- 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
  • Editors' Suggestion
  • Open Access

r-process heating implementation in hydrodynamic simulations with neural networks

Oliver Just1,2, Zewei Xiong1, and Gabriel Martínez-Pinedo1,3

PDF

Phys. Rev. D 113, 083022 – Published 16 April, 2026

DOI: https://doi.org/10.1103/gl2l-7f3g

Abstract

Neutron-rich outflows in neutron-star mergers (NSMs) or other explosive events can be subject to substantial heating through the release of rest-mass energy in the course of the rapid neutron-capture (r-) process. This r-process heating can potentially have a significant impact on the dynamics determining the velocity distribution of the ejecta, but due to the complexity of detailed nuclear networks required to describe the r-process self-consistently, hydrodynamic models of NSMs often neglect r-process heating or include it using crude parametrizations. In this work, we present a conceptually new method, RHINE, for emulating the r-process and concomitant energy release in hydrodynamic simulations via machine learning algorithms. The method requires the evolution of only a few additional quantities characterizing the composition, of which the nuclear rates of change are obtained at each location and time step from neural networks trained by a large set of trajectories from full nuclear network calculations. The scheme is tested by comparing spherically symmetric wind simulations and long-term simulations of NSMs using RHINE with postprocessing results from nucleosynthesis calculations, showing agreement in the released heating energy to within ≲10%. In our NSM models on average about 2.3 MeV, 0.7 MeV, and 2.1 MeV are released per baryon in dynamical ejecta, NS-torus ejecta, and black hole (BH) torus ejecta, respectively. The strongest velocity boost is observed for BH-torus ejecta, which also become 40% more massive with r-process heating. The nucleosynthesis yields are only mildly affected by r-process heating, but the kilonova gets significantly brighter once the BH-torus ejecta become visible. RHINE can be readily implemented in existing hydrodynamics codes using pretrained machine learning data and routines for source-term prediction that we provide online.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (113)

  1. B. P. Abbott, R. Abbott, T. D. Abbott, F. Acernese, K. Ackley, C. Adams, T. Adams, P. Addesso, R. X. Adhikari, V. B. Adya et al., Astrophys. J. Lett. 848, L12 (2017).
  2. S. J. Smartt et al., Nature (London) 551, 75 (2017).
  3. M. Tanaka et al., Publ. Astron. Soc. Jpn. 69, 102 (2017).
  4. A. Frebel, Annu. Rev. Nucl. Part. Sci. 68, 237 (2018).
  5. L. Lombardo, C. J. Hansen, F. Rizzuti, G. Cescutti, L. I. Mashonkina, P. François, P. Bonifacio, E. Caffau, A. Alencastro Puls, R. Fernandes de Melo, A. J. Gallagher, Á. Skúladóttir, A. J. Koch-Hansen, and L. Sbordone, Astron. Astrophys. 693, A293 (2025).
  6. S. Shen, R. J. Cooke, E. Ramirez-Ruiz, P. Madau, L. Mayer, and J. Guedes, Astrophys. J. 807, 115 (2015).
  7. F. van de Voort, E. Quataert, P. F. Hopkins, D. Kereš, and C.-A. Faucher-Giguère, Mon. Not. R. Astron. Soc. 447, 140 (2015).
  8. C. Kobayashi, A. I. Karakas, and M. Lugaro, Astrophys. J. 900, 179 (2020).
  9. L. Baiotti and L. Rezzolla, Rep. Prog. Phys. 80, 096901 (2017).
  10. M. D. Duez and Y. Zlochower, Rep. Prog. Phys. 82, 016902 (2018).
  11. M. Shibata and K. Hotokezaka, Annu. Rev. Nucl. Part. Sci. 69, 41 (2019).
  12. S. Bernuzzi, Gen. Relativ. Gravit. 52, 108 (2020).
  13. H. T. Janka and A. Bauswein, Handbook of Nuclear Physics (Springer, Singapore, 2023).
  14. F. Foucart, Liv. Rev. Comput. Astrophys. 9, 1 (2023).
  15. D. M. Siegel, Nat. Rev. Phys. 4, 306 (2022).
  16. S. Rosswog and O. Korobkin, Ann. Phys. (Berlin) 536, 2200306 (2024).
  17. C. J. Horowitz et al., J. Phys. G 46 083001 (2019).
  18. M. Arnould and S. Goriely, Prog. Part. Nucl. Phys. 112, 103766 (2020).
  19. J. J. Cowan, C. Sneden, J. E. Lawler, A. Aprahamian, M. Wiescher, K. Langanke, G. Martínez-Pinedo, and F. K. Thielemann, Rev. Mod. Phys. 93, 015002 (2021).
  20. E. M. Holmbeck, T. M. Sprouse, and M. R. Mumpower, The Eur. Phys. J. A 59, 28 (2023).
  21. B. D. Metzger, G. Martínez-Pinedo, S. Darbha, E. Quataert, A. Arcones, D. Kasen, R. Thomas, P. Nugent, I. V. Panov, and N. T. Zinner, Mon. Not. R. Astron. Soc. 406, 2650 (2010).
  22. L. F. Roberts, D. Kasen, W. H. Lee, and E. Ramirez-Ruiz, Astrophys. J. Lett. 736, L21 (2011).
  23. B. D. Metzger, Living Rev. Relativity 23, 1 (2019).
  24. E. Burns, Living Rev. Relativity 23, 4 (2020).
  25. D. Kasen, R. Fernández, and B. D. Metzger, Mon. Not. R. Astron. Soc. 450, 1777 (2015).
  26. J. M. Miller, B. R. Ryan, J. C. Dolence, A. Burrows, C. J. Fontes, C. L. Fryer, O. Korobkin, J. Lippuner, M. R. Mumpower, and R. T. Wollaeger, Phys. Rev. D 100, 023008 (2019).
  27. K. Kawaguchi, S. Fujibayashi, M. Shibata, M. Tanaka, and S. Wanajo, Astrophys. J. 913, 100 (2021).
  28. O. Just, V. Vijayan, Z. Xiong, S. Goriely, T. Soultanis, A. Bauswein, J. Guilet, H.-T. Janka, and G. Martínez-Pinedo, Astrophys. J. Lett. 951, L12 (2023).
  29. H. Klion, A. Tchekhovskoy, D. Kasen, A. Kathirgamaraju, E. Quataert, and R. Fernández, Mon. Not. R. Astron. Soc. 510, 2968 (2022).
  30. L. J. Shingles, C. E. Collins, V. Vijayan, A. Flörs, O. Just, G. Leck, Z. Xiong, A. Bauswein, G. Martínez-Pinedo, and S. A. Sim, Astrophys. J. Lett. 954, L41 (2023).
  31. L. Combi and D. M. Siegel, Phys. Rev. Lett. 131, 231402 (2023).
  32. S. Curtis, P. Bosch, P. Mösta, D. Radice, S. Bernuzzi, A. Perego, R. Haas, and E. Schnetter, Astrophys. J. Lett. 961, L26 (2024).
  33. F. Magistrelli, S. Bernuzzi, A. Perego, and D. Radice, Astrophys. J. Lett. 974, L5 (2024).
  34. W. H. Lee, E. Ramirez-Ruiz, and D. Page, Astrophys. J. 632, 421 (2005).
  35. R. Fernández and B. D. Metzger, Mon. Not. R. Astron. Soc. 435, 502 (2013).
  36. D. M. Siegel and B. D. Metzger, Astrophys. J. 858, 52 (2018).
  37. M. Haddadi, M. D. Duez, F. Foucart, T. Ramirez, R. Fernández, A. L. Knight, J. Jesse, F. Hébert, L. E. Kidder, H. P. Pfeiffer, and M. A. Scheel, Classical Quantum Gravity 40, 085008 (2023).
  38. R. Fernández, O. Just, Z. Xiong, and G. Martínez-Pinedo, Phys. Rev. D 110, 023001 (2024).
  39. S. Rosswog, O. Korobkin, A. Arcones, F.-K. Thielemann, and T. Piran, Mon. Not. R. Astron. Soc. 439, 744 (2014).
  40. R. Fernández, E. Quataert, J. Schwab, D. Kasen, and S. Rosswog, Mon. Not. R. Astron. Soc. 449, 390 (2015).
  41. O. Just, A. Bauswein, R. A. Pulpillo, S. Goriely, and H.-T. Janka, Mon. Not. R. Astron. Soc. 448, 541 (2015).
  42. M.-R. Wu, R. Fernández, G. Martínez-Pinedo, and B. D. Metzger, Mon. Not. R. Astron. Soc. 463, 2323 (2016).
  43. D. Desai, B. D. Metzger, and F. Foucart, Mon. Not. R. Astron. Soc. 485, 4404 (2019).
  44. F. Foucart, P. Mösta, T. Ramirez, A. J. Wright, S. Darbha, and D. Kasen, Phys. Rev. D 104, 123010 (2021).
  45. S. Darbha, D. Kasen, F. Foucart, and D. J. Price, Astrophys. J. 915, 69 (2021).
  46. A. Sneppen, D. Watson, A. Bauswein, O. Just, R. Kotak, E. Nakar, D. Poznanski, and S. Sim, Nature (London) 614, 436 (2023).
  47. S. Rosswog, Mon. Not. R. Astron. Soc. 376, L48 (2007).
  48. B. D. Metzger, A. Arcones, E. Quataert, and G. Martínez-Pinedo, Mon. Not. R. Astron. Soc. 402, 2771 (2010).
  49. W. Ishizaki, K. Kiuchi, K. Ioka, and S. Wanajo, Astrophys. J. 922, 185 (2021).
  50. C. Musolino, R. Duqué, and L. Rezzolla, Astrophys. J. Lett. 966, L31 (2024).
  51. K. Kawaguchi, N. Domoto, S. Fujibayashi, H. Hamidani, K. Hayashi, M. Shibata, M. Tanaka, and S. Wanajo, Mon. Not. R. Astron. Soc. 535, 3711 (2024).
  52. L.-T. Ma, K.-C. Pan, M.-R. Wu, and R. Fernández, Astrophys. J. 999, 247 (2026).
  53. F. Rosenblatt, Psychol. Rev. 65, 386 (1958).
  54. Y. LeCun, B. Boser, J. S. Denker, D. Henderson, R. E. Howard, W. Hubbard, and L. D. Jackel, Neural Comput. 1, 541 (1989).
  55. K. Hornik, M. Stinchcombe, and H. White, Neural Netw. 2, 359 (1989).
  56. G. Cybenko, Math. Control Signals Syst. 2, 303 (1989).
  57. J. Schmidhuber, Neural Netw. 61, 85 (2015).
  58. T. Soultanis, K. Maltsev, A. Bauswein, K. Chatziioannou, F. K. Röpke, and N. Stergioulas, Phys. Rev. D 111, 023002 (2025).
  59. G. M. Mirouh, G. C. Angelou, D. R. Reese, and G. Costa, Mon. Not. R. Astron. Soc. 483, L28 (2018).
  60. L. Hendriks and C. Aerts, Publ. Astron. Soc. Pac. 131, 108001 (2019).
  61. B. T. H. Tsang, D. Vartanyan, and A. Burrows, Astrophys. J. Lett. 937, L15 (2022).
  62. K. Maltsev, F. R. N. Schneider, I. Mandel, B. Müller, A. Heger, F. K. Röpke, and E. Laplace, Astron. Astrophys. 700, A20 (2025).
  63. A. Harada, S. Nishikawa, and S. Yamada, Astrophys. J. 925, 117 (2022).
  64. S. Abbar, M.-R. Wu, and Z. Xiong, Phys. Rev. D 109, 043024 (2024).
  65. S. Richers, J. Froustey, S. Ghosh, F. Foucart, and J. Gomez, Phys. Rev. D 110, 103019 (2024).
  66. D. Fan, D. E. Willcox, C. DeGrendele, M. Zingale, and A. Nonaka, Astrophys. J. 940, 134 (2022).
  67. M. Li, T. M. Sprouse, B. S. Meyer, and M. R. Mumpower, Phys. Lett. B 848, 138385 (2024).
  68. A. Grichener, M. Renzo, W. E. Kerzendorf, R. Farmer, S. E. de Mink, E. P. Bellinger, C.-k. Chan, N. Chen, E. Farag, and S. Justham, Astrophys. J. Suppl. Ser. 279, 49 (2025).
  69. Y. Saito, I. Dillmann, R. Krücken, M. R. Mumpower, and R. Surman, J. Phys. G 52, 055201 (2025).
  70. D. M. Siegel and B. D. Metzger, Phys. Rev. Lett. 119, 231102 (2017).
  71. S. Fujibayashi, M. Shibata, S. Wanajo, K. Kiuchi, K. Kyutoku, and Y. Sekiguchi, Phys. Rev. D 101, 083029 (2020).
  72. A. Bauswein, S. Goriely, and H.-T. Janka, Astrophys. J. 773, 78 (2013).
  73. K. Hotokezaka, K. Kiuchi, K. Kyutoku, H. Okawa, Y.-i. Sekiguchi, M. Shibata, and K. Taniguchi, Phys. Rev. D 87, 024001 (2013).
  74. D. Radice, F. Galeazzi, J. Lippuner, L. F. Roberts, C. D. Ott, and L. Rezzolla, Mon. Not. R. Astron. Soc. 460, 3255 (2016).
  75. F. Foucart, R. Haas, M. D. Duez, E. O’Connor, C. D. Ott, L. Roberts, L. E. Kidder, J. Lippuner, H. P. Pfeiffer, and M. A. Scheel, Phys. Rev. D 93, 044019 (2016).
  76. O. Just, M. Obergaulinger, and H.-T. Janka, Mon. Not. R. Astron. Soc. 453, 3386 (2015).
  77. W. R. Hix and B. S. Meyer, Nucl. Phys. A777, 188 (2006).
  78. M. Wang, W. Huang, F. Kondev, G. Audi, and S. Naimi, Chin. Phys. C 45, 030003 (2021).
  79. P. Möller, J. R. Nix, W. D. Myers, and W. J. Swiatecki, At. Data Nucl. Data Tables 59, 185 (1995).
  80. A. W. Steiner, M. Hempel, and T. Fischer, Astrophys. J. 774, 17 (2013).
  81. M. Obergaulinger, Astrophysical magnetohydrodynamics and radiative transfer, Dissertation, Technische Universität München, München, 2008.
  82. F. X. Timmes and F. D. Swesty, Astrophys. J. Suppl. Ser. 126, 501 (2000).
  83. F. X. Timmes and D. Arnett, Astrophys. J. Suppl. Ser. 125, 277 (1999).
  84. I. Goodfellow, Y. Bengio, and A. Courville, Deep Learning (MIT Press, Cambridge, MA, 2016).
  85. C. E. Collins, A. Bauswein, S. A. Sim, V. Vijayan, G. Martínez-Pinedo, O. Just, L. J. Shingles, and M. Kromer, Mon. Not. R. Astron. Soc. 521, 1858 (2023).
  86. J. D. J. Mendoza-Temis, M. R. Wu, K. Langanke, G. Martínez-Pinedo, A. Bauswein, and H. T. Janka, Phys. Rev. C 92, 055805 (2015).
  87. P. Möller, B. Pfeiffer, and K. L. Kratz, Phys. Rev. C 67, 055802 (2003).
  88. S. Goriely, N. Chamel, and J. M. Pearson, Phys. Rev. C 82, 035804 (2010).
  89. T. Marketin, L. Huther, and G. Martínez-Pinedo, Phys. Rev. C 93, 025805 (2016).
  90. S. E. Woosley, W. D. Arnett, and D. D. Clayton, Astrophys. J. Suppl. Ser. 26, 231 (1973).
  91. B. S. Meyer, T. D. Krishnan, and D. D. Clayton, Astrophys. J. 462, 825 (1996).
  92. B. S. Meyer, T. D. Krishnan, and D. D. Clayton, Astrophys. J. 498, 808 (1998).
  93. A. Perego, D. Vescovi, A. Fiore, L. Chiesa, C. Vogl, S. Benetti, S. Bernuzzi, M. Branchesi, E. Cappellaro, S. Cristallo, A. Flörs, W. E. Kerzendorf, and D. Radice, Astrophys. J. 925, 22 (2022).
  94. A. Sneppen, O. Just, A. Bauswein, R. Damgaard, D. Watson, L. J. Shingles, C. E. Collins, S. A. Sim, Z. Xiong, G. Martinez-Pinedo, T. Soultanis, and V. Vijayan, Phys. Rev. D 113, 063038 (2026).
  95. O. Just, S. Goriely, H. T. Janka, S. Nagataki, and A. Bauswein, Mon. Not. R. Astron. Soc. 509, 1377 (2022).
  96. J. Frank, A. R. King, and D. J. Raine, Accretion Power in Astrophysics, 3rd ed. (Cambridge University Press, Cambridge, England, 2002).
  97. S. Goriely, Eur. Phys. J. A 51, 22 (2015).
  98. R. Ardevol-Pulpillo, H.-T. Janka, O. Just, and A. Bauswein, Mon. Not. R. Astron. Soc. 485, 4754 (2019).
  99. E. O’Connor, Astrophys. Journal, Suppl. Ser. 219, 24 (2015).
  100. B. D. Metzger, A. L. Piro, and E. Quataert, Mon. Not. R. Astron. Soc. 396, 304 (2009).
  101. A. Neuweiler, T. Dietrich, M. Bulla, S. V. Chaurasia, S. Rosswog, and M. Ujevic, Phys. Rev. D 107, 023016 (2023).
  102. O. Just, I. Kullmann, S. Goriely, A. Bauswein, H. T. Janka, and C. E. Collins, Mon. Not. R. Astron. Soc. 510, 2820 (2022).
  103. J. Barnes, D. Kasen, M.-R. Wu, and G. Martínez-Pinedo, Astrophys. J. 829, 110 (2016).
  104. T. Dieselhorst, W. Cook, S. Bernuzzi, and D. Radice, Symmetry 13, 2157 (2021).
  105. O. Just, Z. Xiong, and G. Martínez-Pinedo, Rhine: R-process heating implementation in hydrodynamic simulations with neural networks, 10.5281/zenodo.15864447 (2025).
  106. O. Just, Z. Xiong, and G. Martínez-Pinedo, RHINE,https://git.gsi.de/nucastro_public/rhine (2025).
  107. S. van der Walt, S. C. Colbert, and G. Varoquaux, Comput. Sci. Eng. 13, 22 (2011).
  108. J. D. Hunter, Comput. Sci. Eng. 9, 90 (2007).
  109. P. Virtanen, R. Gommers, T. E. Oliphant, M. Haberland, T. Reddy, D. Cournapeau, E. Burovski, P. Peterson, W. Weckesser, J. Bright, S. J. van der Walt, M. Brett, J. Wilson, K. J. Millman, N. Mayorov, A. R. J. Nelson et al., Nat. Methods 17, 261 (2020).
  110. Wes McKinney, in Proceedings of the 9th python in Science Conference, edited by Stéfan van der Walt and Jarrod Millman (SciPy Proceedings, Austin, 2010), pp. 56–61.
  111. 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, in Advances in Neural Information Processing Systems 32 (Curran Associates, Inc., Vancouver, 2019), pp. 8024–8035.
  112. T. Plewa and E. Müller, Astron. Astrophys. 342, 179 (1999), https://ui.adsabs.harvard.edu/abs/1999A%26A...342..179P/exportcitation.
  113. T. A. Thompson, A. Burrows, and B. S. Meyer, Astrophys. J. 562, 887 (2001).

Outline

Information

Phys. Rev. D 113, 083022– Published 16 April, 2026
  • Received 11 July 2025
  • Accepted 16 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.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation