- 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

Lattice fermion formulation via physics-informed neural networks: Ginsparg-Wilson relation and overlap fermions

Tatsuhiro Misumi*

  • *Contact author: misumi@phys.kindai.ac.jp
PDF

Phys. Rev. D 114, 014511 – Published 16 July, 2026

DOI: https://doi.org/10.1103/756m-stz5

Abstract

We propose a novel, machine-learning-based framework for constructing lattice fermions using physics-informed neural networks (PINNs). Our approach treats the formulation of the Dirac operator as an optimization problem guided by physical requirements, such as symmetries, locality and doubler-decoupling conditions. We first demonstrate that, when trained to satisfy the Ginsparg-Wilson (GW) relation as a soft constraint, a neural network reproduces the overlap fermion operator to high numerical accuracy and learns an effective sign-function mapping without explicitly using a prescribed polynomial or rational approximation. Second, we extend the framework from operator construction to machine-assisted algebraic discovery. Within a generalized polynomial ansatz, the network autonomously drives higher-order terms to zero and recovers the standard Ginsparg-Wilson relation. Remarkably, by changing the initial search bias, the same framework also finds a distinct solution corresponding to a Fujikawa-type generalized GW relation.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (74)

  1. L. H. Karsten and J. Smit, Lattice fermions: Species doubling, chiral invariance, and the triangle anomaly, Nucl. Phys. B183, 103 (1981).
  2. H. B. Nielsen and M. Ninomiya, Absence of neutrinos on a lattice. 1. Proof by homotopy theory, Nucl. Phys. B185, 20 (1981).
  3. H. B. Nielsen and M. Ninomiya, Absence of neutrinos on a lattice. 2. Intuitive topological proof, Nucl. Phys. B193, 173 (1981).
  4. H. B. Nielsen and M. Ninomiya, No go theorem for regularizing chiral fermions, Phys. Lett. 105B, 219 (1981).
  5. K. G. Wilson, Quarks nd strings on a lattice, in New Phenomena in Subnuclear Physics: Proceedings, International School of Subnuclear Physics, Erice, Sicily, 1975. Part A (Plenum Press, New York, NY, 1975), pp. 69–142, 10.1007/978-1-4613-4208-3-6.
  6. J. B. Kogut and L. Susskind, Hamiltonian formulation of Wilson’s lattice gauge theories, Phys. Rev. D 11, 395 (1975).
  7. L. Susskind, Lattice fermions, Phys. Rev. D 16, 3031 (1977).
  8. P. H. Ginsparg and K. G. Wilson, A remnant of chiral symmetry on the lattice, Phys. Rev. D 25, 2649 (1982).
  9. M. Luscher, Exact chiral symmetry on the lattice and the Ginsparg-Wilson relation, Phys. Lett. B 428, 342 (1998).
  10. H. Neuberger, More about exactly massless quarks on the lattice, Phys. Lett. B 427, 353 (1998).
  11. D. B. Kaplan, A method for simulating chiral fermions on the lattice, Phys. Lett. B 288, 342 (1992).
  12. Y. Shamir, Chiral fermions from lattice boundaries, Nucl. Phys. B406, 90 (1993).
  13. V. Furman and Y. Shamir, Axial symmetries in lattice QCD with Kaplan fermions, Nucl. Phys. B439, 54 (1995).
  14. T.-W. Chiu, T.-H. Hsieh, C.-H. Huang, and T.-R. Huang, A note on the Zolotarev optimal rational approximation for the overlap Dirac operator, Phys. Rev. D 66, 114502 (2002).
  15. T.-W. Chiu, Optimal domain wall fermions, Phys. Rev. Lett. 90, 071601 (2003).
  16. C. Gattringer, A new approach to Ginsparg-Wilson fermions, Phys. Rev. D 63, 114501 (2001).
  17. W. Bietenholz, Solutions of the Ginsparg-Wilson relation and improved domain wall fermions, Eur. Phys. J. C 6, 537 (1999).
  18. W. Bietenholz and I. Hip, The scaling of exact and approximate Ginsparg-Wilson fermions, Nucl. Phys. B570, 423 (2000).
  19. S. Dürr and G. Koutsou, Brillouin improvement for Wilson fermions, Phys. Rev. D 83, 114512 (2011).
  20. S. Dürr, G. Koutsou, and T. Lippert, Meson and baryon dispersion relations with brillouin fermions, Phys. Rev. D 86, 114514 (2012).
  21. Y.-G. Cho, S. Hashimoto, J.-I. Noaki, A. Juttner, and M. Marinkovic, O(a2)-improved actions for heavy quarks and scaling studies on quenched lattices, Proc. Sci. LATTICE2013 (2014) 255 [arXiv:1312.4630].
  22. Y.-G. Cho, S. Hashimoto, A. Juttner, T. Kaneko, M. Marinkovic, J.-I. Noaki, and J. T. Tsang, Improved lattice fermion action for heavy quarks, J. High Energy Phys. 05 (2015) 072.
  23. S. Dürr and G. Koutsou, On the suitability of the brillouin action as a kernel to the overlap procedure, arXiv:1701.00726.
  24. M. Creutz, T. Kimura, and T. Misumi, Index theorem and overlap formalism with naive and minimally doubled fermions, J. High Energy Phys. 12 (2010) 041.
  25. T. Misumi, New fermion discretizations and their applications, Proc. Sci. LATTICE2012 (2012) 005 [arXiv:1211.6999].
  26. 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).
  27. D. Boyda et al., Applications of machine learning to lattice quantum field theory, in Snowmass 2021 (2022), 2; arXiv:2202.05838.
  28. A. Tomiya, Machine learning for lattice QCD, J. Phys. Soc. Jpn. 94, 031006 (2025).
  29. U. Wenger, Machine learning for four-dimensional SU(3) lattice gauge theories, Proc. Sci. LATTICE2025 (2026) 0020.
  30. K. Holland, A. Ipp, D. I. Müller, and U. Wenger, Fixed point actions from convolutional neural networks, Proc. Sci. LATTICE2023 (2024) 038 [arXiv:2311.17816].
  31. K. Holland, A. Ipp, D. I. Müller, and U. Wenger, Machine learning a fixed point action for SU(3) gauge theory with a gauge equivariant convolutional neural network, Phys. Rev. D 110, 074502 (2024).
  32. U. Wenger, K. Holland, A. Ipp, and D. I. Müller, HMC and gradient flow with machine-learned classically perfect fixed point actions, Proc. Sci. LATTICE2024 (2025) 466 [arXiv:2502.03315].
  33. K. Holland, A. Ipp, D. I. Müller, and U. Wenger, Machine-learned renormalization-group-Improved gauge actions and classically perfect gradient flows, Phys. Rev. Lett. 136, 031901 (2026).
  34. S. Yasunaga, K. Yoshimura, A. Tomiya, and Y. Nagai, Parameter optimization of domain-wall fermion using machine learning, in 42th International Symposium on Lattice Field Theory (2026), 3; arXiv:2603.16329.
  35. H. Liu, K. Simonyan, and Y. Yang, DARTS: Differentiable architecture search, arXiv:1806.09055.
  36. K. Fujikawa, Algebraic generalization of the Ginsparg-Wilson relation, Nucl. Phys. B589, 487 (2000).
  37. M. Clancy, D. B. Kaplan, and H. Singh, Generalized Ginsparg-Wilson relations, Phys. Rev. D 109, 014502 (2024).
  38. P. Hasenfratz and F. Niedermayer, Perfect lattice action for asymptotically free theories, Nucl. Phys. B414, 785 (1994).
  39. W. Bietenholz and U. J. Wiese, Perfect lattice actions for quarks and gluons, Nucl. Phys. B464, 319 (1996).
  40. W. Bietenholz, Optimised Dirac operators on the lattice: Construction, properties and applications, Fortschr. Phys. 56, 107 (2008).
  41. L. H. Karsten, Lattice fermions in Euclidean space-time, Phys. Lett. 104B, 315 (1981).
  42. F. Wilczek, On lattice fermions, Phys. Rev. Lett. 59, 2397 (1987).
  43. M. Creutz, Four-dimensional graphene and chiral fermions, J. High Energy Phys. 04 (2008) 017.
  44. A. Borici, Creutz fermions on an orthogonal lattice, Phys. Rev. D 78, 074504 (2008).
  45. P. F. Bedaque, M. I. Buchoff, B. C. Tiburzi, and A. Walker-Loud, Broken symmetries from minimally doubled fermions, Phys. Lett. B 662, 449 (2008).
  46. P. F. Bedaque, M. I. Buchoff, B. C. Tiburzi, and A. Walker-Loud, Search for fermion actions on hyperdiamond lattices, Phys. Rev. D 78, 017502 (2008).
  47. S. Capitani, J. Weber, and H. Wittig, Minimally doubled fermions at one loop, Phys. Lett. B 681, 105 (2009).
  48. T. Kimura and T. Misumi, Characters of lattice fermions based on the hyperdiamond lattice, Prog. Theor. Phys. 124, 415 (2010).
  49. T. Kimura and T. Misumi, Lattice fermions based on higher-dimensional hyperdiamond lattices, Prog. Theor. Phys. 123, 63 (2010).
  50. M. Creutz and T. Misumi, Classification of minimally doubled fermions, Phys. Rev. D 82, 074502 (2010).
  51. M. Creutz, Minimal doubling and point splitting, Proc. Sci. LATTICE2010 (2010) 078 [arXiv:1009.3154].
  52. S. Capitani, M. Creutz, J. Weber, and H. Wittig, Renormalization of minimally doubled fermions, J. High Energy Phys. 09 (2010) 027.
  53. B. C. Tiburzi, Chiral lattice fermions, minimal doubling, and the axial anomaly, Phys. Rev. D 82, 034511 (2010).
  54. S. Kamata and H. Tanaka, Minimal doubling fermion and hermiticity, Prog. Theor. Exp. Phys. 2013, 023B05 (2013).
  55. T. Misumi, Phase structure for lattice fermions with flavored chemical potential terms, J. High Energy Phys. 08 (2012) 068.
  56. T. Misumi, T. Kimura, and A. Ohnishi, QCD phase diagram with 2-flavor lattice fermion formulations, Phys. Rev. D 86, 094505 (2012).
  57. S. Capitani, Reducing the number of counterterms with new minimally doubled actions, Phys. Rev. D 89, 014501 (2014).
  58. S. Capitani, New chiral lattice actions of the Borici-Creutz type, Phys. Rev. D 89, 074508 (2014).
  59. T. Misumi, Fermion actions extracted from lattice super Yang-Mills theories, J. High Energy Phys. 12 (2013) 063.
  60. J. H. Weber, S. Capitani, and H. Wittig, Numerical studies of minimally doubled fermions, Proc. Sci. LATTICE2013 (2014) 122 [arXiv:1312.0488].
  61. J. H. Weber, Properties of minimally doubled fermions., Ph.D. thesis, Mainz U., 2015, arXiv:1706.07104.
  62. S. Dürr and J. H. Weber, Dispersion relation and spectral range of Karsten-Wilczek and Borici-Creutz fermions, Phys. Rev. D 102, 014516 (2020).
  63. Y.-Y. Li, J. Wang, and Y.-Z. You, Quantum many-body lattice C-R-T symmetry: Fractionalization, anomaly, and symmetric mass generation, arXiv:2412.19691.
  64. A. Chatterjee, S. D. Pace, and S.-H. Shao, Quantized axial charge of staggered fermions and the chiral anomaly, Phys. Rev. Lett. 134, 021601 (2025).
  65. S. Catterall and A. Pradhan, Symmetries and anomalies of Hamiltonian staggered fermions, Phys. Rev. D 113, 014504 (2026).
  66. T. Yamaoka, Quantized axial charge in the Hamiltonian approach to Wilson fermions, J. High Energy Phys. 10 (2025) 102.
  67. T. Onogi and T. Yamaoka, Non-singlet conserved charges and anomalies in 3+1D staggered fermions, arXiv:2509.04906.
  68. S. Aoki, Y. Kikukawa, and T. Takemoto, Chiral anomaly of Kogut-Susskind fermion in (3+1)-dimensional Hamiltonian formalism, Phys. Rev. D 113, 034514 (2026).
  69. N. Seiberg and W. Zhang, Tori, klein bottles, and modulo 8 parity/time-reversal anomalies of 2+1d staggered fermions, J. High Energy Phys. 05 (2026) 264.
  70. T. Misumi, Minimal-doubling and single-Weyl fermion Hamiltonians, Phys. Rev. D 113, 074521 (2026).
  71. T. Misumi, T. Onogi, and T. Yamaoka, Taste-splitting mass and edge modes in 3+1˜D staggered fermions, arXiv:2604.02078.
  72. P. Hernandez, K. Jansen, and M. Luscher, Locality properties of neuberger’s lattice Dirac operator, Nucl. Phys. B552, 363 (1999).
  73. I. Horvath, Ginsparg-Wilson relation and ultralocality, Phys. Rev. Lett. 81, 4063 (1998).
  74. W. Bietenholz, On the absence of ultralocal Ginsparg-Wilson fermions, arXiv:hep-lat/9901005.

Outline

Information

Phys. Rev. D 114, 014511– Published 16 July, 2026
  • Received 13 May 2026
  • Accepted 24 June 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