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  • Letter
  • Open Access

Learning trivializing gradient flows for lattice gauge theories

Simone Bacchio1, Pan Kessel2,3, Stefan Schaefer4, and Lorenz Vaitl2

  • 1Computation-based Science and Technology Research Center, The Cyprus Institute, Nicosia, Cyprus
  • 2Machine Learning Group, Technische Universität Berlin, Berlin, Germany
  • 3BIFOLD Berlin Institute for the Foundations of Learning and Data, Berlin, Germany
  • 4John von Neumann-Institut für Computing NIC, Deutsches Elektronen-Synchrotron DESY, Germany

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Phys. Rev. D 107, L051504 – Published 20 March, 2023

DOI: https://doi.org/10.1103/PhysRevD.107.L051504

Abstract

We propose a unifying approach that starts from the perturbative construction of trivializing maps by Lüscher and then improves on it by learning. The resulting continuous normalizing flow model can be implemented using common tools of lattice field theory and requires several orders of magnitude fewer parameters than any existing machine learning approach. Specifically, our model can achieve competitive performance with as few as 14 parameters while existing deep-learning models have around 1 million parameters for SU(3) Yang-Mills theory on a 162 lattice. This has obvious consequences for training speed and interpretability. It also provides a plausible path for scaling machine-learning approaches toward realistic theories.

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References (44)

  1. U. Wolff, Collective Monte Carlo Updating for Spin Systems, Phys. Rev. Lett. 62, 361 (1989).
  2. R. H. Swendsen and J.-S. Wang, Nonuniversal Critical Dynamics in Monte Carlo Simulations, Phys. Rev. Lett. 58, 86 (1987).
  3. M. Lüscher, Trivializing maps, the Wilson flow and the HMC algorithm, Commun. Math. Phys. 293, 899 (2010).
  4. H. Nicolai, Supersymmetry and functional integration measures, Nucl. Phys. B176, 419 (1980).
  5. G. P. Engel and S. Schaefer, Testing trivializing maps in the hybrid Monte Carlo algorithm, Comput. Phys. Commun. 182, 2107 (2011).
  6. 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).
  7. G. Kanwar, M. S. Albergo, D. Boyda, K. Cranmer, D. C. Hackett, S. Racanière, D. J. Rezende, and P. E. Shanahan, Equivariant Flow-Based Sampling for Lattice Gauge Theory, Phys. Rev. Lett. 125, 121601 (2020).
  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. M. S. Albergo, G. Kanwar, S. Racanière, D. J. Rezende, J. M. Urban, D. Boyda, K. Cranmer, D. C. Hackett, and P. E. Shanahan, Flow-based sampling for fermionic lattice field theories, Phys. Rev. D 104, 114507 (2021).
  10. D. C. Hackett, C.-C. Hsieh, M. S. Albergo, D. Boyda, J.-W. Chen, K.-F. Chen, K. Cranmer, G. Kanwar, and P. E. Shanahan, Flow-based sampling for multimodal distributions in lattice field theory, arXiv:2107.00734.
  11. M. S. Albergo, D. Boyda, K. Cranmer, D. C. Hackett, G. Kanwar, S. Racanière, D. J. Rezende, F. Romero-López, P. E. Shanahan, and J. M. Urban, Flow-based sampling in the lattice Schwinger model at criticality, Phys. Rev. D 106, 014514 (2022).
  12. R. Abbott et al., Gauge-equivariant flow models for sampling in lattice field theories with pseudofermions, Phys. Rev. D 106, 074506 (2022).
  13. R. Abbott et al., Sampling QCD field configurations with gauge-equivariant flow models, Proc. Sci. LATTICE2022 (2023) 036.
  14. L. Del Debbio, J. M. Rossney, and M. Wilson, Efficient modeling of trivializing maps for lattice ϕ4 theory using normalizing flows: A first look at scalability, Phys. Rev. D 104, 094507 (2021).
  15. L. Del Debbio, J. M. Rossney, and M. Wilson, Machine learning trivializing maps: A first step towards understanding how flow-based samplers scale up, Proc. Sci. LATTICE2021 (2022) 059.
  16. K. A. Nicoli, C. J. Anders, L. Funcke, T. Hartung, K. Jansen, P. Kessel, S. Nakajima, and P. Stornati, Machine learning of thermodynamic observables in the presence of mode collapse, Proc. Sci. LATTICE2021 (2022) 338.
  17. 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).
  18. 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.
  19. M. Gerdes, P. de Haan, C. Rainone, R. Bondesan, and M. C. N. Cheng, Learning lattice quantum field theories with equivariant continuous flows, arXiv:2207.00283.
  20. J. Finkenrath, Tackling critical slowing down using global correction steps with equivariant flows: The case of the Schwinger model, arXiv:2201.02216.
  21. M. Caselle, E. Cellini, A. Nada, and M. Panero, Stochastic normalizing flows as non-equilibrium transformations, J. High Energy Phys. 07 (2022) 015.
  22. M. Favoni, A. Ipp, and D. I. Müller, Applications of lattice gauge equivariant neural networks, EPJ Web Conf. 274, 09001 (2022).
  23. M. Favoni, A. Ipp, D. I. Müller, and D. Schuh, Lattice Gauge Equivariant Convolutional Neural Networks, Phys. Rev. Lett. 128, 032003 (2022).
  24. R. Abbott et al., Aspects of scaling and scalability for flow-based sampling of lattice QCD, arXiv:2211.07541.
  25. W. Grathwohl, R. T. Chen, J. Bettencourt, I. Sutskever, and D. Duvenaud, FFJORD: Free-form continuous dynamics for scalable reversible generative models, arXiv:1810.01367.
  26. J. Köhler, L. Klein, and F. Noé, Equivariant flows: Exact likelihood generative learning for symmetric densities, in International Conference on Machine Learning (PMLR, 2020), pp. 5361–5370.
  27. See Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevD.107.L051504 for more details on proofs, derivations and training models.
  28. B. Sengupta, K. J. Friston, and W. D. Penny, Efficient gradient computation for dynamical models, NeuroImage 98, 521 (2014).
  29. L. Falorsi, Continuous normalizing flows on manifolds, arXiv:2104.14959.
  30. E. Mathieu and M. Nickel, Riemannian continuous normalizing flows, Adv. Neural Inf. Process. Syst. 33, 2503 (2020).
  31. I. Katsman, A. Lou, D. Lim, Q. Jiang, S. N. Lim, and C. M. De Sa, Equivariant manifold flows, Adv. Neural Inf. Process. Syst. 34, 10600 (2021).
  32. A. Lou, D. Lim, I. Katsman, L. Huang, Q. Jiang, S. N. Lim, and C. M. De Sa, Neural manifold ordinary differential equations, Adv. Neural Inf. Process. Syst. 33, 17548 (2020).
  33. P. E. Crouch and R. Grossman, Numerical integration of ordinary differential equations on manifolds, J. Nonlinear Sci. 3, 1 (1993).
  34. D. P. Kingma and J. Ba, Adam: A method for stochastic optimization, arXiv:1412.6980.
  35. L. Richter, A. Boustati, N. Nüsken, F. Ruiz, and O. D. Akyildiz, VarGrad: A low-variance gradient estimator for variational inference, Adv. Neural Inf. Process. Syst. 33, 13481 (2020).
  36. L. Vaitl, K. A. Nicoli, S. Nakajima, and P. Kessel, Path-gradient estimators for continuous normalizing flows, in International Conference on Machine Learning (PMLR, 2022), pp. 21945–21959.
  37. L. Vaitl, K. A. Nicoli, S. Nakajima, and P. Kessel, Gradients should stay on path: Better estimators of the reverse- and forward KL divergence for normalizing flows, Mach. Learn.: Sci. Technol. 3, 045006 (2022).
  38. K. A. Nicoli, S. Nakajima, N. Strodthoff, W. Samek, K.-R. Müller, and P. Kessel, Asymptotically unbiased estimation of physical observables with neural samplers, Phys. Rev. E 101, 023304 (2020).
  39. D. Albandea, L. Del Debbio, P. Hernández, R. Kenway, J. M. Rossney, and A. Ramos, Learning trivializing flows, 39th International Symposium on Lattice Field Theory (2022), arXiv:2211.12806.
  40. M. Lüscher, Schwarz-preconditioned HMC algorithm for two-flavour lattice QCD, Comput. Phys. Commun. 165, 199 (2005).
  41. S. Schaefer, R. Sommer, and F. Virotta (ALPHA Collaboration), Critical slowing down and error analysis in lattice QCD simulations, Nucl. Phys. B845, 93 (2011).
  42. S. Bacchio, J. Finkenrath, and C. Stylianou, Lyncs-API: A python API for Lattice QCD applications, Proc. Sci. LATTICE2021 (2022) 542.
  43. S. Yamamoto, S. Bacchio, and J. Finenrath, Running HMC simulation with python via QUDA, Proc. Sci. LATTICE2022 (2023) 346.
  44. M. A. Clark, R. Babich, K. Barros, R. C. Brower, and C. Rebbi, Solving lattice QCD systems of equations using mixed precision solvers on GPUs, Comput. Phys. Commun. 181, 1517 (2010).

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Phys. Rev. D 107, L051504– Published 20 March, 2023
  • Received 5 January 2023
  • Accepted 28 February 2023
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