- Physical Review Journals
All JournalsPhysics Magazine

Physical Review Letters

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Monte Carlo Event Generation with Continuous Normalizing Flows

E. Bothmann1,2, T. Janßen2,3,*, M. Knobbe4, B. Schmitzer3,5, and F. Sinz3,5

  • *Contact author: tjansse2@uni-goettingen.de
PDF

Phys. Rev. Lett. 137, 091902 – Published 27 August, 2026

DOI: https://doi.org/10.1103/jt6y-h375

Abstract

We apply continuous normalizing flows trained with the flow matching method to the problem of phase-space sampling in Monte Carlo event generation for high-energy collider physics. Focusing on lepton-pair and top-quark pair production with multiple jets, the two computationally most expensive processes at the Large Hadron Collider, we train helicity-conditioned continuous normalizing flows to remap the random numbers used in matrix element evaluation. Compared to standard methods, we achieve unweighting efficiency improvements by factors of up to 184 and 25 for the two processes at their respective highest jet number, at the cost of an increased evaluation time. When combining the advantages of continuous normalizing flows with the fast evaluation times of coupling-layer-based flows, using the RegFlow approach, we find parton-level unweighted event generation walltime gains of about a factor of 10 at the highest jet numbers. These substantial gains highlight the promise of samplers based on machine learning for next-generation collider experiments.

View figure in article

Physics Subject Headings (PhySH)

See Also

Efficient many-jet event generation with flow matching

E. Bothmann, T. Janßen, M. Knobbe, B. Schmitzer, and F. Sinz
Phys. Rev. D 114, 034055 (2026)

Article Text

References (65)

  1. G. Aad et al. (ATLAS Collaboration), The ATLAS experiment at the CERN large hadron collider, J. Instrum. 3, S08003 (2008).
  2. S. Chatrchyan et al. (CMS Collaboration), The CMS experiment at the CERN LHC, J. Instrum. 3, S08004 (2008).
  3. A. Abada et al. (FCC Collaboration), FCC physics opportunities: Future circular collider conceptual design report volume 1, Eur. Phys. J. C 79, 474 (2019).
  4. A. Abada et al. (FCC Collaboration), FCC-ee: The lepton collider: Future circular collider conceptual design report volume 2, Eur. Phys. J. Spec. Top. 228, 261 (2019).
  5. A. Abada et al. (FCC Collaboration), FCC-hh: The hadron collider: Future circular collider conceptual design report volume 3, Eur. Phys. J. Spec. Top. 228, 755 (2019).
  6. A. Buckley et al., General-purpose event generators for LHC physics, Phys. Rep. 504, 145 (2011).
  7. European Strategy Group, 2020 Update of the European Strategy for Particle Physics (CERN Council, Geneva, 2020), 10.17181/ESU2020.
  8. M. Narain et al., The future of US particle physics—The snowmass 2021 energy frontier report, arXiv:2211.11084.
  9. J. Albrecht et al. (HEP Software Foundation Collaboration), A roadmap for HEP software and computing R&D for the 2020s, Comput. Software Big Sci. 3, 7 (2019).
  10. S. Amoroso et al. (HSF Physics Event Generator WG Collaboration), Challenges in Monte Carlo event generator software for high-luminosity LHC, Comput. Software Big Sci. 5, 12 (2021).
  11. G. Aad et al. (ATLAS Collaboration), Measurement of the production cross section for a Higgs boson in association with a vector boson in the H→WW*→ℓνℓν channel in pp collisions at s=13  TeV with the ATLAS detector, Phys. Lett. B 798, 134949 (2019).
  12. G. Aad et al. (ATLAS Collaboration), Measurement of the associated production of a Higgs boson decaying into b-quarks with a vector boson at high transverse momentum in pp collisions at s=13  TeV with the ATLAS detector, Phys. Lett. B 816, 136204 (2021).
  13. M. Aaboud et al. (ATLAS Collaboration), Measurements of top-quark pair spin correlations in the eμ channel at s=13  TeV using pp collisions in the ATLAS detector, Eur. Phys. J. C 80, 754 (2020).
  14. G. Aad et al. (ATLAS Collaboration), Measurement of the tt¯ production cross-section in the lepton+jets channel at s=13  TeV with the ATLAS experiment, Phys. Lett. B 810, 135797 (2020).
  15. G. Aad et al. (ATLAS Collaboration), Modelling and computational improvements to the simulation of single vector-boson plus jet processes for the ATLAS experiment, J. High Energy Phys. 8 (2022) 089.
  16. E. Bothmann, A. Buckley, I. A. Christidi, C. Gütschow, S. Höche, M. Knobbe, T. Martin, and M. Schönherr, Accelerating LHC event generation with simplified pilot runs and fast PDFs, Eur. Phys. J. C 82, 1128 (2022).
  17. R. Kleiss and R. Pittau, Weight optimization in multichannel Monte Carlo, Comput. Phys. Commun. 83, 141 (1994).
  18. G. P. Lepage, A new algorithm for adaptive multidimensional integration, J. Comput. Phys. 27, 192 (1978).
  19. T. Ohl, vegas revisited: Adaptive Monte Carlo integration beyond factorization, Comput. Phys. Commun. 120, 13 (1999).
  20. G. P. Lepage, Adaptive multidimensional integration: vegas enhanced, J. Comput. Phys. 439, 110386 (2021).
  21. S. Höche, S. Prestel, and H. Schulz, Simulation of vector boson plus many jet final states at the high luminosity LHC, Phys. Rev. D 100, 014024 (2019).
  22. E. Tabak and E. Vanden-Eijnden, Density estimation by dual ascent of the log-likelihood, Commun. Math. Sci. 8, 217 (2010).
  23. E. G. Tabak and C. V. Turner, A family of nonparametric density estimation algorithms, Commun. Pure Appl. Math. 66, 145 (2013).
  24. L. Dinh, D. Krueger, and Y. Bengio, NICE: Non-linear independent components estimation, in 3rd International Conference on Learning Representations, Workshop Track Proceedings (2015), arXiv:1410.8516.
  25. M. D. Klimek and M. Perelstein, Neural network-based approach to phase space integration, SciPost Phys. 9, 053 (2020).
  26. E. Bothmann, T. Janßen, M. Knobbe, T. Schmale, and S. Schumann, Exploring phase space with neural importance sampling, SciPost Phys. 8, 069 (2020).
  27. C. Gao, S. Höche, J. Isaacson, C. Krause, and H. Schulz, Event generation with normalizing flows, Phys. Rev. D 101, 076002 (2020).
  28. T. Heimel, R. Winterhalder, A. Butter, J. Isaacson, C. Krause, F. Maltoni, O. Mattelaer, and T. Plehn, MadNIS—Neural multi-channel importance sampling, SciPost Phys. 15, 141 (2023).
  29. R. Verheyen, Event generation and density estimation with surjective normalizing flows, SciPost Phys. 13, 047 (2022).
  30. T. Heimel, N. Huetsch, F. Maltoni, O. Mattelaer, T. Plehn, and R. Winterhalder, The MadNIS reloaded, SciPost Phys. 17, 023 (2024).
  31. T. Heimel, O. Mattelaer, T. Plehn, and R. Winterhalder, Differentiable MadNIS-lite, SciPost Phys. 18, 017 (2025).
  32. S. Badger et al., Machine learning and LHC event generation, SciPost Phys. 14, 079 (2023).
  33. R. T. Q. Chen, Y. Rubanova, J. Bettencourt, and D. Duvenaud, Neural ordinary differential equations, in Proceedings of the 32nd International Conference on Neural Information Processing Systems (Curran Associates Inc., Red Hook, NY, USA, 2018), pp. 6572–6583, arXiv:1806.07366.
  34. Y. Lipman, R. T. Q. Chen, H. Ben-Hamu, M. Nickel, and M. Le, Flow matching for generative modeling, in The Eleventh International Conference on Learning Representations (2023), arXiv:2210.02747.
  35. M. S. Albergo and E. Vanden-Eijnden, Building normalizing flows with stochastic interpolants, in The Eleventh International Conference on Learning Representations (2023), arXiv:2209.15571.
  36. M. S. Albergo, N. M. Boffi, and E. Vanden-Eijnden, Stochastic interpolants: A unifying framework for flows and diffusions, arXiv:2303.08797.
  37. X. Liu, C. Gong, and Q. Liu, Flow straight and fast: Learning to generate and transfer data with rectified flow, arXiv:2209.03003.
  38. T. Müller, B. Mcwilliams, F. Rousselle, M. Gross, and J. Novák, Neural importance sampling, ACM Trans. Graph. 38, 1 (2019).
  39. C. Durkan, A. Bekasov, I. Murray, and G. Papamakarios, Neural spline flows, in Neural spline flows, in Proceedings of the 33rd International Conference on Neural Information Processing Systems (NeurIPS 2019) (Curran Associates, Inc., Red Hook, NY, 2019).
  40. E. Bothmann, T. Janßen, M. Knobbe, B. Schmitzer, and F. Sinz, companion paper, Efficient many-jet event generation with flow matching, Phys. Rev. D 114, 034055 (2026).
  41. A. Buckley, J. Ferrando, S. Lloyd, K. Nordström, B. Page, M. Rüfenacht, M. Schönherr, and G. Watt, lhapdf6: Parton density access in the LHC precision era, Eur. Phys. J. C 75, 132 (2015).
  42. S. Kullback and R. A. Leibler, On information and sufficiency, Ann. Math. Stat. 22, 79 (1951).
  43. A. Tong, K. Fatras, N. Malkin, G. Huguet, Y. Zhang, J. Rector-Brooks, G. Wolf, and Y. Bengio, Improving and generalizing flow-based generative models with minibatch optimal transport, arXiv:2302.00482.
  44. E. Bothmann, T. Childers, W. Giele, F. Herren, S. Hoeche, J. Isaacson, M. Knobbe, and R. Wang, Efficient phase-space generation for hadron collider event simulation, SciPost Phys. 15, 169 (2023).
  45. E. Bothmann, T. Childers, W. Giele, S. Höche, J. Isaacson, and M. Knobbe, A portable parton-level event generator for the high-luminosity LHC, SciPost Phys. 17, 081 (2024).
  46. E. Bothmann, W. Giele, S. Hoeche, J. Isaacson, and M. Knobbe, Many-gluon tree amplitudes on modern GPUs: A case study for novel event generators, SciPost Phys. Codebases 2022, 3 (2022).
  47. E. Bothmann, J. Isaacson, M. Knobbe, S. Höche, and W. Giele, QCD tree amplitudes on modern GPUs: A case study for novel event generators, Proc. Sci. ICHEP2022 (2022) 222.
  48. T. Gleisberg, S. Höche, F. Krauss, M. Schönherr, S. Schumann, F. Siegert, and J. Winter, Event generation with sherpa 1.1, J. High Energy Phys. 02 (2009) 007.
  49. E. Bothmann et al. (sherpa Collaboration), Event generation with sherpa 2.2, SciPost Phys. 7, 034 (2019).
  50. E. Bothmann et al. (sherpa Collaboration), Event generation with sherpa 3, J. High Energy Phys. 12 (2024) 156.
  51. C. Bierlich et al., A comprehensive guide to the physics and usage of pythia 8.3, SciPost Phys. Codebases 2022, 8 (2022).
  52. T. Sjöstrand, S. Ask, J. R. Christiansen, R. Corke, N. Desai, P. Ilten, S. Mrenna, S. Prestel, C. O. Rasmussen, and P. Z. Skands, An introduction to pythia 8.2, Comput. Phys. Commun. 191, 159 (2015).
  53. T. Sjostrand, S. Mrenna, and P. Z. Skands, pythia 6.4 physics and manual, J. High Energy Phys. 05 (2006) 026.
  54. T. Sjostrand, S. Mrenna, and P. Z. Skands, A brief introduction to pythia 8.1, Comput. Phys. Commun. 178, 852 (2008).
  55. T. Melia, Dyck words and multiquark primitive amplitudes, Phys. Rev. D 88, 014020 (2013).
  56. T. Melia, Dyck words and multi-quark amplitudes, Proc. Sci. RADCOR2013 (2013) 031.
  57. H. Johansson and A. Ochirov, Color-kinematics duality for QCD amplitudes, J. High Energy Phys. 01 (2016) 170.
  58. S. Höche, S. Prestel, and H. Schulz, Simulation of vector boson plus many jet final states at the high luminosity LHC, Phys. Rev. D 100, 014024 (2019).
  59. E. Bothmann, T. Childers, C. Gütschow, S. Höche, P. Hovland, J. Isaacson, M. Knobbe, and R. Latham, Efficient precision simulation of processes with many-jet final states at the LHC, Phys. Rev. D 109, 014013 (2024).
  60. R. D. Ball et al. (NNPDF Collaboration), Parton distributions for the LHC Run II, J. High Energy Phys. 04 (2015) 040.
  61. Z. Bern, L. J. Dixon, F. Febres Cordero, S. Höche, H. Ita, D. A. Kosower, D. Maître, and K. J. Ozeren, Next-to-leading order W+5-Jet production at the LHC, Phys. Rev. D 88, 014025 (2013).
  62. M. Tancik, P. P. Srinivasan, B. Mildenhall, S. Fridovich-Keil, N. Raghavan, U. Singhal, R. Ramamoorthi, J. T. Barron, and R. Ng, Fourier features let networks learn high frequency functions in low dimensional domains, in Proceedings of the 34th International Conference on Neural Information Processing Systems (Curran Associates Inc., Red Hook, NY, USA, 2020), arXiv:2006.10739.
  63. I. Loshchilov and F. Hutter, Decoupled weight decay regularization, in Proceedings of the 7th International Conference on Learning Representations (2019), arXiv:1711.05101.
  64. D. Rehman, O. Davis, J. Lu, J. Tang, M. Bronstein, Y. Bengio, A. Tong, and A. J. Bose, Efficient regression-based training of normalizing flows for Boltzmann generators, arXiv:2506.01158.
  65. E. Bothmann, T. Janßen, M. Knobbe, B. Schmitzer, and F. Sinz, Efficient many-jet event generation with flow matching, Zenodo (2026), 10.5281/zenodo.19949391.

Outline

Information

Phys. Rev. Lett. 137, 091902– Published 27 August, 2026
  • Received 25 July 2025
  • Revised 11 January 2026
  • Accepted 27 March 2026
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP3.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation