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  • Open Access

Nonperturbative trivializing flows for lattice gauge theories

Mathis Gerdes*

Pim de Haan

Roberto Bondesan

Miranda C. N. Cheng†

  • CuspAI, Amsterdam, The Netherlands

  • *Contact author: m.gerdes@uva.nl
  • †On leave from CNRS, France.
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Phys. Rev. D 112, 094516 – Published 26 November, 2025

DOI: https://doi.org/10.1103/31d5-hvp6

Abstract

Continuous normalizing flows are known to be highly expressive and flexible, which allows for easier incorporation of large symmetries and makes them a powerful computational tool for lattice field theories. Building on previous work, we present a general continuous normalizing flow architecture for matrix Lie groups that is equivariant under group transformations. We apply this to lattice gauge theories in two dimensions as a proof of principle and demonstrate competitive performance, showing its potential as a tool for future lattice computations.

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References (42)

  1. C. Morningstar, The Monte Carlo method in quantum field theory, arXiv:hep-lat/0702020.
  2. A. Sokal, Monte Carlo methods in statistical mechanics: Foundations and new algorithms, in Functional Integration: Basics and Applications (Springer, New York, 1997), pp. 131–192.
  3. G. Papamakarios, E. Nalisnick, D. J. Rezende, S. Mohamed, and B. Lakshminarayanan, Normalizing flows for probabilistic modeling and inference, J. Mach. Learn. Res. 22, 1 (2021).
  4. M. S. Albergo, G. Kanwar, and P. E. Shanahan, Flow-based generative models for Markov chain Monte Carlo in lattice field theory, Phys. Rev. D 100, 034515 (2019).
  5. M. S. Albergo, D. Boyda, D. C. Hackett, G. Kanwar, K. Cranmer, S. Racanière, D. J. Rezende, and P. E. Shanahan, Introduction to normalizing flows for lattice field theory, arXiv:2101.08176.
  6. M. C. N. Cheng and N. Stratikopoulou, Lecture notes on normalizing flows for lattice quantum field theories, arXiv:2504.18126.
  7. R. T. Q. Chen, Y. Rubanova, J. Bettencourt, and D. Duvenaud, Neural ordinary differential equations, arXiv:1806.07366.
  8. D. Boyda, G. Kanwar, S. Racanière, D. J. Rezende, M. S. Albergo, K. Cranmer, D. C. Hackett, and P. E. Shanahan, Sampling using $SU(N)$ gauge equivariant flows, Phys. Rev. D 103, 074504 (2021).
  9. P. de Haan, C. Rainone, M. C. N. Cheng, and R. Bondesan, Scaling up machine learning for quantum field theory with equivariant continuous flows, arXiv:2110.02673.
  10. M. Gerdes, P. de Haan, C. Rainone, R. Bondesan, and M. C. N. Cheng, Learning lattice quantum field theories with equivariant continuous flows, SciPost Phys. 15, 238 (2023).
  11. M. Caselle, E. Cellini, and A. Nada, Sampling the lattice nambu-goto string using continuous normalizing flows, J. High Energy Phys. 02 (2024) 048.
  12. Y. Nagai and A. Tomiya, Gauge covariant neural network for quarks and gluons, Phys. Rev. D 111, 074501 (2025).
  13. R. Abbott et al., Normalizing flows for lattice gauge theory in arbitrary space-time dimension, arXiv:2305.02402.
  14. R. Abbott, A. Botev, D. Boyda, D. C. Hackett, G. Kanwar, S. Racanière, D. J. Rezende, F. Romero-López, P. E. Shanahan, and J. M. Urban, Applications of flow models to the generation of correlated lattice QCD ensembles, Phys. Rev. D 109, 094514 (2024).
  15. M. Lüscher, Trivializing maps, the Wilson flow and the HMC algorithm, Commun. Math. Phys. 293, 899 (2010).
  16. S. Bacchio, P. Kessel, S. Schaefer, and L. Vaitl, Learning trivializing gradient flows for lattice gauge theories, Phys. Rev. D 107, L051504 (2023).
  17. M. Gerdes, bijx: Bijections & normalizing flows with JAX/NNX, https://github.com/mathisgerdes/bijx.
  18. M. Gerdes, P. de Haan, R. Bondesan, and M. C. N. Cheng, Non-perturbative trivializing flows for lattice gauge theories, 10.5281/zenodo.15576712 (2025).
  19. K. A. Nicoli, C. J. Anders, L. Funcke, T. Hartung, K. Jansen, P. Kessel, S. Nakajima, and P. Stornati, Estimation of thermodynamic observables in lattice field theories with deep generative models, Phys. Rev. Lett. 126, 032001 (2021).
  20. D. Albandea, L. Del Debbio, P. Hernández, R. Kenway, J. Marsh Rossney, and A. Ramos, Learning trivializing flows, Eur. Phys. J. C 83, 676 (2023).
  21. R. T. Q. Chen, Y. Rubanova, J. Bettencourt, and D. K. Duvenaud, Neural Ordinary Differential Equations, in Advances in Neural Information Processing Systems (Curran Associates, Inc., 2018), Vol. 31.
  22. L. Falorsi, Continuous normalizing flows on manifolds, arXiv:2104.14959.
  23. A. Lou, D. Lim, I. Katsman, L. Huang, Q. Jiang, S.-N. Lim, and C. De Sa, Neural manifold ordinary differential equations, arXiv:2006.10254.
  24. L. Falorsi and P. Forré, Neural ordinary differential equations on manifolds, arXiv:2006.06663.
  25. M. Nakahara, Geometry, Topology and Physics, 2nd ed. (CRC Press, Bristol, 2003).
  26. P. E. Crouch and R. Grossman, Numerical integration of ordinary differential equations on manifolds, J. Nonlinear Sci. 3, 1 (1993).
  27. M. Wandelt, Geometric integration on Lie groups and its applications in Lattice QCD, Ph.D. thesis, Universitätsbibliothek Wuppertal, 2021.
  28. R. Abbott, M. S. Albergo, D. Boyda, D. C. Hackett, G. Kanwar, F. Romero-López, P. E. Shanahan, and J. M. Urban, Multiscale normalizing flows for gauge theories, Proc. Sci., LATTICE2023 (2024) 035 [arXiv:2404.10819].
  29. H. Wu, J. Köhler, and F. Noé, Stochastic Normalizing flows, arXiv:2002.06707.
  30. L. I. Midgley, V. Stimper, G. N. C. Simm, B. Schölkopf, and J. M. Hernández-Lobato, Flow annealed importance sampling bootstrap, arXiv:2208.01893.
  31. M. Caselle, E. Cellini, A. Nada, and M. Panero, Stochastic normalizing flows as non-equilibrium transformations, J. High Energy Phys. 07 (2022) 015.
  32. A. Bulgarelli, E. Cellini, and A. Nada, Sampling SU(3) pure gauge theory with stochastic normalizing flows, Proc. Sci., LATTICE2024 (2025) 040 [arXiv:2409.18861].
  33. A. Bulgarelli, E. Cellini, and A. Nada, Scaling of stochastic normalizing flows in SU(3) lattice gauge theory, Phys. Rev. D 111, 074517 (2025).
  34. R. Abbott et al., Aspects of scaling and scalability for flow-based sampling of lattice QCD, Eur. Phys. J. A 59, 257 (2023).
  35. L. Del Debbio, J. M. Rossney, and M. Wilson, Efficient modelling of trivializing maps for lattice ϕ theory using normalizing flows: A first look at scalability, Phys. Rev. D 104, 094507 (2021).
  36. M. Favoni, A. Ipp, D. I. Müller, and D. Schuh, Lattice gauge equivariant convolutional neural networks, Phys. Rev. Lett. 128, 032003 (2022).
  37. J. Bradbury et al., JAX: Composable transformations of Python+NumPy programs (2018).
  38. J. J. Duistermaat and J. A. C. Kolk, Lie Groups, Universitext (Springer, Berlin, Heidelberg, 2000).
  39. F. Mezzadri, How to generate random matrices from the classical compact groups, Not. Am. Math. Soc. 54, 592 (2007).
  40. D. Bump, The Weyl Integration Formula, in Lie Groups, Graduate Texts in Mathematics, edited by D. Bump (Springer, New York, 2004), pp. 112–116.
  41. A. A. Migdal, Recursion equations in gauge theories, Sov. Phys. JETP 42, 413 (1975).
  42. D. J. Gross and W. Taylor, Two-dimensional QCD is a string theory, Nucl. Phys. B400, 181 (1993).

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Phys. Rev. D 112, 094516– Published 26 November, 2025
  • Received 2 June 2025
  • Accepted 11 November 2025
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