1782952386860,"version":"3.54.5"},"publisher-location":"Cham","reference-count":56,"publisher":"Springer International Publishing","isbn-type":[{"value":"9783030779566","type":"print"},{"value":"9783030779573","type":"electronic"}],"license":[{"start":{"date-parts":[[2021,1,1]],"date-time":"2021-01-01T00:00:00Z","timestamp":1609459200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2021,1,1]],"date-time":"2021-01-01T00:00:00Z","timestamp":1609459200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2021]]},"DOI":"10.1007\/978-3-030-77957-3_14","type":"book-chapter","created":{"date-parts":[[2021,6,26]],"date-time":"2021-06-26T22:02:27Z","timestamp":1624744947000},"page":"276-297","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Learning Physics from Data: A Thermodynamic Interpretation"],"prefix":"10.1007","author":[{"given":"Francisco","family":"Chinesta","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"El\u00edas","family":"Cueto","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Miroslav","family":"Grmela","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Beatriz","family":"Moya","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Michal","family":"Pavelka","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Martin","family":"\u0160\u00edpka","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2021,6,27]]},"reference":[{"key":"14_CR1","unstructured":"Gesamtausgabe, L.: Ludwig Boltzmann Gesamtausgabe - Collected Works (1983)"},{"key":"14_CR2","unstructured":"Gorban, A.N., Grechuk, B., Tyukin, I.Y.: Augmented artificial intelligence: a conceptual framework (2018)"},{"key":"14_CR3","first-page":"76","volume":"7","author":"DD Kosambi","year":"1943","unstructured":"Kosambi, D.D.: J. Indian Math. Soc. 7, 76 (1943)","journal-title":"J. Indian Math. Soc."},{"key":"14_CR4","unstructured":"Golub, G., Van\u00a0Loan, C.: Matrix Computations. Johns Hopkins Studies in the Mathematical Sciences. Johns Hopkins University Press (2013). https:\/\/books.google.cz\/books?id=X5YfsuCWpxMC"},{"issue":"5500","key":"14_CR5","doi-asserted-by":"publisher","first-page":"2323","DOI":"10.1126\/science.290.5500.2323","volume":"290","author":"ST Roweis","year":"2000","unstructured":"Roweis, S.T., Saul, L.K.: Science 290(5500), 2323 (2000)","journal-title":"Science"},{"issue":"1","key":"14_CR6","doi-asserted-by":"publisher","first-page":"501","DOI":"10.1146\/annurev-statistics-031017-100045","volume":"5","author":"L Wasserman","year":"2018","unstructured":"Wasserman, L.: Ann. Rev. Stat. Appl. 5(1), 501 (2018)","journal-title":"Ann. Rev. Stat. Appl."},{"key":"14_CR7","doi-asserted-by":"publisher","unstructured":"Brunton, S., Proctor, J., Kutz, J.: Proceedings of the National Academy of Sciences (2016). https:\/\/doi.org\/10.1073\/pnas.1517384113","DOI":"10.1073\/pnas.1517384113"},{"key":"14_CR8","doi-asserted-by":"crossref","unstructured":"Kaiser, E., Kutz, J., Brunton, S.: Discovering conservation laws from data for control (2018)","DOI":"10.1109\/CDC.2018.8618963"},{"issue":"9","key":"14_CR9","doi-asserted-by":"publisher","first-page":"4847","DOI":"10.4249\/scholarpedia.4847","volume":"5","author":"Y Kevrekidis","year":"2010","unstructured":"Kevrekidis, Y., Samaey, G.: Scholarpedia 5(9), 4847 (2010)","journal-title":"Scholarpedia"},{"key":"14_CR10","doi-asserted-by":"publisher","unstructured":"Weinan, E., Commun. Math. Stat. 5, 1 (2017). https:\/\/doi.org\/10.1007\/s40304-017-0103-z","DOI":"10.1007\/s40304-017-0103-z"},{"key":"14_CR11","doi-asserted-by":"publisher","unstructured":"Weinan, E., Han, J., Li, Q.: Res. Math. Sci. 6(1), 10 (2018). https:\/\/doi.org\/10.1007\/s40687-018-0172-y","DOI":"10.1007\/s40687-018-0172-y"},{"key":"14_CR12","doi-asserted-by":"publisher","unstructured":"Moya, B., Gonzalez, D., Alfaro, I., Chinesta, F., Cueto, E.: Comput. Mech. 64(2), 511 (2019). https:\/\/doi.org\/10.1007\/s00466-019-01705-3","DOI":"10.1007\/s00466-019-01705-3"},{"key":"14_CR13","unstructured":"Maes, C., Neto\u010dn\u00fd, K.: eprint arXiv:cond-mat\/0202501 (2002)"},{"key":"14_CR14","doi-asserted-by":"publisher","unstructured":"Grmela, M., \u00d6ttinger, H.C.: Phys. Rev. E 56, 6620 (1997). https:\/\/doi.org\/10.1103\/PhysRevE.56.6620","DOI":"10.1103\/PhysRevE.56.6620"},{"key":"14_CR15","doi-asserted-by":"publisher","first-page":"6633","DOI":"10.1103\/PhysRevE.56.6633","volume":"56","author":"HC \u00d6ttinger","year":"1997","unstructured":"\u00d6ttinger, H.C., Grmela, M.: Phys. Rev. E 56, 6633 (1997). https:\/\/doi.org\/10.1103\/PhysRevE.56.6633","journal-title":"Phys. Rev. E"},{"key":"14_CR16","doi-asserted-by":"crossref","unstructured":"\u00d6ttinger, H.: Beyond Equilibrium Thermodynamics. Wiley (2005)","DOI":"10.1002\/0471727903"},{"key":"14_CR17","doi-asserted-by":"publisher","unstructured":"Pavelka, M., Klika, V., Grmela, M.: Multiscale Thermo-Dynamics (De Gruyter, Berlin. Boston (2018). https:\/\/doi.org\/10.1515\/9783110350951. http:\/\/www.degruyter.com\/view\/books\/9783110350951\/9783110350951\/9783110350951.xml","DOI":"10.1515\/9783110350951"},{"key":"14_CR18","doi-asserted-by":"crossref","unstructured":"Pavelka, M., Klika, V., Grmela, M.: Phys. Rev. E 90 (2014)","DOI":"10.1103\/PhysRevE.90.062131"},{"key":"14_CR19","doi-asserted-by":"publisher","unstructured":"Onsager, L.: Phys. Rev. 37, 405 (1931). https:\/\/doi.org\/10.1103\/PhysRev.37.405. http:\/\/link.aps.org\/doi\/10.1103\/PhysRev.37.405","DOI":"10.1103\/PhysRev.37.405"},{"key":"14_CR20","doi-asserted-by":"publisher","first-page":"2265","DOI":"10.1103\/PhysRev.38.2265","volume":"38","author":"L Onsager","year":"1931","unstructured":"Onsager, L.: Phys. Rev. 38, 2265 (1931). https:\/\/doi.org\/10.1103\/PhysRev.38.2265","journal-title":"Phys. Rev."},{"key":"14_CR21","doi-asserted-by":"publisher","first-page":"343","DOI":"10.1103\/RevModPhys.17.343","volume":"17","author":"HBG Casimir","year":"1945","unstructured":"Casimir, H.B.G.: Rev. Mod. Phys. 17, 343 (1945). https:\/\/doi.org\/10.1103\/RevModPhys.17.343","journal-title":"Rev. Mod. Phys."},{"key":"14_CR22","unstructured":"de Groot, S.R., Mazur, P.: Non-equilibrium Thermodynamics. Dover Publications, New York (1984)"},{"key":"14_CR23","doi-asserted-by":"crossref","unstructured":"Grmela, M., Klika, V., Pavelka, M.: Phys. Rev. E 92 (2015)","DOI":"10.1103\/PhysRevE.92.032111"},{"issue":"2","key":"14_CR24","doi-asserted-by":"publisher","first-page":"282","DOI":"10.1007\/s10955-016-1694-6","volume":"166","author":"M Grmela","year":"2017","unstructured":"Grmela, M.: J. Stat. Phys. 166(2), 282 (2017)","journal-title":"J. Stat. Phys."},{"key":"14_CR25","doi-asserted-by":"crossref","unstructured":"Grmela, M.: J. Phys. Commun 2 (2018)","DOI":"10.1088\/2399-6528\/aab642"},{"key":"14_CR26","doi-asserted-by":"publisher","first-page":"379","DOI":"10.1002\/j.1538-7305.1948.tb01338.x","volume":"27","author":"CE Shannon","year":"1948","unstructured":"Shannon, C.E.: Bell Syst. Tech. J. 27, 379 (1948)","journal-title":"Bell Syst. Tech. J."},{"issue":"4","key":"14_CR27","doi-asserted-by":"publisher","first-page":"620","DOI":"10.1103\/PhysRev.106.620","volume":"106","author":"ET Jaynes","year":"1957","unstructured":"Jaynes, E.T.: Phys. Rev. 106(4), 620 (1957)","journal-title":"Phys. Rev."},{"key":"14_CR28","unstructured":"Gorban, A., Karlin, I.: Invariant Manifolds for Physical and Chemical Kinetics. Lecture Notes in Physics. Springer (2005). http:\/\/books.google.cz\/books?id=hjvjPmL5rPwC"},{"key":"14_CR29","doi-asserted-by":"publisher","first-page":"715","DOI":"10.3390\/e21070715","volume":"21","author":"V Klika","year":"2019","unstructured":"Klika, V., Pavelka, M., V\u00e1gner, P., Grmela, M.: Entropy 21, 715 (2019). https:\/\/doi.org\/10.3390\/e21070715","journal-title":"Entropy"},{"key":"14_CR30","unstructured":"Chapman, S., Cowling, T., Burnett, D., Cercignani, C.: The Mathematical Theory of Non-uniform Gases: An Account of the Kinetic Theory of Viscosity, Thermal Conduction and Diffusion in Gases. Cambridge Mathematical Library. Cambridge University Press (1990). https:\/\/books.google.cz\/books?id=Cbp5JP2OTrwC"},{"key":"14_CR31","doi-asserted-by":"crossref","unstructured":"Callen, H.: Thermodynamics: An Introduction to the Physical Theories of Equilibrium Thermostatics and Irreversible Thermodynamics. Wiley (1960). http:\/\/books.google.cz\/books?id=mf5QAAAAMAAJ","DOI":"10.1119\/1.1935945"},{"key":"14_CR32","doi-asserted-by":"publisher","first-page":"569","DOI":"10.1007\/s10955-013-0778-9","volume":"152","author":"B Turkington","year":"2013","unstructured":"Turkington, B.: J. Stat. Phys. 152, 569 (2013)","journal-title":"J. Stat. Phys."},{"key":"14_CR33","unstructured":"Pavelka, M., Klika, V., Grmela, M.: J. Stat. Phys. Accepted (2020)"},{"key":"14_CR34","unstructured":"Ehrenfest, P., Ehrenfest, T.: The Conceptual Foundations of the Statistical Approach in Mechanics. Dover Books on Physics. Dover Publications (1990)"},{"key":"14_CR35","doi-asserted-by":"crossref","unstructured":"Gorban, A.N., Karlin, I.V., \u00d6ttinger, H.C., Tatarinova, L.L.: Phys. Rev. E 63 (2001)","DOI":"10.1103\/PhysRevE.63.066124"},{"issue":"3\u20134","key":"14_CR36","doi-asserted-by":"publisher","first-page":"399","DOI":"10.1016\/S0378-4371(03)00510-7","volume":"327","author":"IV Karlin","year":"2003","unstructured":"Karlin, I.V., Tatarinova, L.L., Gorban, A.N., \u00d6ttinger, H.C.: Physica A: Stat. Mech. Appl. 327(3\u20134), 399 (2003)","journal-title":"Physica A: Stat. Mech. Appl."},{"key":"14_CR37","doi-asserted-by":"crossref","unstructured":"Pavelka, M., Klika, V., Grmela, M.: Entropy 20 (2018)","DOI":"10.3390\/e20060457"},{"key":"14_CR38","doi-asserted-by":"publisher","unstructured":"Pavelka, M., Klika, V., Grmela, M.: Physica D: Nonlinear Phenomena 399, 193 (2019). https:\/\/doi.org\/10.1016\/j.physd.2019.06.006. http:\/\/www.sciencedirect.com\/science\/article\/pii\/S0167278918305232","DOI":"10.1016\/j.physd.2019.06.006"},{"key":"14_CR39","doi-asserted-by":"publisher","unstructured":"Grmela, M.: Comput. Math. Appl. 65(10), 1457 (2013). https:\/\/doi.org\/10.1016\/j.camwa.2012.11.019. http:\/\/www.sciencedirect.com\/science\/article\/pii\/S0898122112006803","DOI":"10.1016\/j.camwa.2012.11.019"},{"issue":"3","key":"14_CR40","doi-asserted-by":"publisher","first-page":"1652","DOI":"10.3390\/e16031652","volume":"16","author":"M Grmela","year":"2014","unstructured":"Grmela, M.: Entropy 16(3), 1652 (2014). https:\/\/doi.org\/10.3390\/e16031652","journal-title":"Entropy"},{"key":"14_CR41","unstructured":"\u010capek, K.l.: R.U.R. (Rossum\u2019s Universal Robots). Penguin Books, London (2004)"},{"key":"14_CR42","first-page":"7","volume":"78","author":"A Chatterjee","year":"2000","unstructured":"Chatterjee, A.: Curr. Sci. 78, 7 (2000)","journal-title":"Curr. Sci."},{"key":"14_CR43","doi-asserted-by":"publisher","first-page":"79","DOI":"10.1016\/0024-3795(91)90021-N","volume":"146","author":"R Brockett","year":"1991","unstructured":"Brockett, R.: Linear Algebra Appl. 146, 79 (1991). https:\/\/doi.org\/10.1016\/0024-3795(91)90021-N","journal-title":"Linear Algebra Appl."},{"key":"14_CR44","doi-asserted-by":"crossref","unstructured":"Bloch,A.M., Krishnaprasad, P.S., Marsden, J.E., Ratiu, T.S.: Annales de l\u2019I.H.P. Analyse non lin\u00e9aire 11(1), 37 (1994)","DOI":"10.1016\/S0294-1449(16)30196-2"},{"issue":"4","key":"14_CR45","first-page":"291","volume":"2","author":"A Absil","year":"2006","unstructured":"Absil, A.: Int. J. Unconvent. Comput. 2(4), 291 (2006)","journal-title":"Int. J. Unconvent. Comput."},{"issue":"3","key":"14_CR46","doi-asserted-by":"publisher","first-page":"375","DOI":"10.1093\/mnras\/181.3.375","volume":"181","author":"R Gingold","year":"1977","unstructured":"Gingold, R., Monaghan, J.: Mon. Not. R. Astron. Soc. 181(3), 375 (1977)","journal-title":"Mon. Not. R. Astron. Soc."},{"key":"14_CR47","doi-asserted-by":"publisher","unstructured":"Espa\u00f1ol, P., Revenga, M.: Phys. Rev. E 67 (2003). https:\/\/doi.org\/10.1103\/PhysRevE.67.026705. https:\/\/link.aps.org\/doi\/10.1103\/PhysRevE.67.026705","DOI":"10.1103\/PhysRevE.67.026705"},{"issue":"1","key":"14_CR48","doi-asserted-by":"publisher","first-page":"103","DOI":"10.1007\/s10483-018-2255-6","volume":"39","author":"M Ellero","year":"2018","unstructured":"Ellero, M., Espa\u0148ol, P.: Appl. Math. Mech. 39(1), 103 (2018)","journal-title":"Appl. Math. Mech."},{"key":"14_CR49","doi-asserted-by":"publisher","unstructured":"Gonz\u00e1lez, D., Chinesta, F., Cueto, E.: Continuum Mechanics and Thermodynamics (2018). https:\/\/doi.org\/10.1007\/s00161-018-0677-z","DOI":"10.1007\/s00161-018-0677-z"},{"key":"14_CR50","doi-asserted-by":"publisher","first-page":"14","DOI":"10.3389\/fmats.2019.00014","volume":"6","author":"D Gonz\u00e1lez","year":"2019","unstructured":"Gonz\u00e1lez, D., Chinesta, F., Cueto, E.: Front. Mat. 6, 14 (2019)","journal-title":"Front. Mat."},{"key":"14_CR51","doi-asserted-by":"crossref","unstructured":"Brockett, R.: Linear Algebra Appl. 122-124, 761 (1989). Special Issue on Linear Systems and Control","DOI":"10.1016\/0024-3795(89)90675-7"},{"key":"14_CR52","unstructured":"Landau, L., Lifshitz, E.: Mechanics. Butterworth-Heinemann (1976)"},{"issue":"1","key":"14_CR53","doi-asserted-by":"publisher","first-page":"319","DOI":"10.5802\/aif.233","volume":"16","author":"V Arnold","year":"1966","unstructured":"Arnold, V.: Annales de l\u2019institut Fourier 16(1), 319 (1966)","journal-title":"Annales de l\u2019institut Fourier"},{"issue":"1","key":"14_CR54","doi-asserted-by":"publisher","first-page":"147","DOI":"10.2307\/1999527","volume":"281","author":"J Marsden","year":"1984","unstructured":"Marsden, J., Ratiu, T., Weinstein, A.: Trans. Am. Math. Soc. 281(1), 147 (1984). https:\/\/doi.org\/10.2307\/1999527","journal-title":"Trans. Am. Math. Soc."},{"key":"14_CR55","doi-asserted-by":"publisher","unstructured":"Simo, J.C., Marsden, J.E., Krishnaprasad, P.S.: Arch. Rat. Mech. Anal. 104(2), 125 (1988). https:\/\/doi.org\/10.1007\/BF00251673. https:\/\/doi.org\/10.1007\/BF00251673","DOI":"10.1007\/BF00251673"},{"key":"14_CR56","unstructured":"Pavelka, M., \u0160\u00edpka, M.: Machine-learning-rigid-body (2020). https:\/\/github.com\/enaipi\/machine-learning-rigid-body.git"}],"container-title":["Springer Proceedings in Mathematics &amp; Statistics","Geometric Structures of Statistical Physics, Information Geometry, and Learning"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/978-3-030-77957-3_14","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,6,26]],"date-time":"2021-06-26T22:05:25Z","timestamp":1624745125000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/978-3-030-77957-3_14"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021]]},"ISBN":["9783030779566","9783030779573"],"references-count":56,"URL":"https:\/\/doi.org\/10.1007\/978-3-030-77957-3_14","relation":{},"ISSN":["2194-1009","2194-1017"],"issn-type":[{"value":"2194-1009","type":"print"},{"value":"2194-1017","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021]]},"assertion":[{"value":"27 June 2021","order":1,"name":"first_online","label":"First Online","group":{"name":"ChapterHistory","label":"Chapter History"}},{"value":"SPIGL","order":1,"name":"conference_acronym","label":"Conference Acronym","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"Workshop on Joint Structures and Common Foundations of Statistical Physics, Information Geometry and Inference for Learning","order":2,"name":"conference_name","label":"Conference Name","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"Les Houches","order":3,"name":"conference_city","label":"Conference City","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"France","order":4,"name":"conference_country","label":"Conference Country","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"2020","order":5,"name":"conference_year","label":"Conference Year","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"27 July 2020","order":7,"name":"conference_start_date","label":"Conference Start Date","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"31 July 2020","order":8,"name":"conference_end_date","label":"Conference End Date","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"1","order":9,"name":"conference_number","label":"Conference Number","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"spigl2020","order":10,"name":"conference_id","label":"Conference ID","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"https:\/\/franknielsen.github.io\/SPIG-LesHouches2020\/","order":11,"name":"conference_url","label":"Conference URL","group":{"name":"ConferenceInfo","label":"Conference Information"}}]}}