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Solving Bemfica-Disconzi-Noronha-Kovtun diffusion using physics-informed neural networks

Vicente Chomalí-Castro1,*, Nick Clarisse1,†, Nicki Mullins2,‡, and Jorge Noronha1,§

  • *Contact author: faridc2@illinois.edu
  • †Contact author: nclari2@illinois.edu
  • ‡Contact author: nmmulli2@ncsu.edu
  • §Contact author: jn0508@illinois.edu
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Phys. Rev. D 114, 076013 – Published 7 October, 2026

DOI: https://doi.org/10.1103/shpm-ksq9

Abstract

In this work, we reformulate the relativistic Bemfica-Disconzi-Noronha-Kovtun (BDNK) diffusion equation in flux-conservative form, and solve the resulting partial differential equations in (1+1)D using both a second-order Kurganov-Tadmor finite volume scheme and physics-informed neural networks (PINNs). In particular, we introduce the SA-PINN-ACTO framework, which combines the self-adaptive PINN (SA-PINN) technique with an exact enforcement of auxiliary conditions through a transform to the output (ACTO), allowing the network to focus solely on minimizing the PDE residual. We test both approaches on smooth and discontinuous initial data, for both trivial and dynamically evolving velocity and temperature BDNK backgrounds, and for two characteristic speeds. The SA-PINN-ACTO method matches the converged Kurganov-Tadmor solutions for smooth profiles, while for discontinuous profiles the errors increase, reflecting an expected limitation of PINNs near sharp gradients.

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References (112)

  1. L. D. Landau and E. M. Lifshitz, Fluid Mechanics: Volume 6 (Elsevier, New York, 1987), Vol. 6.
  2. A. M. Anile, Relativistic Fluids and Magneto-fluids: With Applications in Astrophysics and Plasma Physics (Cambridge Monographs on Mathematical Physics), 1 ed. (Cambridge University Press, Cambridge, England, 1990).
  3. G. S. Rocha, D. Wagner, G. S. Denicol, J. Noronha, and D. H. Rischke, Theories of relativistic dissipative fluid dynamics, Entropy 26, 189 (2024).
  4. U. Frisch, Turbulence: The Legacy of A. N. Kolmogorov (Cambridge University Press, Cambridge, England, 1995).
  5. L. Rezzolla and O. Zanotti, Relativistic Hydrodynamics (Oxford University Press, New York, 2013).
  6. C. Eckart, The Thermodynamics of irreversible processes. 3. Relativistic theory of the simple fluid, Phys. Rev. 58, 919 (1940).
  7. P. Kovtun, Lectures on hydrodynamic fluctuations in relativistic theories, J. Phys. A 45, 473001 (2012).
  8. W. A. Hiscock and L. Lindblom, Generic instabilities in first-order dissipative relativistic fluid theories, Phys. Rev. D 31, 725 (1985).
  9. W. Israel and J. M. Stewart, Transient relativistic thermodynamics and kinetic theory, Ann. Phys. (Amsterdam) 118, 341 (1979).
  10. G. S. Denicol, H. Niemi, E. Molnár, and D. H. Rischke, Derivation of transient relativistic fluid dynamics from the Boltzmann equation, Phys. Rev. D 85, 114047 (2012).
  11. R. Baier, P. Romatschke, D. T. Son, A. O. Starinets, and M. A. Stephanov, Relativistic viscous hydrodynamics, conformal invariance, and holography, J. High Energy Phys. 04 (2008) 100.
  12. P. Romatschke and U. Romatschke, Relativistic Fluid Dynamics In and Out of Equilibrium: And Applications to Relativistic Nuclear Collisions, Cambridge Monographs on Mathematical Physics (Cambridge University Press, Cambridge, England, 2019).
  13. L. Gavassino, M. Antonelli, and B. Haskell, Bulk viscosity in relativistic fluids: From thermodynamics to hydrodynamics, Classical Quantum Gravity 38, 075001 (2021).
  14. M. G. Alford, L. Bovard, M. Hanauske, L. Rezzolla, and K. Schwenzer, Viscous dissipation and heat conduction in binary neutron-star mergers, Phys. Rev. Lett. 120, 041101 (2018).
  15. E. R. Most, S. P. Harris, C. Plumberg, M. G. Alford, J. Noronha, J. Noronha-Hostler, F. Pretorius, H. Witek, and N. Yunes, Projecting the likely importance of weak-interaction-driven bulk viscosity in neutron star mergers, Mon. Not. R. Astron. Soc. 509, 1096 (2021).
  16. T. Celora, I. Hawke, P. C. Hammond, N. Andersson, and G. L. Comer, Formulating bulk viscosity for neutron star simulations, Phys. Rev. D 105, 103016 (2022).
  17. G. Camelio, L. Gavassino, M. Antonelli, S. Bernuzzi, and B. Haskell, Simulating bulk viscosity in neutron stars. II. Evolution in spherical symmetry, Phys. Rev. D 107, 103032 (2023).
  18. G. Camelio, L. Gavassino, M. Antonelli, S. Bernuzzi, and B. Haskell, Simulating bulk viscosity in neutron stars. I. Formalism, Phys. Rev. D 107, 103031 (2023).
  19. E. R. Most, A. Haber, S. P. Harris, Z. Zhang, M. G. Alford, and J. Noronha, Emergence of microphysical bulk viscosity in binary neutron star postmerger dynamics, Astrophys. J. Lett. 967, L14 (2024).
  20. M. Chabanov and L. Rezzolla, Impact of bulk viscosity on the postmerger gravitational-wave signal from merging neutron stars, Phys. Rev. Lett. 134, 071402 (2025).
  21. L. Gavassino and J. Noronha, Relativistic bulk rheology: From neutron star mergers to viscous cosmology, Phys. Rev. D 109, 096040 (2024).
  22. J. L. Ripley, A. Hegade K. R., and N. Yunes, Probing internal dissipative processes of neutron stars with gravitational waves during the inspiral of neutron star binaries, Phys. Rev. D 108, 103037 (2023).
  23. Y. Yang, M. Hippert, E. Speranza, and J. Noronha, Far-from-equilibrium bulk-viscous transport coefficients in neutron star mergers, Phys. Rev. C 109, 015805 (2024).
  24. Y. Yang, M. Hippert, E. Speranza, and J. Noronha, Symmetry energy dependence of the bulk viscosity of nuclear matter, Phys. Rev. Lett. 135, 222702 (2025).
  25. W. A. Hiscock and L. Lindblom, Stability and causality in dissipative relativistic fluids, Ann. Phys. (Amsterdam) 151, 466 (1983).
  26. L. Gavassino, Applying the Gibbs stability criterion to relativistic hydrodynamics, Classical Quantum Gravity 38, 21LT02 (2021).
  27. L. Gavassino, M. Antonelli, and B. Haskell, Thermodynamic stability implies causality, Phys. Rev. Lett. 128, 010606 (2022).
  28. L. Gavassino, M. M. Disconzi, and J. Noronha, Universality classes of relativistic fluid dynamics: Foundations, Phys. Rev. Lett. 132, 222302 (2024).
  29. F. S. Bemfica, M. M. Disconzi, and J. Noronha, Causality of the Einstein-Israel-Stewart theory with bulk viscosity, Phys. Rev. Lett. 122, 221602 (2019).
  30. I. Cordeiro, E. Speranza, K. Ingles, F. S. Bemfica, and J. Noronha, Causality bounds on dissipative general-relativistic magnetohydrodynamics, Phys. Rev. Lett. 133, 091401 (2024).
  31. I. Cordeiro, F. S. Bemfica, E. Speranza, and J. Noronha, Nonlinear causality of Israel-Stewart theory with diffusion, Phys. Rev. D 113, 045024 (2026).
  32. F. S. Bemfica, M. M. Disconzi, V. Hoang, J. Noronha, and M. Radosz, Nonlinear constraints on relativistic fluids far from equilibrium, Phys. Rev. Lett. 126, 222301 (2021).
  33. S. Floerchinger and E. Grossi, Causality of fluid dynamics for high-energy nuclear collisions, J. High Energy Phys. 08 (2018) 186.
  34. M. M. Disconzi, V. Hoang, and M. Radosz, Breakdown of smooth solutions to the Müller–Israel–Stewart equations of relativistic viscous fluids, Lett. Math. Phys. 113, 55 (2023).
  35. F. S. Bemfica, Finite-time gradient blow-up and shock formation in Israel-Stewart theory: Bulk, shear, and diffusion regimes, Phys. Rev. E 112, 065105 (2025).
  36. F. S. Bemfica, M. M. Disconzi, and J. Noronha, Causality and existence of solutions of relativistic viscous fluid dynamics with gravity, Phys. Rev. D 98, 104064 (2018).
  37. P. Kovtun, First-order relativistic hydrodynamics is stable, J. High Energy Phys. 10 (2019) 034.
  38. F. S. Bemfica, M. M. Disconzi, and J. Noronha, Nonlinear causality of general first-order relativistic viscous hydrodynamics, Phys. Rev. D 100, 104020 (2019); 105, 069902(E) (2022).
  39. R. E. Hoult and P. Kovtun, Stable and causal relativistic Navier-Stokes equations, J. High Energy Phys. 06 (2020) 067.
  40. F. S. Bemfica, M. M. Disconzi, and J. Noronha, First-order general-relativistic viscous fluid dynamics, Phys. Rev. X 12, 021044 (2022).
  41. F. S. Bemfica, M. M. Disconzi, C. Rodriguez, and Y. Shao, Local existence and uniqueness in Sobolev spaces for first-order conformal causal relativistic viscous hydrodynamics, Commun. Pure Appl. Anal. 20, 2279 (2021).
  42. F. S. Bemfica, M. M. Disconzi, and P. J. Graber, Local well-posedness in Sobolev spaces for first-order barotropic causal relativistic viscous hydrodynamics, Commun. Pure Appl. Anal. 20, 2885 (2021).
  43. M. M. Disconzi and Y. Shao, Strongly hyperbolic quasilinear systems revisited, with applications to relativistic fluid dynamics, Asymptotic Analysis 140, 281 (2024).
  44. M. Sroczinski, Global existence and decay of small solutions for quasilinear second-order uniformly dissipative hyperbolic-hyperbolic systems, J. Differ. Equations 383, 130 (2024).
  45. H. Freistühler and M. Sroczinski, Nonlinear stability of first-order relativistic viscous hydrodynamics, J. Math. Phys. (N.Y.) 66, 093101 (2025).
  46. N. Abboud, E. Speranza, and J. Noronha, Causal and stable first-order chiral hydrodynamics, Phys. Rev. D 109, 094007 (2024).
  47. C. Plumberg, D. Almaalol, T. Dore, J. Noronha, and J. Noronha-Hostler, Causality violations in realistic simulations of heavy-ion collisions, Phys. Rev. C 105, L061901 (2022).
  48. J. Redondo-Yuste, Perturbations of relativistic dissipative stars, Classical Quantum Gravity 42, 075012 (2025).
  49. D. A. Caballero and N. Yunes, Neutron star radial perturbations for causal, viscous, relativistic fluids, Phys. Rev. D 112, 063050 (2025).
  50. J. Redondo-Yuste and V. Cardoso, Superradiant amplification by rotating viscous compact objects, Phys. Rev. D 112, L061501 (2025).
  51. A. Pandya, E. R. Most, and F. Pretorius, Conservative finite volume scheme for first-order viscous relativistic hydrodynamics, Phys. Rev. D 105, 123001 (2022).
  52. A. Pandya, E. R. Most, and F. Pretorius, Causal, stable first-order viscous relativistic hydrodynamics with ideal gas microphysics, Phys. Rev. D 106, 123036 (2022).
  53. H. Bantilan, Y. Bea, and P. Figueras, Evolutions in first-order viscous hydrodynamics, J. High Energy Phys. 08 (2022) 298.
  54. Y. Bea and P. Figueras, Field redefinitions and evolutions in relativistic Navier-Stokes, J. High Energy Phys. 11 (2024) 110.
  55. Y. Bea, Relativistic Navier-Stokes description of the quark-gluon plasma radial flow, arXiv:2503.02931.
  56. J. Bhambure, A. Mazeliauskas, J.-F. Paquet, R. Singh, M. Singh, D. Teaney, and F. Zhou, Relativistic viscous hydrodynamics in the density frame: Numerical tests and comparisons, Phys. Rev. C 111, 064910 (2025).
  57. D. Fantini and M. E. Rubio, Constraint evolution in first-order viscous relativistic fluids, Phys. Rev. D 112, 063038 (2025).
  58. N. Clarisse, E. O. Pinho, T. Patel, F. S. Bemfica, M. Hippert, and J. Noronha, Flux-conservative Bemfica-Disconzi-Noronha-Kovtun hydrodynamics and shock regularization, Phys. Rev. D 113, 024051 (2026).
  59. L. S. Keeble and F. Pretorius, First-order viscous relativistic hydrodynamics on the two-sphere, Phys. Rev. D 112, 124034 (2025).
  60. H. L. H. Shum, F. Abalos, Y. Bea, M. Bezares, P. Figueras, and C. Palenzuela, Neutron star evolution with the Bemfica-Disconzi-Noronha-Kovtun viscous hydrodynamics framework, Phys. Rev. D 113, 084029 (2026).
  61. A. Kurganov and E. Tadmor, New high-resolution central schemes for nonlinear conservation laws and convection–diffusion equations, J. Comput. Phys. 160, 241 (2000).
  62. D. Bazow, U. W. Heinz, and M. Strickland, Massively parallel simulations of relativistic fluid dynamics on graphics processing units with CUDA, Comput. Phys. Commun. 225, 92 (2018).
  63. M. Raissi, P. Perdikaris, and G. 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).
  64. Q. He and A. M. Tartakovsky, Physics-informed neural network method for forward and backward advection-dispersion equations, Water Resour. Res. 57, e2020WR029479 (2021).
  65. D. Zhang, L. Guo, and G. E. Karniadakis, Learning in modal space: Solving time-dependent stochastic PDEs using physics-informed neural networks, SIAM J. Sci. Comput. 42, A639 (2020).
  66. G. Pang, L. Lu, and G. E. Karniadakis, fPINNs: Fractional physics-informed neural networks, SIAM J. Sci. Comput. 41, A2603 (2019).
  67. G. E. Karniadakis, I. G. Kevrekidis, L. Lu, P. Perdikaris, S. Wang, and L. Yang, Physics-informed machine learning, Nat. Rev. Phys. 3, 422 (2021).
  68. S. Scher and G. Messori, Weather and climate forecasting with neural networks: Using general circulation models (GCMs) with different complexity as a study ground, Geosci. Model Dev. 12, 2797 (2019).
  69. M. G. Schultz, C. Betancourt, B. Gong, F. Kleinert, M. Langguth, L. H. Leufen, A. Mozaffari, and S. Stadtler, Can deep learning beat numerical weather prediction?, Phil. Trans. R. Soc. A 379, 20200097 (2021).
  70. J. Pathak, S. Subramanian, P. Harrington, S. Raja, A. Chattopadhyay, M. Mardani, T. Kurth, D. Hall, Z. Li, K. Azizzadenesheli, P. Hassanzadeh, K. Kashinath, and A. Anandkumar, FourCastNet: A global data-driven high-resolution weather model using adaptive fourier neural operators, arXiv:2202.11214.
  71. K. Bi, L. Xie, H. Zhang, X. Chen, X. Gu, and Q. Tian, Accurate medium-range global weather forecasting with 3D neural networks, Nature (London) 619, 533 (2023).
  72. S. J. Wetzel, S. Ha, R. Iten, M. Klopotek, and Z. Liu, Interpretable machine learning in physics: A review, arXiv:2503.23616.
  73. T. Roxlo, Opening the Black Box of Neural Nets: Case Studies in Stop/Top Discrimination (Harvard University, Cambridge, MA, 2018).
  74. H. Huang, B. Xiao, Z. Liu, Z. Wu, Y. Mu, and H. Song, Applications of deep learning to relativistic hydrodynamics, Phys. Rev. Res. 3, 023256 (2021).
  75. H. Hirvonen, K. J. Eskola, and H. Niemi, Deep learning for flow observables in ultrarelativistic heavy-ion collisions, Phys. Rev. C 108, 034905 (2023).
  76. L.-G. Pang, K. Zhou, N. Su, H. Petersen, H. Stöcker, and X.-N. Wang, An equation-of-state-meter of quantum chromodynamics transition from deep learning, Nat. Commun. 9, 210 (2018).
  77. N. Mallick, S. Prasad, A. N. Mishra, R. Sahoo, and G. G. Barnaföldi, Deep learning predicted elliptic flow of identified particles in heavy-ion collisions at the RHIC and LHC energies, Phys. Rev. D 107, 094001 (2023).
  78. J.-A. Sun, L. Yan, C. Gale, and S. Jeon, End-to-end generative diffusion model for heavy-ion collisions, Phys. Rev. C 112, L051903 (2025).
  79. C. Shen and L. Yan, Recent development of hydrodynamic modeling in heavy-ion collisions, Nucl. Sci. Tech. 31, 122 (2020).
  80. W. Busza, K. Rajagopal, and W. van der Schee, Heavy ion collisions: The big picture, and the big questions, Annu. Rev. Nucl. Part. Sci. 68, 339 (2018).
  81. L. Liu, S. Liu, H. Xie, F. Xiong, T. Yu, M. Xiao, L. Liu, and H. Yong, Discontinuity computing using physics-informed neural networks, J. Sci. Comput. 98, 22 (2024).
  82. J. Abbasi, A. D. Jagtap, B. Moseley, A. Hiorth, and P. Østebø Andersen, Challenges and advancements in modeling shock fronts with physics-informed neural networks: A review and benchmarking study, Neurocomputing 657, 131440 (2025).
  83. T. De Ryck, S. Mishra, and R. Molinaro, wPINNs: Weak physics informed neural networks for approximating entropy solutions of hyperbolic conservation laws, SIAM J. Numer. Anal. 62, 811 (2024).
  84. K. Cai and J. Wang, Physics-informed neural networks for solving incompressible Navier–Stokes equations in wind engineering, Phys. Fluids 36, 121303 (2024).
  85. A. Mathews, M. Francisquez, J. W. Hughes, D. R. Hatch, B. Zhu, and B. N. Rogers, Uncovering turbulent plasma dynamics via deep learning from partial observations, Phys. Rev. E 104, 025205 (2021).
  86. A. Ferrer-Sánchez, J. D. Martín-Guerrero, R. R. de Austri-Bazan, A. Torres-Forné, and J. A. Font, Gradient-annihilated PINNs for solving Riemann problems: Application to relativistic hydrodynamics, Comput. Methods Appl. Mech. Eng. 424, 116906 (2024).
  87. J. F. Urbán and J. A. Pons, An approximate Riemann solver approach in physics-informed neural networks for hyperbolic conservation laws, Phys. Fluids 37, 096114 (2025).
  88. L. D. McClenny and U. M. Braga-Neto, Self-adaptive physics-informed neural networks, J. Comput. Phys. 474, 111722 (2023).
  89. N. Mullins, M. Hippert, L. Gavassino, and J. Noronha, Relativistic hydrodynamic fluctuations from an effective action: Causality, stability, and the information current, Phys. Rev. D 108, 116019 (2023).
  90. P. Kovtun, Temperature in relativistic fluids, Phys. Rev. D 107, 086012 (2023).
  91. L. Gavassino, Can we make sense of dissipation without causality?, Phys. Rev. X 12, 041001 (2022).
  92. C. Ratti and R. Bellwied, The Deconfinement Transition of QCD: Theory Meets Experiment, Lecture notes in physics (Springer, Cham, Switzerland, 2021), Vol. 981.
  93. G. S. Denicol, C. Gale, S. Jeon, A. Monnai, B. Schenke, and C. Shen, Net-baryon diffusion in fluid-dynamic simulations of relativistic heavy-ion collisions, Phys. Rev. C 98, 034916 (2018).
  94. P. Kovtun, D. T. Son, and A. O. Starinets, Viscosity in strongly interacting quantum field theories from black hole physics, Phys. Rev. Lett. 94, 111601 (2005).
  95. V. Chomalí-Castro, python simulation codes and data for Kurganov–Tadmor and SA-PINN-ACTO solutions of (1+1)D BDNK diffusion (2026) https://github.com/vchomalicastro/1-1D-BDNK-diffusion-simulations.
  96. L. F. Richardson, The approximate arithmetical solution by finite differences of physical problems involving differential equations, with an application to the stresses in a masonry dam, Phil. Trans. R. Soc. A 210, 307 (1911).
  97. L. F. Richardson and J. A. Gaunt, VIII. The deferred approach to the limit, Phil. Trans. R. Soc. A 226, 299 (1927).
  98. M. Stein, Large sample properties of simulations using Latin hypercube sampling, Technometrics 29, 143 (1987).
  99. A. Paszke, S. Gross, F. Massa, A. Lerer, J. Bradbury, G. Chanan, T. Killeen, Z. Lin, N. Gimelshein, L. Antiga, A. Desmaison, A. Kopf, E. Yang, Z. DeVito, M. Raison, A. Tejani, S. Chilamkurthy, B. Steiner, L. Fang, J. Bai, and S. Chintala, pytorch: An Imperative Style, High-Performance Deep Learning Library, in Advances in Neural Information Processing Systems, edited by H. Wallach, H. Larochelle, A. Beygelzimer, F. d’Alché-Buc, E. Fox, and R. Garnett (Curran Associates, Inc., Red Hook, NY, USA, 2019), Vol. 32.
  100. J. F. Urbán, P. Stefanou, and J. A. Pons, Unveiling the optimization process of physics informed neural networks: How accurate and competitive can PINNs be?, J. Comput. Phys. 523, 113656 (2025).
  101. R. Mattey and S. Ghosh, A novel sequential method to train physics informed neural networks for Allen Cahn and Cahn Hilliard equations, Comput. Methods Appl. Mech. Eng. 390, 114474 (2022).
  102. D. P. Kingma and J. Ba, Adam: A method for stochastic optimization, arXiv:1412.6980.
  103. D. C. Liu and J. Nocedal, On the limited memory BFGS method for large scale optimization, Math. Program. 45, 503 (1989).
  104. B. Hao, C. Liu, U. Braga-Neto, L. Wang, and M. Zhong, Stability in training PINNs for stiff PDEs: Why initial conditions matter, Found. Data Sci. (2026), 10.3934/fods.2026016.
  105. F. Cao, F. Gao, D. Yuan, and J. Liu, Multistep asymptotic pre-training strategy based on PINNs for solving steep boundary singular perturbation problems, Comput. Methods Appl. Mech. Eng. 431, 117222 (2024).
  106. M. Ren, Z. Fang, K. Li, and A. Mukherjee, Improving PINNs by algebraic inclusion of boundary and initial conditions, arXiv:2407.20741.
  107. T. Le-Duc, S. Lee, H. Nguyen-Xuan, and J. Lee, A hierarchically normalized physics-informed neural network for solving differential equations: Application for solid mechanics problems, Engineering Applications of Artificial Intelligence 133, 108400 (2024).
  108. C. Dugas, Y. Bengio, F. Bélisle, C. Nadeau, and R. Garcia, Incorporating second-order functional knowledge for better option pricing, in Advances in Neural Information Processing Systems, edited by T. Leen, T. Dietterich, and V. Tresp (MIT Press, Cambridge, MA, 2000), Vol. 13.
  109. L. Gavassino, Á. D. Kovács, and H. S. Reall, Initial data of effective field theories of relativistic viscous fluids and gravity, Phys. Rev. D 113, 124022 (2026).
  110. R. J. Gladstone, M. A. Nabian, N. Sukumar, A. Srivastava, and H. Meidani, FO-PINN: A first-order formulation for physics-informed neural networks, Engineering Analysis with Boundary Elements 174, 106161 (2025).
  111. C.-W. Shu and S. Osher, Efficient implementation of essentially non-oscillatory shock-capturing schemes, J. Comput. Phys. 77, 439 (1988).
  112. R. Courant, K. Friedrichs, and H. Lewy, On the partial difference equations of mathematical physics, IBM J. Res. Dev. 11, 215 (1967).

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Phys. Rev. D 114, 076013– Published 7 October, 2026
  • Received 5 March 2026
  • Accepted 20 August 2026
Article part of CHORUS
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