Skip to main content
SpringerLink
Log in

Learning Physics from Data: A Thermodynamic Interpretation

  • Conference paper
  • First Online:
Geometric Structures of Statistical Physics, Information Geometry, and Learning (SPIGL 2020)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 361))

Abstract

Experimental data bases are typically very large and high dimensional. To learn from them requires to recognize important features (a pattern), often present at scales different to that of the recorded data. Following the experience collected in statistical mechanics and thermodynamics, the process of recognizing the pattern (the learning process) can be seen as a dissipative time evolution driven by entropy from a detailed level of description to less detailed. This is the way thermodynamics enters machine learning. On the other hand, reversible (typically Hamiltonian) evolution is propagation within the levels of description, that is also to be recognized. This is how Poisson geometry enters machine learning. Learning to handle free surface liquids and damped rigid body rotation serves as an illustration.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 219.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 279.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 279.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    In this work we mean pattern recognition in a broad sense as a process of extracting any information from the data. The dissipative-driven pattern recognition can be then imagined as a retouche of the original data leading to recognition of the important aspects.

  2. 2.

    The Poisson bracket corresponding to the Poisson bivector is \(\{F,G\}=\langle F_x| {^\uparrow \mathbf {L}}| G_x \rangle \), where \(\langle \bullet |\bullet \rangle \) denotes a scalar product.

  3. 3.

    The more detailed level is referred to as the upper while the less detailed (reduced) as lower.

  4. 4.

    Another dynamical system converging to eigenvalues of a matrix was found in [43], where the double bracket dissipation, geometrized in [44], was found.

  5. 5.

    It is often assumed that the reduced manifold keeps the structure of a cotangent bundle, such that a reversible evolution is generated by the canonical Poisson bivector (equipped with entropy) as on the original manifold. Therefore, the reduced dynamics can be interpreted as dynamics of a lower number of (quasi-)particles, since otherwise an another Poisson bivector would have to be sought. This is not, however, strictly necessary nor a limitation of the method, see for instance [49, 50].

  6. 6.

    Corresponding to dissipation potential \(\Xi =\frac{1}{2} y^*_a {{^\downarrow }M}^{ab} y^*_b\).

References

  1. Gesamtausgabe, L.: Ludwig Boltzmann Gesamtausgabe - Collected Works (1983)

    Google Scholar 

  2. Gorban, A.N., Grechuk, B., Tyukin, I.Y.: Augmented artificial intelligence: a conceptual framework (2018)

    Google Scholar 

  3. Kosambi, D.D.: J. Indian Math. Soc. 7, 76 (1943)

    MathSciNet  Google Scholar 

  4. Golub, G., Van Loan, C.: Matrix Computations. Johns Hopkins Studies in the Mathematical Sciences. Johns Hopkins University Press (2013). https://books.google.cz/books?id=X5YfsuCWpxMC

  5. Roweis, S.T., Saul, L.K.: Science 290(5500), 2323 (2000)

    Article  Google Scholar 

  6. Wasserman, L.: Ann. Rev. Stat. Appl. 5(1), 501 (2018)

    Article  Google Scholar 

  7. Brunton, S., Proctor, J., Kutz, J.: Proceedings of the National Academy of Sciences (2016). https://doi.org/10.1073/pnas.1517384113

  8. Kaiser, E., Kutz, J., Brunton, S.: Discovering conservation laws from data for control (2018)

    Google Scholar 

  9. Kevrekidis, Y., Samaey, G.: Scholarpedia 5(9), 4847 (2010)

    Article  Google Scholar 

  10. Weinan, E., Commun. Math. Stat. 5, 1 (2017). https://doi.org/10.1007/s40304-017-0103-z

  11. Weinan, E., Han, J., Li, Q.: Res. Math. Sci. 6(1), 10 (2018). https://doi.org/10.1007/s40687-018-0172-y

  12. 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

  13. Maes, C., Netočný, K.: eprint arXiv:cond-mat/0202501 (2002)

  14. Grmela, M., Öttinger, H.C.: Phys. Rev. E 56, 6620 (1997). https://doi.org/10.1103/PhysRevE.56.6620

  15. Öttinger, H.C., Grmela, M.: Phys. Rev. E 56, 6633 (1997). https://doi.org/10.1103/PhysRevE.56.6633

    Article  MathSciNet  Google Scholar 

  16. Öttinger, H.: Beyond Equilibrium Thermodynamics. Wiley (2005)

    Google Scholar 

  17. 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

  18. Pavelka, M., Klika, V., Grmela, M.: Phys. Rev. E 90 (2014)

    Google Scholar 

  19. 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

  20. Onsager, L.: Phys. Rev. 38, 2265 (1931). https://doi.org/10.1103/PhysRev.38.2265

    Article  Google Scholar 

  21. Casimir, H.B.G.: Rev. Mod. Phys. 17, 343 (1945). https://doi.org/10.1103/RevModPhys.17.343

    Article  Google Scholar 

  22. de Groot, S.R., Mazur, P.: Non-equilibrium Thermodynamics. Dover Publications, New York (1984)

    Google Scholar 

  23. Grmela, M., Klika, V., Pavelka, M.: Phys. Rev. E 92 (2015)

    Google Scholar 

  24. Grmela, M.: J. Stat. Phys. 166(2), 282 (2017)

    Article  MathSciNet  Google Scholar 

  25. Grmela, M.: J. Phys. Commun 2 (2018)

    Google Scholar 

  26. Shannon, C.E.: Bell Syst. Tech. J. 27, 379 (1948)

    Article  Google Scholar 

  27. Jaynes, E.T.: Phys. Rev. 106(4), 620 (1957)

    Article  MathSciNet  Google Scholar 

  28. Gorban, A., Karlin, I.: Invariant Manifolds for Physical and Chemical Kinetics. Lecture Notes in Physics. Springer (2005). http://books.google.cz/books?id=hjvjPmL5rPwC

  29. Klika, V., Pavelka, M., Vágner, P., Grmela, M.: Entropy 21, 715 (2019). https://doi.org/10.3390/e21070715

    Article  Google Scholar 

  30. 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

  31. Callen, H.: Thermodynamics: An Introduction to the Physical Theories of Equilibrium Thermostatics and Irreversible Thermodynamics. Wiley (1960). http://books.google.cz/books?id=mf5QAAAAMAAJ

  32. Turkington, B.: J. Stat. Phys. 152, 569 (2013)

    Article  MathSciNet  Google Scholar 

  33. Pavelka, M., Klika, V., Grmela, M.: J. Stat. Phys. Accepted (2020)

    Google Scholar 

  34. Ehrenfest, P., Ehrenfest, T.: The Conceptual Foundations of the Statistical Approach in Mechanics. Dover Books on Physics. Dover Publications (1990)

    Google Scholar 

  35. Gorban, A.N., Karlin, I.V., Öttinger, H.C., Tatarinova, L.L.: Phys. Rev. E 63 (2001)

    Google Scholar 

  36. Karlin, I.V., Tatarinova, L.L., Gorban, A.N., Öttinger, H.C.: Physica A: Stat. Mech. Appl. 327(3–4), 399 (2003)

    Article  Google Scholar 

  37. Pavelka, M., Klika, V., Grmela, M.: Entropy 20 (2018)

    Google Scholar 

  38. 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

  39. 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

  40. Grmela, M.: Entropy 16(3), 1652 (2014). https://doi.org/10.3390/e16031652

    Article  MathSciNet  Google Scholar 

  41. Čapek, K.l.: R.U.R. (Rossum’s Universal Robots). Penguin Books, London (2004)

    Google Scholar 

  42. Chatterjee, A.: Curr. Sci. 78, 7 (2000)

    Google Scholar 

  43. Brockett, R.: Linear Algebra Appl. 146, 79 (1991). https://doi.org/10.1016/0024-3795(91)90021-N

    Article  MathSciNet  Google Scholar 

  44. Bloch,A.M., Krishnaprasad, P.S., Marsden, J.E., Ratiu, T.S.: Annales de l’I.H.P. Analyse non linéaire 11(1), 37 (1994)

    Google Scholar 

  45. Absil, A.: Int. J. Unconvent. Comput. 2(4), 291 (2006)

    Google Scholar 

  46. Gingold, R., Monaghan, J.: Mon. Not. R. Astron. Soc. 181(3), 375 (1977)

    Article  Google Scholar 

  47. Español, 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

  48. Ellero, M., Espaňol, P.: Appl. Math. Mech. 39(1), 103 (2018)

    Article  MathSciNet  Google Scholar 

  49. González, D., Chinesta, F., Cueto, E.: Continuum Mechanics and Thermodynamics (2018). https://doi.org/10.1007/s00161-018-0677-z

  50. González, D., Chinesta, F., Cueto, E.: Front. Mat. 6, 14 (2019)

    Article  Google Scholar 

  51. Brockett, R.: Linear Algebra Appl. 122-124, 761 (1989). Special Issue on Linear Systems and Control

    Google Scholar 

  52. Landau, L., Lifshitz, E.: Mechanics. Butterworth-Heinemann (1976)

    Google Scholar 

  53. Arnold, V.: Annales de l’institut Fourier 16(1), 319 (1966)

    Article  MathSciNet  Google Scholar 

  54. Marsden, J., Ratiu, T., Weinstein, A.: Trans. Am. Math. Soc. 281(1), 147 (1984). https://doi.org/10.2307/1999527

    Article  Google Scholar 

  55. 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

  56. Pavelka, M., Šípka, M.: Machine-learning-rigid-body (2020). https://github.com/enaipi/machine-learning-rigid-body.git

Download references

Acknowledgements

We are grateful to Václav Klika for discussing the manuscript.

F.Ch. thanks ESI Group through its research chair at “Arts et Métiers ParisTech”, whose first invited position was Prof. M. Grmela, for performing the researches here addressed. F. Ch. also knowledges Dr. Alain de Rouvray by the rich and inspiring discussions on pattern recognition as the first step towards machine learning and artificial intelligence, motivating the preset work. The support from ANR (Agence Nationale de la Recherche, France) through its grant AAPG2018 DataBEST is also gratefully acknowledged.

E.C. also acknowledges the financial support of ESI Group through the project “Simulated Reality”. The support given by the Spanish Ministry of Economy and Competitiveness through Grant number DPI2017-85139-C2-1-R, and by the Regional Government of Aragon and the European Social Fund, research group T88, is also greatly acknowledged.

M.G. was supported by the Natural Sciences and Engineering Research Council of Canada, Grants 3100319 and 3100735.

B.M. acknowledges the support of the Spanish Ministry of Science, Innovation and Universities through grant number PRE2018-083211.

M.P. and M.Š were supported by Czech Science Foundation, Project No. 20-22092S. M.P. was supported by Charles University Research Program No. UNCE/SCI/023.

Author information

Authors and Affiliations

  1. ESI Group Chair @ PIMM Lab, Arts et Metiers Institute of Technology, 155 Boulevard de l’Hopital, 75013, Paris, France

    Francisco Chinesta

  2. Aragon Institute of Engineering Research, Universidad de Zaragoza, Maria de Luna, s.n., 50018, Zaragoza, Spain

    Elías Cueto & Beatriz Moya

  3. École Polytechnique de Montréal, C.P. 6079 succ. Centre-ville, Montréal, Québec, H3C 3A7, Canada

    Miroslav Grmela

  4. Mathematical Institute, Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 75, Prague, Czech Republic

    Michal Pavelka & Martin Šípka

Authors
  1. Francisco Chinesta
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Elías Cueto
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Miroslav Grmela
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Beatriz Moya
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Michal Pavelka
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Martin Šípka
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Francisco Chinesta .

Editor information

Editors and Affiliations

  1. Thales Land & Air Systems, Technical Directorate, Thales, Limours, France

    Frédéric Barbaresco

  2. Sony Computer Science Laboratories Inc., Tokyo, Japan

    Frank Nielsen

Rights and permissions

Reprints and permissions

Copyright information

© 2021 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Chinesta, F., Cueto, E., Grmela, M., Moya, B., Pavelka, M., Šípka, M. (2021). Learning Physics from Data: A Thermodynamic Interpretation. In: Barbaresco, F., Nielsen, F. (eds) Geometric Structures of Statistical Physics, Information Geometry, and Learning. SPIGL 2020. Springer Proceedings in Mathematics & Statistics, vol 361. Springer, Cham. https://doi.org/10.1007/978-3-030-77957-3_14

Download citation

Publish with us

Policies and ethics

Search

Navigation