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Emergence of nonequilibrium latent cycles in unsupervised generative modeling

Marco Baiesi1,2 and Alberto Rosso3

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Phys. Rev. E 114, 015301 – Published 7 July, 2026

DOI: https://doi.org/10.1103/8gqs-fkh6

Abstract

We show that nonequilibrium dynamics can play a constructive role in unsupervised machine learning by inducing the spontaneous emergence of latent-state cycles. We introduce a model in which visible and hidden variables interact through two independently parametrized transition matrices, defining a Markov chain whose steady state is intrinsically out of equilibrium. Likelihood maximization drives this system toward nonequilibrium steady states with finite entropy production, reduced self-transition probabilities, and persistent probability currents in the latent space. These cycles are not imposed by the architecture but arise from training, and models that develop them reproduce the empirical distribution of data classes more faithfully, with a clear correlation between agreement with the data and entropy production. Compared with equilibrium approaches such as restricted Boltzmann machines, our model breaks the detailed balance between the forward and backward conditional transitions and relies on a log-likelihood gradient that depends explicitly on the last two steps of the Markov chain. Hence, this exploration of the interface between nonequilibrium statistical physics and modern machine learning suggests that introducing irreversibility into latent-variable models can improve the fidelity of the generated data distribution.

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References (42)

  1. P. Mehta, M. Bukov, C.-H. Wang, A. G. Day, C. Richardson, C. K. Fisher, and D. J. Schwab, A high-bias, low-variance introduction to machine learning for physicists, Phys. Rep. 810, 1 (2019).
  2. P. Smolensky, Information Processing in Dynamical Systems: Foundations of Harmony Theory, in Parallel Distributed Processing, edited by D. E. Rumelhart and J. L. McClelland (MIT Press, Cambridge, 1986), Vol. 1, pp. 194–281.
  3. G. E. Hinton, A practical guide to training restricted Boltzmann machines, Neural Networks: Tricks of the Trade, 2nd ed. (Springer, Berlin, 2012), pp. 599–619.
  4. C. Malbranke, D. Bikard, S. Cocco, R. Monasson, and J. Tubiana, Machine learning for evolutionary-based and physics-inspired protein design: Current and future synergies, Curr. Opin. Struct. Biol. 80, 102571 (2023).
  5. J. Fernandez-de Cossio-Diaz, P. Hardouin, F.-X. Lyonnet du Moutier, A. Di Gioacchino, B. Marchand, Y. Ponty, B. Sargueil, R. Monasson, and S. Cocco, Designing molecular RNA switches with restricted Boltzmann machines, Nat. Commun. 16, 11223 (2025).
  6. A. Decelle, B. Seoane, and L. Rosset, Unsupervised hierarchical clustering using the learning dynamics of restricted Boltzmann machines, Phys. Rev. E 108, 014110 (2023).
  7. A. Braghetto, E. Orlandini, and M. Baiesi, Interpretable machine learning of amino acid patterns in proteins: A statistical ensemble approach, J. Chem. Theory Comput. 19, 6011 (2023).
  8. C. Roussel, S. Cocco, and R. Monasson, Barriers and dynamical paths in alternating Gibbs sampling of restricted Boltzmann machines, Phys. Rev. E 104, 034109 (2021).
  9. E. Agoritsas, G. Catania, A. Decelle, and B. Seoane, Explaining the effects of non-convergent MCMC in the training of energy-based models, in International Conference on Machine Learning, Honolulu, HI (PMLR, 2023), pp. 322–336.
  10. N. Béreux, A. Decelle, C. Furtlehner, L. Rosset, and B. Seoane, Fast training and sampling of restricted Boltzmann machines, in Proceedings of the 13th International Conference on Learning Representations-ICLR 2025 (ICLR, Singapore, 2025).
  11. M. E. Newman and G. T. Barkema, Monte Carlo Methods in Statistical Physics (Clarendon Press, Oxford, 1999).
  12. G. E. Hinton, P. Dayan, B. J. Frey, and R. M. Neal, The “wake-sleep” algorithm for unsupervised neural networks, Science 268, 1158 (1995).
  13. R. Salakhutdinov and G. Hinton, Deep Boltzmann machines, in Artificial Intelligence and Statistics,Clearwater Beach, FL (PMLR, 2009), pp. 448–455.
  14. J. Bornschein and Y. Bengio, Reweighted wake-sleep, in Proceedings of the International Conference on Learning Representations (ICLR, San Diego, CA, 2015).
  15. D. P. Kingma and M. Welling, Auto-encoding variational Bayes, arXiv:1312.6114.
  16. D. J. Rezende, S. Mohamed, and D. Wierstra, Stochastic backpropagation and approximate inference in deep generative models, in International Conference on Machine Learning, Beijing, China (PMLR, 2014), pp. 1278–1286.
  17. J. Sohl-Dickstein, E. Weiss, N. Maheswaranathan, and S. Ganguli, Deep unsupervised learning using nonequilibrium thermodynamics, in International Conference on Machine Learning, Lille, France (PMLR, 2015), pp. 2256–2265.
  18. Y. Song, J. Sohl-Dickstein, D. P. Kingma, A. Kumar, S. Ermon, and B. Poole, Score-based generative modeling through stochastic differential equations, arXiv:2011.13456.
  19. G. Biroli, T. Bonnaire, V. De Bortoli, and M. Mézard, Dynamical regimes of diffusion models, Nat. Commun. 15, 9957 (2024).
  20. M. Kamb and S. Ganguli, An analytic theory of creativity in convolutional diffusion models, arXiv:2412.20292.
  21. A. Sclocchi, A. Favero, and M. Wyart, A phase transition in diffusion models reveals the hierarchical nature of data, Proc. Natl. Acad. Sci. USA 122, e2408799121 (2025).
  22. K. Ikeda, T. Uda, D. Okanohara, and S. Ito, Speed-accuracy relations for diffusion models: Wisdom from nonequilibrium thermodynamics and optimal transport, Phys. Rev. X 15, 031031 (2025).
  23. M. Baiesi and C. Maes, Life efficiency does not always increase with the dissipation rate, J. Phys. Commun. 2, 045017 (2018).
  24. A. Murugan, D. A. Huse, and S. Leibler, Speed, dissipation, and error in kinetic proofreading, Proc. Natl. Acad. Sci. USA 109, 12034 (2012).
  25. P. Gaspard, Kinetics and thermodynamics of DNA polymerases with exonuclease proofreading, Phys. Rev. E 93, 042420 (2016).
  26. F. Jülicher, A. Ajdari, and J. Prost, Modeling molecular motors, Rev. Mod. Phys. 69, 1269 (1997).
  27. A. Parmeggiani, T. Franosch, and E. Frey, Phase coexistence in driven one-dimensional transport, Phys. Rev. Lett. 90, 086601 (2003).
  28. T. Chou, K. Mallick, and R. K. Zia, Non-equilibrium statistical mechanics: From a paradigmatic model to biological transport, Rep. Prog. Phys. 74, 116601 (2011).
  29. P. Sartori and S. Pigolotti, Thermodynamics of error correction, Phys. Rev. X 5, 041039 (2015).
  30. A. C. Barato, D. Hartich, and U. Seifert, Efficiency of cellular information processing, New J. Phys. 16, 103024 (2014).
  31. P. Diaconis, S. Holmes, and R. M. Neal, Analysis of a nonreversible Markov chain sampler, Ann. Appl. Probab. 10, 726 (2000).
  32. E. P. Bernard, W. Krauth, and D. B. Wilson, Event-chain Monte Carlo algorithms for hard-sphere systems, Phys. Rev. E 80, 056704 (2009).
  33. M. Michel, S. C. Kapfer, and W. Krauth, Generalized event-chain Monte Carlo: Constructing rejection-free global-balance algorithms from infinitesimal steps, J. Chem. Phys. 140, 054116 (2014).
  34. M. Vucelja, Lifting—a nonreversible Markov chain Monte Carlo algorithm, Am. J. Phys. 84, 958 (2016).
  35. G. Montavon and K.-R. Müller, Deep Boltzmann machines and the centering trick, Neural Networks: Tricks of the Trade (Springer, Berlin, 2012), pp. 621–637.
  36. Y. LeCun, L. Bottou, Y. Bengio, and P. Haffner, Gradient-based learning applied to document recognition, Proc. IEEE 86, 2278 (1998).
  37. A. Barra, G. Genovese, P. Sollich, and D. Tantari, Phase transitions in restricted Boltzmann machines with generic priors, Phys. Rev. E 96, 042156 (2017).
  38. J. Tubiana and R. Monasson, Emergence of compositional representations in restricted Boltzmann machines, Phys. Rev. Lett. 118, 138301 (2017).
  39. E. Ventura, S. Cocco, R. Monasson, and F. Zamponi, Unlearning regularization for Boltzmann machines, Mach. Learn.: Sci. Technol. 5, 025078 (2024).
  40. D. S. Salazar, Nonequilibrium thermodynamics of restricted Boltzmann machines, Phys. Rev. E 96, 022131 (2017).
  41. A. Decelle, C. Furtlehner, and B. Seoane, Equilibrium and non-equilibrium regimes in the learning of restricted Boltzmann machines, J. Stat. Mech. (2022) 114009 .
  42. D. Bachtis, G. Biroli, A. Decelle, and B. Seoane, Cascade of phase transitions in the training of energy-based models, Adv. Neural Infor. Proc. Systems 37, 55591 (2024).

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Phys. Rev. E 114, 015301– Published 7 July, 2026
  • Received 12 December 2025
  • Accepted 27 April 2026
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