- Physical Review Journals
All JournalsPhysics Magazine

Physical Review D

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

Physics-informed neural network approach to quark-antiquark color flux tube

Wei Kou1,2,*, Xiaoxuan Lin3,1, Bing’ang Guo1,2, and Xurong Chen1,2,4,†

  • *Contact author: kouwei@impcas.ac.cn
  • †Contact author: xchen@impcas.ac.cn
PDF

Phys. Rev. D 112, 056027 – Published 22 September, 2025

DOI: https://doi.org/10.1103/z61n-r7gw

Abstract

We introduce a physics-informed neural network (PINNs) framework for modeling the spatial distribution of chromodynamic fields induced by quark-antiquark pairs, based on lattice Monte Carlo simulations. In contrast to conventional neural networks, PINNs incorporate physical laws—expressed here as differential equations governing type-II superconductivity—directly into the training objective. By embedding these equations into the loss function, we guide the network to learn physically consistent solutions. Adopting an inverse problem approach, we extract the parameters of the superconducting equations from lattice QCD data and subsequently solve them. To accommodate physical boundary conditions, we recast the system into an integrodifferential form and extend the analysis within the fractional PINNs framework. The accuracy of the reconstructed field distribution is assessed via relative L2-error norms. We further extract physical observables such as the string tension and the mean width of the flux tube, offering quantitative insight into the confinement mechanism. This method enables the reconstruction of color field profiles as functions of quark-antiquark separation without recourse to predefined parametric models. Our results illuminate aspects of the dual Meissner effect and highlight the promise of data-driven strategies in addressing nonperturbative challenges in quantum chromodynamics.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (63)

  1. Y. Nambu, Strings, monopoles and gauge fields, Phys. Rev. D 10, 4262 (1974).
  2. A. A. Abrikosov, On the magnetic properties of superconductors of the second group, Sov. Phys. JETP 5, 1174 (1957), http://www.jetp.ras.ru//cgi-bin/e/index/e/5/6/p1174?a=list.
  3. A. A. Abrikosov, The magnetic properties of superconducting alloys, J. Phys. Chem. Solids 2, 199 (1957).
  4. H. B. Nielsen and P. Olesen, Vortex line models for dual strings, Nucl. Phys. B61, 45 (1973).
  5. K.-I. Kondo, S. Kato, A. Shibata, and T. Shinohara, Quark confinement: Dual superconductor picture based on a non-Abelian Stokes theorem and reformulations of Yang–Mills theory, Phys. Rep. 579, 1 (2015).
  6. V. Singh, D. A. Browne, and R. W. Haymaker, Structure of Abrikosov vortices in SU(2) lattice gauge theory, Phys. Lett. B 306, 115 (1993).
  7. K. Schilling, G. S. Bali, and C. Schlichter, A Ginzburg-Landau analysis of the color electric flux tube, Nucl. Phys. B, Proc. Suppl. 73, 638 (1999).
  8. M. N. Chernodub, F. V. Gubarev, M. I. Polikarpov, and V. I. Zakharov, Towards Abelian—like formulation of the dual gluodynamics, Nucl. Phys. B600, 163 (2001).
  9. M. N. Chernodub, K. Ishiguro, Y. Mori, Y. Nakamura, M. I. Polikarpov, T. Sekido, T. Suzuki, and V. I. Zakharov, Vacuum type of SU(2) gluodynamics in maximally Abelian and Landau gauges, Phys. Rev. D 72, 074505 (2005).
  10. P. Cea, L. Cosmai, and A. Papa, Chromoelectric flux tubes and coherence length in QCD, Phys. Rev. D 86, 054501 (2012).
  11. T. Suzuki, M. Hasegawa, K. Ishiguro, Y. Koma, and T. Sekido, Gauge invariance of color confinement due to the dual Meissner effect caused by Abelian monopoles, Phys. Rev. D 80, 054504 (2009).
  12. J. R. Clem, Simple model for the vortex core in a type II superconductor, J. Low Temp. Phys. 18, 427 (1975).
  13. A. J. Larkoski, I. Moult, and B. Nachman, Jet substructure at the large hadron collider: A review of recent advances in theory and machine learning, Phys. Rep. 841, 1 (2020).
  14. D. Guest, K. Cranmer, and D. Whiteson, Deep learning and its application to LHC physics, Annu. Rev. Nucl. Part. Sci. 68, 161 (2018).
  15. A. Radovic, M. Williams, D. Rousseau, M. Kagan, D. Bonacorsi, A. Himmel, A. Aurisano, K. Terao, and T. Wongjirad, Machine learning at the energy and intensity frontiers of particle physics, Nature (London) 560, 41 (2018).
  16. K. Albertsson et al., Machine learning in high energy physics community white paper, J. Phys. Conf. Ser. 1085, 022008 (2018).
  17. G. Carleo, I. Cirac, K. Cranmer, L. Daudet, M. Schuld, N. Tishby, L. Vogt-Maranto, and L. Zdeborová, Machine learning and the physical sciences, Rev. Mod. Phys. 91, 045002 (2019).
  18. D. Bourilkov, Machine and deep learning applications in particle physics, Int. J. Mod. Phys. A 34, 1930019 (2020).
  19. M. D. Schwartz, Modern machine learning and particle physics (2021), 10.1162/99608f92.beeb1183.
  20. G. Karagiorgi, G. Kasieczka, S. Kravitz, B. Nachman, and D. Shih, Machine learning in the search for new fundamental physics, arXiv:2112.03769.
  21. A. Boehnlein et al., Colloquium: Machine learning in nuclear physics, Rev. Mod. Phys. 94, 031003 (2022).
  22. P. Shanahan et al., Snowmass 2021 Computational Frontier CompF03 Topical Group Report: Machine Learning, arXiv:2209.07559.
  23. Z. Yang, Y. He, W. Chen, W.-Y. Ke, L.-G. Pang, and X.-N. Wang, Deep learning assisted jet tomography for the study of Mach cones in QGP, Eur. Phys. J. C 83, 652 (2023).
  24. F.-P. Li, H.-L. Lü, L.-G. Pang, and G.-Y. Qin, Deep-learning quasi-particle masses from QCD equation of state, Phys. Lett. B 844, 138088 (2023).
  25. W.-B. He, Y.-G. Ma, L.-G. Pang, H.-C. Song, and K. Zhou, High-energy nuclear physics meets machine learning, Nucl. Sci. Technol. 34, 88 (2023).
  26. K. Zhou, L. Wang, L.-G. Pang, and S. Shi, Exploring QCD matter in extreme conditions with machine learning, Prog. Part. Nucl. Phys. 135, 104084 (2024).
  27. K. Zhou, L. Pang, S. Shi, and H. Stoecker, Deep learning for inverse problems in nuclear physics, Proc. Sci., FAIRness2022 (2023) 064.
  28. L.-G. Pang, Studying high-energy nuclear physics with machine learning, Int. J. Mod. Phys. E 33, 2430009 (2024).
  29. Y.-G. Ma, L.-G. Pang, R. Wang, and K. Zhou, Phase transition study meets machine learning, Chin. Phys. Lett. 40, 122101 (2023).
  30. O.-Y. Luo, X. Chen, F.-P. Li, X.-H. Li, and K. Zhou, Neural network modeling of heavy-quark potential from holography, Eur. Phys. J. C 85, 637 (2025).
  31. B. Chen, X. Chen, X. Li, Z.-R. Zhu, and K. Zhou, Exploring transport properties of quark-gluon plasma with a machine-learning assisted holographic approach, Phys. Rev. D 111, 086033 (2025).
  32. M. Omana Kuttan, J. Steinheimer, K. Zhou, A. Redelbach, and H. Stoecker, Extraction of global event features in heavy-ion collision experiments using PointNet, Proc. Sci., FAIRness2022 (2023) 040.
  33. S. Shi, K. Zhou, J. Zhao, S. Mukherjee, and P. Zhuang, From lattice QCD to in-medium heavy-quark interactions via deep learning, Proc. Sci., LATTICE2021 (2022) 537.
  34. S. Shi, K. Zhou, J. Zhao, S. Mukherjee, and P. Zhuang, From lattice QCD to in-medium heavy-quark interactions via deep learning, EPJ Web Conf. 259, 04003 (2022).
  35. S. Shi, K. Zhou, J. Zhao, S. Mukherjee, and P. Zhuang, Heavy quark potential in the quark-gluon plasma: Deep neural network meets lattice quantum chromodynamics, Phys. Rev. D 105, 014017 (2022).
  36. M. Mansouri, K. Bitaghsir Fadafan, and X. Chen, Holographic complex potential of a quarkonium from deep learning, arXiv:2406.06285.
  37. X. Chen and M. Huang, Flavor dependent critical endpoint from holographic QCD through machine learning, J. High Energy Phys. 02 (2025) 123.
  38. X. Chen and M. Huang, Machine learning holographic black hole from lattice QCD equation of state, Phys. Rev. D 109, L051902 (2024).
  39. X.-Y. Wang, C. Dong, and X. Liu, Analysis of strong coupling constant with machine learning and its application, Chin. Phys. Lett. 41, 031201 (2023).
  40. M. P. Bento, H. B. Câmara, and J. F. Seabra, Unraveling particle dark matter with physics-informed neural networks, Phys. Lett. B 868, 139690 (2025).
  41. G. Aarts, K. Fukushima, T. Hatsuda, A. Ipp, S. Shi, L. Wang, and K. Zhou, Physics-driven learning for inverse problems in quantum chromodynamics, Nat. Rev. Phys. 7, 154 (2025).
  42. M. Raissi, P. Perdikaris, and G. E. Karniadakis, Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations, J. Comput. Phys. 378, 686 (2019).
  43. A. G. Baydin, B. A. Pearlmutter, A. A. Radul, and J. M. Siskind, Automatic differentiation in machine learning: A survey, J. Mach. Learn. Res. 18, 1 (2018), https://jmlr.org/papers/v18/17-468.html.
  44. C. Amalinadhi, P. S. Palar, R. Stevenson, and L. Zuhal, On physics-informed deep learning for solving Navier-Stokes equations, in AIAA SCITECH 2022 Forum, San Diego, California, USA (2022), p. 1436.
  45. H. Eivazi, M. Tahani, P. Schlatter, and R. Vinuesa, Physics-informed neural networks for solving Reynolds-averaged Navier–Stokes equations, Phys. Fluids 34, 075117(2022)
  46. M. Baker, P. Cea, V. Chelnokov, L. Cosmai, F. Cuteri, and A. Papa, The confining color field in SU(3) gauge theory, Eur. Phys. J. C 80, 514 (2020).
  47. L. Lu, X. Meng, Z. Mao, and G. E. Karniadakis, deepxde: A deep learning library for solving differential equations, SIAM Rev. 63, 208 (2021).
  48. M. S. Cardaci, P. Cea, L. Cosmai, R. Falcone, and A. Papa, Chromoelectric flux tubes in QCD, Phys. Rev. D 83, 014502 (2011).
  49. P. Cea, L. Cosmai, F. Cuteri, and A. Papa, Flux tubes and coherence length in the SU(3) vacuum, arXiv:1310.8423.
  50. P. Cea, L. Cosmai, F. Cuteri, and A. Papa, London penetration depth and coherence length of SU(3) vacuum flux tubes, Proc. Sci., LATTICE2014 (2014) 350 [arXiv:1410.4394].
  51. P. Cea, L. Cosmai, F. Cuteri, and A. Papa, Flux tubes in the SU(3) vacuum: London penetration depth and coherence length, Phys. Rev. D 89, 094505 (2014).
  52. M. Baker, P. Cea, V. Chelnokov, L. Cosmai, and A. Papa, Unveiling the flux tube structure in full QCD, Eur. Phys. J. C 85, 29 (2025).
  53. M. Baker, P. Cea, V. Chelnokov, L. Cosmai, and A. Papa, Investigating the flux tube structure within full QCD, Proc. Sci., LATTICE2024 (2025) 390 [arXiv:2411.01886]
  54. W. Kou and X. Chen, Machine learning insights into quark–antiquark interactions: Probing field distributions and string tension in QCD, Eur. Phys. J. C 85, 261 (2025).
  55. N. Thuerey, B. Holzschuh, P. Holl, G. Kohl, M. Lino, Q. Liu, P. Schnell, and F. Trost, Physics-based deep learning, arXiv:2109.05237.
  56. A. Paszke, S. Gross, S. Chintala, G. Chanan, E. Yang, Z. DeVito, Z. Lin, A. Desmaison, L. Antiga, and A. Lerer, Automatic differentiation in pytorch, in Proceedings of the 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA (2017).
  57. A. Paszke, pytorch: An imperative style, high-performance deep learning library, arXiv:1912.01703.
  58. M. Abadi, P. Barham, J. Chen, Z. Chen, A. Davis, J. Dean, M. Devin, S. Ghemawat, G. Irving, M. Isard et al., tensorflow: A system for Large-Scale machine learning, in Proceedings of the 12th USENIX Symposium on Operating Systems Design and Implementation (OSDI 16), Savannah, GA, USA (2016), pp. 265–283.
  59. M. Lin, Automatic functional differentiation in jax, arXiv:2311.18727.
  60. PaddlePaddle Contributors, Paddlepaddle: An open-source deep learning platform, https://github.com/PaddlePaddle/Paddle (2025) (accessed: 2025-05-22).
  61. G. Pang, L. Lu, and G. E. Karniadakis, fpinns: Fractional physics-informed neural networks, SIAM J. Sci. Comput. 41, A2603 (2019).
  62. L. Yuan, Y.-Q. Ni, X.-Y. Deng, and S. Hao, A-pinn: Auxiliary physics informed neural networks for forward and inverse problems of nonlinear integro-differential equations, J. Comput. Phys. 462, 111260 (2022).
  63. F. M. Rohrhofer, S. Posch, C. Gößnitzer, and B. C. Geiger, Data vs. physics: The apparent pareto front of physics-informed neural networks, IEEE Access 11, 86252 (2023).

Outline

Information

Phys. Rev. D 112, 056027– Published 22 September, 2025
  • Received 8 June 2025
  • Accepted 3 September 2025
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