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Efficient many-jet event generation with flow matching

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

  • *Contact author: timo.janssen@theorie.physik.uni-goettingen.de
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Phys. Rev. D 114, 034055 – Published 27 August, 2026

DOI: https://doi.org/10.1103/j71d-h5w2

Abstract

We apply for the first time, to the best of our knowledge, the flow matching method to the problem of phase-space sampling for event generation in high-energy collider physics. By training the model to remap the random numbers used to generate the momenta and helicities of the scattering matrix elements as implemented in the portable partonic event generator pepper, we find substantial efficiency improvements in the studied processes. We focus our study on the highest final-state multiplicities in Drell-Yan and top-antitop pair production used in simulated samples for the Large Hadron Collider, which computationally are the most relevant ones. We find that the unweighting efficiencies improve by factors of 184 and 25, respectively, when compared to the standard approach of using a vegas-based optimization. We also compare continuous normalizing flows trained with flow matching against the previously studied normalizing flows based on coupling layers and find that the former leads to better results, faster training and a better scaling behavior across the studied multiplicity range, while the latter evaluate faster. When combining the advantages of both methods using the regflow approach, we find parton-level unweighted event generation walltime gains of about a factor of 10 at the highest final-state multiplicities.

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Physics Subject Headings (PhySH)

See Also

Monte Carlo Event Generation with Continuous Normalizing Flows

E. Bothmann, T. Janßen, M. Knobbe, B. Schmitzer, and F. Sinz
Phys. Rev. Lett. 137, 091902 (2026)

Article Text

References (80)

  1. European Strategy Group, 2020 Update of the European Strategy for Particle Physics, CERN Council, Geneva (2020), 10.17181/ESU2020.
  2. M. Narain et al., The future of US particle physics—The Snowmass 2021 energy frontier report, arXiv:2211.11084.
  3. HEP Software Foundation Collaboration, A roadmap for HEP software and computing R&D for the 2020s, Comput. Software Big Sci. 3, 7 (2019).
  4. HSF Physics Event Generator WG Collaboration, Challenges in Monte Carlo Event generator software for high-luminosity LHC, Comput. Software Big Sci. 5, 12 (2021).
  5. HSF Physics Event Generator WG Collaboration, HL-LHC computing review stage-2, Common software projects: Event Generators, arXiv:2109.14938.
  6. ATLAS Collaboration, Modelling and computational improvements to the simulation of single vector-boson plus jet processes for the ATLAS experiment, J. High Energy Phys. 08 (2022) 089.
  7. National Energy Research Scientific Computing Center (NERSC, “Perlmutter architecture,” https://docs.nersc.gov/systems/perlmutter/architecture 2025.
  8. 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).
  9. R. Kleiss and R. Pittau, Weight optimization in multichannel Monte Carlo, Comput. Phys. Commun. 83, 141 (1994).
  10. G. P. Lepage, A new algorithm for adaptive multidimensional integration, J. Comput. Phys. 27, 192 (1978).
  11. T. Ohl, vegas revisited: Adaptive Monte Carlo integration beyond factorization, Comput. Phys. Commun. 120, 13 (1999).
  12. G. P. Lepage, Adaptive multidimensional integration: vegas enhanced, J. Comput. Phys. 439, 110386 (2021).
  13. S. Jadach, Foam: A general purpose cellular Monte Carlo event generator, Comput. Phys. Commun. 152, 55 (2003).
  14. T. Hahn, CUBA: A Library for multidimensional numerical integration, Comput. Phys. Commun. 168, 78 (2005).
  15. A. van Hameren, PARNI for importance sampling and density estimation, Acta Phys. Pol. B 40, 259 (2009).
  16. 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).
  17. M. D. Klimek and M. Perelstein, Neural network-based approach to phase space integration, SciPost Phys. 9, 053 (2020).
  18. E. Bothmann, T. Janßen, M. Knobbe, T. Schmale, and S. Schumann, Exploring phase space with neural importance sampling, SciPost Phys. 8, 069 (2020).
  19. C. Gao, S. Höche, J. Isaacson, C. Krause, and H. Schulz, Event generation with normalizing flows, Phys. Rev. D 101, 076002 (2020).
  20. 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).
  21. R. Verheyen, Event generation and density estimation with surjective normalizing flows, SciPost Phys. 13, 047 (2022).
  22. T. Heimel, N. Huetsch, F. Maltoni, O. Mattelaer, T. Plehn, and R. Winterhalder, The MadNIS reloaded, SciPost Phys. 17, 023 (2024).
  23. T. Heimel, O. Mattelaer, T. Plehn, and R. Winterhalder, Differentiable MadNIS-Lite, SciPost Phys. 18, 017 (2025).
  24. S. Badger et al., Machine learning and LHC event generation, SciPost Phys. 14, 079 (2023).
  25. 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), pp. 6572–6583.
  26. 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), https://openreview.net/forum?id=PqvMRDCJT9t.
  27. M. S. Albergo and E. Vanden-Eijnden, Building normalizing flows with stochastic interpolants, in The Eleventh International Conference on Learning Representations (2023), https://openreview.net/forum?id=li7qeBbCR1t.
  28. M. S. Albergo, N. M. Boffi, and E. Vanden-Eijnden, Stochastic interpolants: A unifying framework for flows and diffusions, J. Mach. Learn. Res. 26, 1 (2025).
  29. X. Liu, C. Gong, and Q. Liu, Flow straight and fast: Learning to generate and transfer data with rectified flow, arXiv:2209.03003.
  30. V. Mikuni and B. Nachman, Score-based generative models for calorimeter shower simulation, Phys. Rev. D 106, 092009 (2022).
  31. E. Buhmann, C. Ewen, D. A. Faroughy, T. Golling, G. Kasieczka, M. Leigh et al., EPiC-ly fast particle cloud generation with flow-matching and diffusion, arXiv:2310.00049.
  32. A. Butter, N. Huetsch, S. Palacios Schweitzer, T. Plehn, P. Sorrenson, and J. Spinner, Jet diffusion versus JetGPT—modern networks for the LHC, SciPost Phys. Core 8, 026 (2025).
  33. L. Favaro, R. Kogler, A. Paasch, S. Palacios Schweitzer, T. Plehn, and D. Schwarz, How to unfold top decays, SciPost Phys. Core 8, 053 (2025).
  34. 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).
  35. 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).
  36. 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.
  37. T. Müller, B. Mcwilliams, F. Rousselle, M. Gross, and J. Novák, Neural importance sampling, ACM Trans. Graph. 38, 1 (2019).
  38. C. Durkan, A. Bekasov, I. Murray, and G. Papamakarios, Neural spline flows, in Advances in Neural Information Processing Systems, edited by H. Wallach, H. Larochelle, A. Beygelzimer, F. d’Alché-Buc, E. Fox, and R. Garnett, Vol. 32 (Curran Associates, Inc., 2019).
  39. D. Rehman, O. Davis, J. Lu, J. Tang, M. Bronstein, Y. Bengio et al., Efficient regression-based training of normalizing flows for Boltzmann generators, arXiv:2506.01158.
  40. E. Bothmann, T. Janßen, M. Knobbe, B. Schmitzer, and F. Sinz, companion Letter, Monte Carlo event generation with continuous normalizing flows, Phys. Rev. Lett. 137, 091902 (2026).
  41. J. von Neumann, John von Neumann Collected Works, Vol. 5: Design of Computers, Theory of Automata and Numerical Analysis (Pergamon Press, Oxford, England, 1961), pp. 768–770.
  42. L. Kish, Survey Sampling (Wiley, New York, 1965).
  43. E. Tabak and E. Vanden-Eijnden, Density estimation by dual ascent of the log-likelihood, Commun. Math. Sci. 8, 217 (2010).
  44. E. G. Tabak and C. V. Turner, A family of nonparametric density estimation algorithms, Commun. Pure Appl. Math. 66, 145 (2013).
  45. B. Stienen and R. Verheyen, Phase space sampling and inference from weighted events with autoregressive flows, SciPost Phys. 10, 038 (2021).
  46. N. Deutschmann and N. Götz, Accelerating HEP simulations with Neural Importance Sampling, J. High Energy Phys. 03 (2024) 083.
  47. C. Finlay, J.-H. Jacobsen, L. Nurbekyan, and A. M. Oberman, How to train your neural ode: The world of jacobian and kinetic regularization, in Proceedings of the 37th International Conference on Machine Learning, ICML’20, JMLR.org, (2020).
  48. D. Onken, S. W. Fung, X. Li, and L. Ruthotto, OT-Flow: Fast and accurate continuous normalizing flows via optimal transport, in AAAI Conference on Artificial Intelligence (2021), Vol. 35, pp. 9223–9232, https://ojs.aaai.org/index.php/AAAI/article/view/17113.
  49. A. Tong, K. Fatras, N. Malkin, G. Huguet, Y. Zhang, J. Rector-Brooks et al., Improving and generalizing flow-based generative models with minibatch optimal transport, arXiv:2302.00482.
  50. A.-A. Pooladian, H. Ben-Hamu, C. Domingo-Enrich, B. Amos, Y. Lipman, and R. T. Q. Chen, Multisample flow matching: Straightening flows with minibatch couplings, in Proceedings of the 40th International Conference on Machine Learning, ICML’23, JMLR.org, (2023).
  51. S. La Cagnina, C. Grunwald, T. Janßen, K. Kröninger, and S. Schumann, Phase space sampling with Markov Chain Monte Carlo methods, EPJ Web Conf. 337, 01319 (2025).
  52. T. Janßen, R. Poncelet, and S. Schumann, Sampling NNLO QCD phase space with normalizing flows, J. High Energy Phys. 09 (2025) 194.
  53. 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).
  54. 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.
  55. T. Gleisberg and S. Hoeche, comix, a new matrix element generator, J. High Energy Phys. 12 (2008) 039.
  56. 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.
  57. sherpa Collaboration, Event Generation with sherpa 2.2, SciPost Phys. 7, 034 (2019).
  58. sherpa Collaboration, Event generation with sherpa 3, J. High Energy Phys. 12 (2024) 156.
  59. T. Melia, Dyck words and multiquark primitive amplitudes, Phys. Rev. D 88, 014020 (2013).
  60. T. Melia, Dyck words and multi-quark amplitudes, Proc. Sci. RADCOR2013 (2013) 031.
  61. H. Johansson and A. Ochirov, Color-kinematics duality for QCD amplitudes, J. High Energy Phys. 01 (2016) 170.
  62. The HDF Group, Hierarchical data format, version 5, https://www.hdfgroup.org/HDF5/.
  63. highfive—HDF5 header-only c++ Library, https://bluebrain.github.io/HighFive/.
  64. 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).
  65. 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).
  66. NNPDF Collaboration, Parton distributions for the LHC Run II, J. High Energy Phys. 04 (2015) 040.
  67. 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).
  68. 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).
  69. T. Akhound-Sadegh, J. Rector-Brooks, A. Joey Bose, S. Mittal, P. Lemos, C.-H. Liu et al., Iterated denoising energy matching for sampling from Boltzmann densities, in Proceedings of the 41st International Conference on Machine Learning, ICML’24, JMLR.org (2024).
  70. C. Gao, J. Isaacson, and C. Krause, i-flow: High-dimensional integration and sampling with normalizing flows, Mach. Learn. Sci. Tech. 1, 045023 (2020).
  71. C. Durkan, A. Bekasov, I. Murray, and G. Papamakarios, nflows: Normalizing flows in pytorch, Zenodo (2020).10.5281/zenodo.4296287
  72. I. Loshchilov and F. Hutter, Decoupled weight decay regularization, in 7th International Conference on Learning Representations (International Conference on Learning Representations (ICLR), New Orleans, Louisiana, USA, 2019).
  73. J. Wildberger, M. Dax, S. Buchholz, S. R. Green, J. H. Macke, and B. Schölkopf, Flow matching for scalable simulation-based inference, in Proceedings of the 37th International Conference on Neural Information Processing Systems, (Curran Associates Inc., Red Hook, NY, 2023).
  74. M. Tancik, P. P. Srinivasan, B. Mildenhall, S. Fridovich-Keil, N. Raghavan, U. Singhal et al., 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, 2020).
  75. J. Dormand and P. Prince, A family of embedded runge-kutta formulae, J. Comput. Appl. Math. 6, 19 (1980).
  76. M. Poli, S. Massaroli, A. Yamashita, H. Asama, J. Park, and S. Ermon, torchdyn: Implicit models and neural numerical methods in pytorch, https://github.com/DiffEqML/torchdyn.
  77. J. Bradbury, R. Frostig, P. Hawkins, M. J. Johnson, C. Leary, D. Maclaurin et al., JAX: Composable transformations of python+numpy programs, 2018.
  78. P. Kidger, On neural differential equations, Ph.D. thesis, University of Oxford, 2021.
  79. P. Kidger and C. Garcia, Equinox: Neural networks in jax via callable PyTrees and filtered transformations, Differentiable Programming workshop at Neural Information Processing Systems 2021 (2021).
  80. 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

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Phys. Rev. D 114, 034055– Published 27 August, 2026
  • Received 25 July 2025
  • Accepted 18 March 2026
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