Exploring molecular excited states holds immense significance across organic chemistry, chemical biology, and materials science. Understanding the photophysical properties of molecular chromophores is crucial for designing nature-inspired functional molecules, with applications ranging from photosynthesis to pharmaceuticals. Non-adiabatic molecular dynamics simulations are powerful tools to investigate the photochemistry of molecules and materials, but demand extensive computing resources, especially for complex molecules and environments. To address these challenges, the integration of machine learning has emerged. Machine learning algorithms can be used to analyse vast datasets and accelerate discoveries by identifying relationships between geometrical features and ground as well as excited-state properties. However, challenges persist, including the acquisition of accurate excited-state data and managing the complexity of the data. This article provides an overview of recent and best practices in machine learning for non-adiabatic molecular dynamics, focusing on pre-processing, surface fitting, and post-processing of data.

Aix Marseille University CNRS ICR Marseille France

Center for Scalable Data Analytics and Artificial Intelligence Dresden Leipzig Germany

Computer Chemistry Center Friedrich Alexander Universität Erlangen Nürnberg Nägelsbachstraße 25 91052 Erlangen Germany

Department of Chemistry and Chemical Biology Northeastern University 805 Columbus Avenue Boston MA 02120 USA

Department of Chemistry University College London 20 Gordon Street London WC1H 0AJ UK

Department of Physical Chemistry University of Chemistry and Technology Technická 5 162 28 Prague Czech Republic

Hoffmann Institute of Advanced Materials Shenzhen Polytechnic University 7098 Liuxian Boulevard Shenzhen Guangdong 518055 P R China

Institute of Theoretical Chemistry Faculty of Chemistry University of Vienna Währinger Straße 17 1090 Wien Austria

Research Platform on Accelerating Photoreaction Discovery University of Vienna Währinger Straße 17 1090 Vienna Austria

Vienna Doctoral School in Chemistry University of Vienna Währinger Straße 42 1090 Vienna Austria

Wilhelm Ostwald Institute Leipzig Universityy Linnéstraße 2 04103 Leipzig Germany

Zobrazit více v PubMed

Gómez S., Galván I. F., Lindh R. and González L., in Motivation and Basic Concepts, John Wiley & Sons, Ltd, 2020, ch. 1, pp. 1–12

Cwiklik L. Pederzoli M. Baig M. W. Kývala M. Pittner J. J. Biophys. J. 2020;118:79a. doi: 10.1016/j.bpj.2019.11.603. DOI

Mathew S. Yella A. Gao P. Humphry-Baker R. Curchod B. F. E. Ashari-Astani N. Tavernelli I. Rothlisberger U. Nazeeruddin M. K. Grätzel M. Nat. Chem. 2014;6:242–247. doi: 10.1038/nchem.1861. PubMed DOI

Westermayr J. Gilkes J. Barrett R. Maurer R. J. Nat. Comput. Sci. 2023;3:139–148. doi: 10.1038/s43588-022-00391-1. PubMed DOI

Mai S., Marquetand P. and González L., in Surface hopping molecular dynamics, John Wiley & Sons, Ltd, 2020, pp. 499–530

Toldo J. M. do Casal M. T. Ventura E. do Monte S. A. Barbatti M. Phys. Chem. Chem. Phys. 2023;25:8293–8316. doi: 10.1039/D3CP00247K. PubMed DOI PMC

Westermayr J. and Marquetand P., Machine Learning in Chemistry: The Impact of Artificial Intelligence, The Royal Society of Chemistry, 2020, pp. 76–108

Westermayr J. Marquetand P. Chem. Rev. 2020;121:9873–9926. doi: 10.1021/acs.chemrev.0c00749. PubMed DOI PMC

Li J. Lopez S. A. Acc. Chem. Res. 2022;55:1972–1984. doi: 10.1021/acs.accounts.2c00288. PubMed DOI

Quantum Chemistry in the Age of Machine Learning, ed. P. O. Dral, Elsevier, 1st edn, 2022 PubMed

Zhang C. Zhong Y. Tao Z.-G. Qin X. Shang H. Lan Z. Prezhdo O. V. Gong X.-G. Chu W. Xiang H. Nat. Commun. 2025;16:2033. doi: 10.1038/s41467-025-57328-1. PubMed DOI PMC

Editorials, Nat. Synth., 2023, 2, 459

Unke O. T. Chmiela S. Sauceda H. E. Gastegger M. Poltavsky I. Schütt K. T. Tkatchenko A. Müller K.-R. Chem. Rev. 2021;121:10142. doi: 10.1021/acs.chemrev.0c01111. PubMed DOI PMC

Dral P. O. Barbatti M. Nat. Rev. Chem. 2021;5:388–405. doi: 10.1038/s41570-021-00278-1. PubMed DOI

Unke O. T. Chmiela S. Sauceda H. E. Gastegger M. Poltavsky I. Schütt K. T. Tkatchenko A. Müller K.-R. Chem. Rev. 2021;121:10142–10186. doi: 10.1021/acs.chemrev.0c01111. PubMed DOI PMC

Li J. Reiser P. Boswell B. R. Eberhard A. Burns N. Z. Friederich P. Lopez S. A. Chem. Sci. 2021;12:5302–5314. doi: 10.1039/D0SC05610C. PubMed DOI PMC

Westermayr J. Gastegger M. Menger M. F. S. J. Mai S. González L. Marquetand P. Chem. Sci. 2019;10:8100–8107. doi: 10.1039/C9SC01742A. PubMed DOI PMC

Akimov A. V. J. Phys. Chem. Lett. 2021;12:12119–12128. doi: 10.1021/acs.jpclett.1c03823. PubMed DOI

Akimov A. V. J. Phys. Chem. Lett. 2018;9:6096–6102. doi: 10.1021/acs.jpclett.8b02826. PubMed DOI

Westermayr J., Gastegger M. and Marquetand P., SchNarc, 2021, https://github.com/schnarc/SchNarc PubMed PMC

Westermayr J. Marquetand P. Mach. Learn.: Sci. Technol. 2020;1:043001.

Li J. Reiser P. Boswell B. R. Eberhard A. Burns N. Z. Friederich P. Lopez S. A. Chem. Sci. 2021;12:5302–5314. doi: 10.1039/D0SC05610C. PubMed DOI PMC

Li C. Hou S. Xie C. J. Chem. Theory Comput. 2023;19:3063–3079. doi: 10.1021/acs.jctc.2c01074. PubMed DOI

Mausenberger S. Müller C. Tkatchenko A. Marquetand P. González L. Westermayr J. Chem. Sci. 2024;15:15880–15890. doi: 10.1039/D4SC04164J. PubMed DOI PMC

Zhang L. Pios S. V. Martyka M. Ge F. Hou Y.-F. Chen Y. Chen L. Jankowska J. Barbatti M. Dral P. O. J. Chem. Theory Comput. 2024;20:5043–5057. doi: 10.1021/acs.jctc.4c00468. PubMed DOI

Barrett R., Ortner C. and Westermayr J., arXiv, 2025, arXiv:2502.12870, 10.48550/arXiv.2502.12870 DOI

Tiefenbacher M. X. Bachmair B. Chen C. G. Westermayr J. Marquetand P. Dietschreit J. C. B. González L. Digital Discovery. 2025;4:1478–1491. doi: 10.1039/D5DD00044K. PubMed DOI PMC

Curchod B. F. E. Martínez T. J. Chem. Rev. 2018;118:3305–3336. doi: 10.1021/acs.chemrev.7b00423. PubMed DOI

Crespo-Otero R. Barbatti M. Chem. Rev. 2018;118:7026–7068. doi: 10.1021/acs.chemrev.7b00577. PubMed DOI

Nelson T. R. White A. J. Bjorgaard J. A. Sifain A. E. Zhang Y. Nebgen B. Fernandez-Alberti S. Mozyrsky D. Roitberg A. E. Tretiak S. Chem. Rev. 2020;120:2215–2287. doi: 10.1021/acs.chemrev.9b00447. PubMed DOI

Tapavicza E. Bellchambers G. D. Vincent J. C. Furche F. Phys. Chem. Chem. Phys. 2013;15:18336–18348. doi: 10.1039/C3CP51514A. PubMed DOI

Cigrang L. L. E. Curchod B. F. E. Ingle R. A. Kelly A. Mannouch J. R. Accomasso D. Alijah A. Barbatti M. Chebbi W. Došlić N. Eklund E. C. Fernandez-Alberti S. Freibert A. González L. Granucci G. Hernández F. J. Hernández-Rodríguez J. Jain A. Janoš J. Kassal I. Kirrander A. Lan Z. Larsson H. R. Lauvergnat D. Le Dé B. Lee Y. Maitra N. T. Min S. K. Peláez D. Picconi D. Qiu Z. Raucci U. Robertson P. Sangiogo Gil E. Sapunar M. Schürger P. Sinnott P. Tretiak S. Tikku A. Vindel-Zandbergen P. Worth G. A. Agostini F. Gómez S. Ibele L. M. Prlj A. J. Phys. Chem. A. 2025;129(31):7023–7050. doi: 10.1021/acs.jpca.5c02171. PubMed DOI PMC

Li J. Lopez S. A. Chem. Phys. Rev. 2023;4:031309. doi: 10.1063/5.0159247. DOI

Beck M. H. Jäckle A. Worth G. A. Meyer H. D. Phys. Rep. 2000;324:1–105. doi: 10.1016/S0370-1573(99)00047-2. DOI

Worth G. A. and Lasorne B., Quantum Chemistry and Dynamics of Excited States, John Wiley & Sons Ltd, 2020, pp. 413–433

Curchod B. F. E. Martínez T. J. Chem. Rev. 2018;118:3305–3336. doi: 10.1021/acs.chemrev.7b00423. PubMed DOI

Levine B. G. Coe J. D. Virshup A. M. Martínez T. J. Chem. Phys. 2008;347:3–16. doi: 10.1016/j.chemphys.2008.01.014. DOI

Ben-Nun M. Martínez T. J. Chem. Phys. Lett. 1998;298:57–65. doi: 10.1016/S0009-2614(98)01115-4. DOI

Kirrander A. and Vacher M., Quantum Chemistry and Dynamics of Excited States, John Wiley & Sons Ltd, 2020, pp. 469–497

Tully J. C. Faraday Discuss. 1998;110:407–419. doi: 10.1039/A801824C. DOI

Tully J. C. J. Chem. Phys. 2012;137:22A301. doi: 10.1063/1.4757762. PubMed DOI

Tully J. C. Catal. Lett. 1991;9:205–217. doi: 10.1007/BF00773179. DOI

Plasser F. Gómez S. Menger M. F. S. J. Mai S. González L. Phys. Chem. Chem. Phys. 2019;21:57–69. doi: 10.1039/C8CP05662E. PubMed DOI

Mai S. Marquetand P. González L. Wiley Interdiscip. Rev.:Comput. Mol. Sci. 2018;8:e1370. PubMed PMC

Mai S., Bachmair B., Gagliardi L., Gallmetzer H. G., Grünewald L., Hennefarth M., Machholdt Høyer N., Korsaye F. A., Mausenberger S., Oppel M., Piteša T., Polonius S., Sangiogo Gil E., Shu Y., Singer N. K., Tiefenbacher M. X., Truhlar D., Vörös D., Zhang L. and Gonzalez L., sharc-md/sharc4: SHARC Release 4.0, 2025, 10.5281/zenodo.15496427 DOI

Barbatti M. Ruckenbauer M. Plasser F. Pittner J. Granucci G. Persico M. Lischka H. Wiley Interdiscip. Rev.:Comput. Mol. Sci. 2014;4:26–33.

Barbatti M. Bondanza M. Crespo-Otero R. Demoulin B. Dral P. O. Granucci G. Kossoski F. Lischka H. Mennucci B. Mukherjee S. Pederzoli M. Persico M. Pinheiro Jr M. Pittner J. Plasser F. Sangiogo Gil E. Stojanovic L. J. Chem. Theory Comput. 2022;18:6851–6865. doi: 10.1021/acs.jctc.2c00804. PubMed DOI PMC

Du L. Lan Z. J. Chem. Theory Comput. 2015;11:1360–1374. doi: 10.1021/ct501106d. PubMed DOI

Plasser F. Granucci G. Pittner J. Barbatti M. Persico M. Lischka H. J. Chem. Phys. 2012;137:22A514. doi: 10.1063/1.4738960. PubMed DOI

Mai S. Marquetand P. González L. Int. J. Quantum Chem. 2015;115:1215–1231. doi: 10.1002/qua.24891. DOI

Cui G. Thiel W. J. Chem. Phys. 2014;141:124101. doi: 10.1063/1.4894849. PubMed DOI

Jaeger H. M. Fischer S. Prezhdo O. V. J. Chem. Phys. 2012;137:22A545. doi: 10.1063/1.4757100. PubMed DOI

Persico M. and Granucci G., Photochemistry: A modern theoretical perspective, Springer, 2018

Mai S. González L. Angew. Chem., Int. Ed. 2020;59:16832–16846. doi: 10.1002/anie.201916381. PubMed DOI PMC

Papineau T. V. Jacquemin D. Vacher M. J. Phys. Chem. Lett. 2024;15:636–643. doi: 10.1021/acs.jpclett.3c03014. PubMed DOI

Janoš J. Slavíček P. J. Chem. Theory Comput. 2023;19:8273–8284. doi: 10.1021/acs.jctc.3c00908. PubMed DOI PMC

Westermayr J. Gastegger M. Schütt K. T. Maurer R. J. J. Chem. Phys. 2021;154:230903. doi: 10.1063/5.0047760. PubMed DOI

Plasser F. Crespo-Otero R. Pederzoli M. Pittner J. Lischka H. Barbatti M. J. Chem. Theory Comput. 2014;10:1395–1405. doi: 10.1021/ct4011079. PubMed DOI

Barbatti M. and Crespo-Otero R., Density-Functional Methods for Excited States, Springer International Publishing, Cham, 2016, pp. 415–444

Huix-Rotllant M., Ferré N. and Barbatti M., Quantum Chemistry and Dynamics of Excited States, John Wiley & Sons Ltd, 2020, pp. 13–46

Casanova D. Krylov A. I. Phys. Chem. Chem. Phys. 2020;22:4326–4342. doi: 10.1039/C9CP06507E. PubMed DOI

Lefrancois D. Tuna D. Martínez T. J. Dreuw A. J. Chem. Theory Comput. 2017;13:4436–4441. doi: 10.1021/acs.jctc.7b00634. PubMed DOI

Lee S. Shostak S. Filatov M. Choi C. H. J. Phys. Chem. A. 2019;123:6455–6462. doi: 10.1021/acs.jpca.9b06142. PubMed DOI

Christiansen O. Koch H. Jørgensen P. Chem. Phys. Lett. 1995;243:409–418. doi: 10.1016/0009-2614(95)00841-Q. DOI

Dreuw A. Wormit M. Wiley Interdiscip. Rev.: Comput. Mol. Sci. 2015;5:82–95.

Levine B. G. Ko C. Quenneville J. Martínez T. J. Mol. Phys. 2006;104:1039–1051. doi: 10.1080/00268970500417762. DOI

Huix-Rotllant M., Nikiforov A., Thiel W. and Filatov M., Density-Functional Methods for Excited States, Springer International Publishing, Cham, 2016, pp. 445–476

Stojanović L. Bai S. Nagesh J. Izmaylov A. F. Crespo-Otero R. Lischka H. Barbatti M. Molecules. 2016;21:1603. doi: 10.3390/molecules21111603. PubMed DOI PMC

Horbatenko Y. Sadiq S. Lee S. Filatov M. Choi C. H. J. Chem. Theory Comput. 2021;17:848–859. doi: 10.1021/acs.jctc.0c01074. PubMed DOI

Lee I. S. Filatov M. Min S. K. J. Chem. Theory Comput. 2019;15:3021–3032. doi: 10.1021/acs.jctc.9b00132. PubMed DOI

Park W. Lee S. Huix-Rotllant M. Filatov M. Choi C. H. J. Phys. Chem. Lett. 2021;12:4339–4346. doi: 10.1021/acs.jpclett.1c00712. PubMed DOI

Andersson K. Malmqvist P. A. Roos B. O. Sadlej A. J. Wolinski K. J. Phys. Chem. 1990;94:5483–5488. doi: 10.1021/j100377a012. DOI

Angeli C. Cimiraglia R. Malrieu J.-P. J. Chem. Phys. 2002;117:9138–9153. doi: 10.1063/1.1515317. DOI

Lischka H. Nachtigallová D. Aquino A. J. A. Szalay P. G. Plasser F. Machado F. B. C. Barbatti M. Chem. Rev. 2018;118:7293–7361. doi: 10.1021/acs.chemrev.8b00244. PubMed DOI

Szalay P. G. Müller T. Gidofalvi G. Lischka H. Shepard R. Chem. Rev. 2012;112:108–181. doi: 10.1021/cr200137a. PubMed DOI

David Sherrill C. Schaefer H. F. Adv. Quant. Chem. 1999;34:143–269. doi: 10.1016/S0065-3276(08)60532-8. DOI

Shiozaki T. Győrffy W. Celani P. Werner H.-J. J. Chem. Phys. 2011;135:081106. doi: 10.1063/1.3633329. PubMed DOI

Battaglia S. Lindh R. J. Chem. Theory Comput. 2020;16:1555–1567. doi: 10.1021/acs.jctc.9b01129. PubMed DOI PMC

Battaglia S. Lindh R. J. Chem. Phys. 2021;154:034102. doi: 10.1063/5.0030944. PubMed DOI

Levine D. S. Hait D. Tubman N. M. Lehtola S. Whaley K. B. Head-Gordon M. J. Chem. Theory Comput. 2020;16:2340–2354. doi: 10.1021/acs.jctc.9b01255. PubMed DOI

Park J. W. J. Chem. Theory Comput. 2021;17:4092–4104. doi: 10.1021/acs.jctc.1c00272. PubMed DOI

Polyak I. Hutton L. Crespo-Otero R. Barbatti M. Knowles P. J. J. Chem. Theory Comput. 2019;15:3929–3940. doi: 10.1021/acs.jctc.9b00396. PubMed DOI

Gómez S. Ibele L. M. González L. Phys. Chem. Chem. Phys. 2019;21:4871–4878. doi: 10.1039/C8CP07766E. PubMed DOI

Li Manni G. Carlson R. K. Luo S. Ma D. Olsen J. Truhlar D. G. Gagliardi L. J. Chem. Theory Comput. 2014;10:3669–3680. doi: 10.1021/ct500483t. PubMed DOI

Gagliardi L. Truhlar D. G. Li Manni G. Carlson R. K. Hoyer C. E. Bao J. L. Acc. Chem. Res. 2017;50:66–73. doi: 10.1021/acs.accounts.6b00471. PubMed DOI

Lykhin A. O. Truhlar D. G. Gagliardi L. J. Am. Chem. Soc. 2021;143:5878–5889. doi: 10.1021/jacs.1c00989. PubMed DOI

Westermayr J. Gastegger M. Vörös D. Panzenboeck L. Joerg F. González L. Marquetand P. Nat. Chem. 2022;14:914–919. doi: 10.1038/s41557-022-00950-z. PubMed DOI

Kidwell N. M. Li H. Wang X. Bowman J. M. Lester M. I. Nat. Chem. 2016;8:509–514. doi: 10.1038/nchem.2488. PubMed DOI

Atalar K. Rath Y. Crespo-Otero R. Booth G. H. Faraday Discuss. 2024;254:542–569. doi: 10.1039/D4FD00062E. PubMed DOI

de Souza B. Angew. Chem., Int. Ed. 2025:e202500393. PubMed

Grimme S. J. Chem. Theory Comput. 2019;15:2847–2862. doi: 10.1021/acs.jctc.9b00143. PubMed DOI

Pracht P. Grimme S. Bannwarth C. Bohle F. Ehlert S. Feldmann G. Gorges J. Müller M. Neudecker T. Plett C. Spicher S. Steinbach P. Wesolowski P. A. Zeller F. J. Chem. Phys. 2024;160:114110. doi: 10.1063/5.0197592. PubMed DOI

Steinmetzer J. Kupfer S. Gräfe S. Int. J. Quantum Chem. 2021;121:e26390. doi: 10.1002/qua.26390. DOI

Sanz García J. Maskri R. Mitrushchenkov A. Joubert-Doriol L. J. Chem. Theory Comput. 2024;20:5643–5654. doi: 10.1021/acs.jctc.4c00326. PubMed DOI

Pracht P. Bannwarth C. J. Chem. Theory Comput. 2022;18:6370–6385. doi: 10.1021/acs.jctc.2c00578. PubMed DOI

Vandaele E. Mališ M. Luber S. J. Chem. Theory Comput. 2024;20:856–872. doi: 10.1021/acs.jctc.3c01008. PubMed DOI

Keal T. W. Koslowski A. Thiel W. Theor. Chem. Acc. 2007;118:837–844.

Galvan I. F. Delcey M. G. Pedersen T. B. Aquilante F. Lindh R. J. Chem. Theory Comput. 2016;12:3636–3653. doi: 10.1021/acs.jctc.6b00384. PubMed DOI

Pinheiro Jr M. Zhang S. Dral P. O. Barbatti M. Sci. Data. 2023;10:95. doi: 10.1038/s41597-023-01998-3. PubMed DOI PMC

Suchan J. Hollas D. Curchod B. F. E. Slavíček P. Faraday Discuss. 2018;212:307–330. doi: 10.1039/C8FD00088C. PubMed DOI

Crespo-Otero R. Barbatti M. Theor. Chem. Acc. 2012;131:7026–7068.

Sršeň Š. Sita J. Slavíček P. Ladányi V. Heger D. J. Chem. Theory Comput. 2020;16:6428–6438. doi: 10.1021/acs.jctc.0c00579. PubMed DOI

Barbatti M. Sen K. Int. J. Quantum Chem. 2016;116:762–771. doi: 10.1002/qua.25049. DOI

Ceriotti M. Bussi G. Parrinello M. J. Chem. Theory Comput. 2010;6:1170–1180. doi: 10.1021/ct900563s. DOI

Zobel J. P. Nogueira J. J. González L. Phys. Chem. Chem. Phys. 2019;21:13906–13915. doi: 10.1039/C8CP03273D. PubMed DOI

Li C. Voth G. A. J. Chem. Theory Comput. 2022;18:599–604. doi: 10.1021/acs.jctc.1c01085. PubMed DOI PMC

Althorpe S. C. Eur. Phys. J. B. 2021;94:155. doi: 10.1140/epjb/s10051-021-00155-2. DOI

Ceriotti M. Manolopoulos D. E. Parrinello M. J. Chem. Phys. 2011;134:84104. doi: 10.1063/1.3556661. PubMed DOI

Pios S. V. Gelin M. F. Ullah A. Dral P. O. Chen L. J. Phys. Chem. Lett. 2024;15:2325–2331. doi: 10.1021/acs.jpclett.4c00107. PubMed DOI

Chen W.-K. Liu X.-Y. Fang W.-H. Dral P. O. Cui G. J. Phys. Chem. Lett. 2018;9:6702–6708. doi: 10.1021/acs.jpclett.8b03026. PubMed DOI

Hu D. Xie Y. Li X. Li L. Lan Z. J. Phys. Chem. Lett. 2018;9:2725–2732. doi: 10.1021/acs.jpclett.8b00684. PubMed DOI

Martyka M. Zhang L. Ge F. Hou Y.-F. Jankowska J. Barbatti M. Dral P. O. npj Comput. Mater. 2025;11:1–12. doi: 10.1038/s41524-025-01636-z. PubMed DOI PMC

Dral P. O. Barbatti M. Thiel W. J. Phys. Chem. Lett. 2018;9:5660–5663. doi: 10.1021/acs.jpclett.8b02469. PubMed DOI PMC

Li J. Stein R. Adrion D. M. Lopez S. A. J. Am. Chem. Soc. 2021;143:20166–20175. doi: 10.1021/jacs.1c07725. PubMed DOI

Smith J. S. Nebgen B. Lubbers N. Isayev O. Roitberg A. E. J. Chem. Phys. 2018;148:241733. doi: 10.1063/1.5023802. PubMed DOI

Behler J. Int. J. Quantum Chem. 2015;115:1032–1050. doi: 10.1002/qua.24890. DOI

Hu D. Xie Y. Li X. Li L. Lan Z. J. Phys. Chem. Lett. 2018;9:2725–2732. doi: 10.1021/acs.jpclett.8b00684. PubMed DOI

Zhang L. Hou Y.-F. Ge F. Dral P. O. Phys. Chem. Chem. Phys. 2023;25:23467–23476. doi: 10.1039/D3CP03515H. PubMed DOI

Behler J. J. Phys.:Condens. Matter. 2014;26:183001. doi: 10.1088/0953-8984/26/18/183001. PubMed DOI

Musil F. Willatt M. J. Langovoy M. A. Ceriotti M. J. Chem. Theory Comput. 2019;15:906–915. doi: 10.1021/acs.jctc.8b00959. PubMed DOI

Abdar M. Pourpanah F. Hussain S. Rezazadegan D. Liu L. Ghavamzadeh M. Fieguth P. Cao X. Khosravi A. Acharya U. R. et al. . Inf. Fusion. 2021;76:243–297. doi: 10.1016/j.inffus.2021.05.008. DOI

Gawlikowski J. Tassi C. R. N. Ali M. Lee J. Humt M. Feng J. Kruspe A. Triebel R. Jung P. Roscher R. et al. . Artif. Intell. Rev. 2023;56:1513–1589. doi: 10.1007/s10462-023-10562-9. DOI

Deringer V. L. Bartók A. P. Bernstein N. Wilkins D. M. Ceriotti M. Csányi G. Chem. Rev. 2021;121:10073–10141. doi: 10.1021/acs.chemrev.1c00022. PubMed DOI PMC

Kellner M. Ceriotti M. Mach. Learn.: Sci. Technol. 2024;5:035006.

Bigi F. Chong S. Ceriotti M. Grasselli F. Mach. Learn.: Sci. Technol. 2024;5:045018.

Chong S. Bigi F. Grasselli F. Loche P. Kellner M. Ceriotti M. Faraday Discuss. 2025;256:322–344. doi: 10.1039/D4FD00101J. PubMed DOI PMC

Mukherjee S. Barbatti M. Res. Chem. 2022;4:100521.

Batzner S. Musaelian A. Sun L. Geiger M. Mailoa J. P. Kornbluth M. Molinari N. Smidt T. E. Kozinsky B. Nat. Commun. 2022;13:2453. doi: 10.1038/s41467-022-29939-5. PubMed DOI PMC

Batatia I., Batzner S., Kovács D. P., Musaelian A., Simm G. N., Drautz R., Ortner C., Kozinsky B. and Csányi G., arXiv, 2022, arXiv:2205.06643, 10.48550/arXiv.2205.06643 PubMed DOI PMC

Schütt K., Unke O. and Gastegger M., Proceedings of the 38th International Conference on Machine Learning, 2021, pp. 9377–9388

Christensen A. S. von Lilienfeld O. A. Mach. Learn.: Sci. Technol. 2020;1:045018.

Artrith N. Butler K. T. Coudert F.-X. Han S. Isayev O. Jain A. Walsh A. Nat. Chem. 2021;13:505–508. doi: 10.1038/s41557-021-00716-z. PubMed DOI

Conical Intersections: Electronic Structure, Dynamics & Spectroscopy, ed. W. Domcke, D. Yarkony and H. Köppel, World Scientific, 2004

Westermayr J. Faber F. A. Christensen A. S. von Lilienfeld O. A. Marquetand P. Mach. Learn.: Sci. Technol. 2020;1:025009.

Westermayr J. Gastegger M. Marquetand P. J. Phys. Chem. Lett. 2020;11:3828–3834. doi: 10.1021/acs.jpclett.0c00527. PubMed DOI PMC

Richardson J. O. J. Chem. Phys. 2023;158:011102. doi: 10.1063/5.0133191. PubMed DOI

An H. Baeck K. K. Chem. Phys. Lett. 2018;696:100–105. doi: 10.1016/j.cplett.2018.02.036. DOI

Baeck K. K. An H. J. Chem. Phys. 2017;146:064107. doi: 10.1063/1.4975323. PubMed DOI

do Casal M. T. Toldo J. M. Jr M. P. Barbatti M. Open Res. Europe. 2021;1:49. PubMed PMC

Shu Y. Zhang L. Chen X. Sun S. Huang Y. Truhlar D. G. J. Chem. Theory Comput. 2022;18:1320–1328. doi: 10.1021/acs.jctc.1c01080. PubMed DOI

Shu Y. Zhang L. Chen X. Sun S. Huang Y. Truhlar D. G. J. Chem. Theory Comput. 2022;18:1320–1328. doi: 10.1021/acs.jctc.1c01080. PubMed DOI

Landau L. Phys. Z. Sowjetunion. 1932;2:46–51.

Zener C. Fowler R. H. Proc. R. Soc. London, Ser. A. 1932;137:696–702.

Suchan J. Janoš J. Slavíček P. J. Chem. Theory Comput. 2020;16:5809–5820. doi: 10.1021/acs.jctc.0c00512. PubMed DOI

Ishida T. Nanbu S. Nakamura H. Int. Rev. Phys. Chem. 2017;36:229–285.

Yu L. Xu C. Lei Y. Zhu C. Wen Z. Phys. Chem. Chem. Phys. 2014;16:25883–25895. doi: 10.1039/C4CP03498H. PubMed DOI

Shchepanovska D. Shannon R. J. Curchod B. F. E. Glowacki D. R. J. Phys. Chem. A. 2021;125:3473–3488. doi: 10.1021/acs.jpca.1c01260. PubMed DOI

Yue L. Yu L. Xu C. Lei Y. Liu Y. Zhu C. ChemPhysChem. 2017;18:1274–1287. doi: 10.1002/cphc.201700049. PubMed DOI

Yue L. Yu L. Xu C. Zhu C. Liu Y. Phys. Chem. Chem. Phys. 2020;22:11440–11451. doi: 10.1039/D0CP00811G. PubMed DOI

Mead C. A. Truhlar D. G. J. Chem. Phys. 1982;77:6090–6098. doi: 10.1063/1.443853. DOI

Shu Y. Varga Z. Kanchanakungwankul S. Zhang L. Truhlar D. G. J. Phys. Chem. A. 2022;126:992–1018. doi: 10.1021/acs.jpca.1c10583. PubMed DOI

Zhu X. Yarkony D. R. J. Chem. Phys. 2014;140:024112. doi: 10.1063/1.4857335. PubMed DOI

Shen Y. Yarkony D. R. J. Phys. Chem. A. 2020;124:4539–4548. doi: 10.1021/acs.jpca.0c02763. PubMed DOI

Abrol R. Kuppermann A. J. Chem. Phys. 2002;116:1035–1062. doi: 10.1063/1.1419257. DOI

Köppel H. Faraday Discuss. 2004;127:35–47. doi: 10.1039/B314471B. PubMed DOI

Zhu X. Yarkony D. R. J. Chem. Phys. 2010;132:104101. doi: 10.1063/1.3324982. PubMed DOI

Varga Z. Parker K. A. Truhlar D. G. Phys. Chem. Chem. Phys. 2018;20:26643–26659. doi: 10.1039/C8CP03410A. PubMed DOI

Atchity G. J. Ruedenberg K. Theor. Chem. Acc. 1997;97:47–58.

Pacher T. Köppel H. Cederbaum L. S. J. Chem. Phys. 1991;95:6668–6680. doi: 10.1063/1.461537. DOI

Yang K. R. Xu X. Truhlar D. G. Chem. Phys. Lett. 2013;573:84–89. doi: 10.1016/j.cplett.2013.04.036. DOI

Wittenbrink N. Venghaus F. Williams D. Eisfeld W. J. Chem. Phys. 2016;145:184108. doi: 10.1063/1.4967258. PubMed DOI

Subotnik J. E. Yeganeh S. Cave R. J. Ratner M. A. J. Chem. Phys. 2008;129:244101. doi: 10.1063/1.3042233. PubMed DOI

Hoyer C. E. Parker K. Gagliardi L. Truhlar D. G. J. Chem. Phys. 2016;144:194101. doi: 10.1063/1.4948728. PubMed DOI

Sršeň Š. von Lilienfeld O. A. Slavíček P. Phys. Chem. Chem. Phys. 2024;26:4306–4319. doi: 10.1039/D3CP05685F. PubMed DOI PMC

Williams D. M. Eisfeld W. J. Chem. Phys. 2018;149:204106. doi: 10.1063/1.5053664. PubMed DOI

Guan Y. Zhang D. H. Guo H. Yarkony D. R. Phys. Chem. Chem. Phys. 2019;21:14205–14213. doi: 10.1039/C8CP06598E. PubMed DOI

Shu Y. Truhlar D. G. J. Chem. Theory Comput. 2020;16:6456–6464. doi: 10.1021/acs.jctc.0c00623. PubMed DOI

Shu Y. Varga Z. Sampaio De Oliveira-Filho A. G. Truhlar D. G. J. Chem. Theory Comput. 2021;17:1106–1116. doi: 10.1021/acs.jctc.0c01110. PubMed DOI

Axelrod S. Shakhnovich E. Gómez-Bombarelli R. Nat. Commun. 2022;13:3440. doi: 10.1038/s41467-022-30999-w. PubMed DOI PMC

Shu Y. Kryven J. Sampaio de Oliveira-Filho A. G. Zhang L. Song G.-L. Li S. L. Meana-Pañeda R. Fu B. Bowman J. M. Truhlar D. G. J. Chem. Phys. 2019;151:104311. doi: 10.1063/1.5111547. PubMed DOI

Guan Y. Guo H. Yarkony D. R. J. Chem. Theory Comput. 2020;16:302–313. doi: 10.1021/acs.jctc.9b00898. PubMed DOI

Werner H.-J. Knowles P. J. Knizia G. Manby F. R. Schütz M. Wiley Interdiscip. Rev.: Comput. Mol. Sci. 2012;2:242–253.

Christopoulou G. Freibert A. Worth G. A. J. Chem. Phys. 2021;154:124127. doi: 10.1063/5.0043720. PubMed DOI

Opalka D. Domcke W. J. Chem. Phys. 2013;138:224103. doi: 10.1063/1.4808358. PubMed DOI

Wang T. Y. Neville S. P. Schuurman M. S. J. Phys. Chem. Lett. 2023;14:7780–7786. doi: 10.1021/acs.jpclett.3c01649. PubMed DOI PMC

Shu Y. Varga Z. Parameswaran A. M. Truhlar D. G. J. Chem. Theory Comput. 2024;20:7042–7051. doi: 10.1021/acs.jctc.4c00424. PubMed DOI

Gutleb T. S., Barrett R., Westermayr J. and Ortner C., arXiv, 2024, arXiv:2407.03731, 10.48550/arXiv.2407.03731 DOI

Zaheer M. Kottur S. Ravanbakhsh S. Poczos B. Salakhutdinov R. R. Smola A. J. Adv. Neural Inf. Process. Syst. 2017;30:3391–3401.

Batatia I. Kovacs D. P. Simm G. Ortner C. Csanyi G. Adv. Neural Inf. Process. Syst. 2022:11423–11436.

Li J. Stein R. Adrion D. M. Lopez S. A. J. Am. Chem. Soc. 2021;143:20166–20175. doi: 10.1021/jacs.1c07725. PubMed DOI

Schütt K. T. Sauceda H. E. Kindermans P.-J. Tkatchenko A. Müller K.-R. J. Chem. Phys. 2018;148:241722. doi: 10.1063/1.5019779. PubMed DOI

Musil F. Grisafi A. Bartók A. P. Ortner C. Csányi G. Ceriotti M. Chem. Rev. 2021;121:9759–9815. doi: 10.1021/acs.chemrev.1c00021. PubMed DOI

Musil F. Grisafi A. Bartók A. P. Ortner C. Csányi G. Ceriotti M. Chem. Rev. 2021;121:9759–9815. doi: 10.1021/acs.chemrev.1c00021. PubMed DOI

Hansen K. Montavon G. Biegler F. Fazli S. Rupp M. Scheffler M. von Lilienfeld O. A. Tkatchenko A. Müller K.-R. J. Chem. Theory Comput. 2013;9:3404–3419. doi: 10.1021/ct400195d. PubMed DOI

Dral P. O. Owens A. Yurchenko S. N. Thiel W. J. Chem. Phys. 2017;146:244108. doi: 10.1063/1.4989536. PubMed DOI

Bartók A. P. Csányi G. Int. J. Quantum Chem. 2015;115:1051–1057. doi: 10.1002/qua.24927. DOI

Rupp M. Tkatchenko A. Müller K.-R. von Lilienfeld O. A. Phys. Rev. Lett. 2012;108:058301. doi: 10.1103/PhysRevLett.108.058301. PubMed DOI

Chmiela S. Sauceda H. E. Poltavsky I. Müller K. R. Tkatchenko A. Comput. Phys. Commun. 2019;240:38–45. doi: 10.1016/j.cpc.2019.02.007. DOI

Hou Y.-F. Ge F. Dral P. O. J. Chem. Theory Comput. 2023;19:2369–2379. doi: 10.1021/acs.jctc.2c01038. PubMed DOI

Westermayr J. Faber F. A. Christensen A. S. von Lilienfeld O. A. Marquetand P. Mach. Learn.: Sci. Technol. 2020;1:025009.

Pinheiro M. Ge F. Ferré N. Dral P. O. Barbatti M. Chem. Sci. 2021;12:14396–14413. doi: 10.1039/D1SC03564A. PubMed DOI PMC

Bartók A. P. Kondor R. Csányi G. Phys. Rev. B:Condens. Matter Mater. Phys. 2013;87:184115. doi: 10.1103/PhysRevB.87.184115. DOI

Behler J. J. Chem. Phys. 2011;134:074106. doi: 10.1063/1.3553717. PubMed DOI

Christensen A. S. Bratholm L. A. Faber F. A. Anatole von Lilienfeld O. J. Chem. Phys. 2020;152:044107. doi: 10.1063/1.5126701. PubMed DOI

Wilson A. and Nickisch H., Proceedings of the 32nd International Conference on Machine Learning, Lille, France, 2015, pp. 1775–1784

Wilson A. G., Hu Z., Salakhutdinov R. and Xing E. P., Proceedings of the 19th International Conference on Artificial Intelligence and Statistics, Cadiz, Spain, 2016, pp. 370–378

Manzhos S. Ihara M. J. Phys. Chem. A. 2023;127:7823–7835. doi: 10.1021/acs.jpca.3c02949. PubMed DOI

Rupp M. Int. J. Quantum Chem. 2015;115:1058–1073. doi: 10.1002/qua.24954. DOI

Dral P. O. Ge F. Xue B.-X. Hou Y.-F. Pinheiro Jr M. Huang J. Barbatti M. Top. Curr. Chem. 2021;379:27. doi: 10.1007/s41061-021-00339-5. PubMed DOI PMC

Rupp M. von Lilienfeld O. A. Burke K. J. Chem. Phys. 2018;148:241401. doi: 10.1063/1.5043213. PubMed DOI

Pedregosa F. Varoquaux G. Gramfort A. Michel V. Thirion B. Grisel O. Blondel M. Prettenhofer P. Weiss R. Dubourg V. Vanderplas J. Passos A. Cournapeau D. Brucher M. Perrot M. Duchesnay E. J. Mach. Learn. Res. 2011;12:2825–2830.

Huo H. Rupp M. Mach. Learn.: Sci. Technol. 2022;3:045017.

Dral P. O. J. Comput. Chem. 2019;40:2339–2347. doi: 10.1002/jcc.26004. PubMed DOI

Dral P. O., Zheng P., Xue B.-X., Ge F., Hou Y.-F., Max Pinheiro J., Su Y., Dai Y. and Chen Y., MLatom: a Package for Atomistic Simulations with Machine Learning, version 2.3.3., Xiamen University, Xiamen, China, 2013–2023

Bartók A. P. Payne M. C. Kondor R. Csányi G. Phys. Rev. Lett. 2010;104:136403. doi: 10.1103/PhysRevLett.104.136403. PubMed DOI

Deringer V. L. Bartók A. P. Bernstein N. Wilkins D. M. Ceriotti M. Csányi G. Chem. Rev. 2021;121:10073–10141. doi: 10.1021/acs.chemrev.1c00022. PubMed DOI PMC

Christensen A. S., Faber F. A., Huang B., Bratholm L. A., Tkatchenko A., Müller K.-R. and von Lilienfeld O. A., ML: A Python Toolkit for Quantum Machine Learning, 2017, https://github.com/qmlcode/qml

Chmiela S. Vassilev-Galindo V. Unke O. T. Kabylda A. Sauceda H. E. Tkatchenko A. Müller K. R. Sci. Adv. 2023;9:eadf0873. doi: 10.1126/sciadv.adf0873. PubMed DOI PMC

Sauceda H. E. Gálvez-González L. E. Chmiela S. Paz-Borbón L. O. Müller K.-R. Tkatchenko A. Nat. Commun. 2022;13:3733. doi: 10.1038/s41467-022-31093-x. PubMed DOI PMC

Chmiela S. Sauceda H. E. Müller K.-R. Tkatchenko A. Nat. Commun. 2018;9:3887. doi: 10.1038/s41467-018-06169-2. PubMed DOI PMC

Bigi F. Pozdnyakov S. N. Ceriotti M. J. Chem. Phys. 2024;161:44116. doi: 10.1063/5.0208746. PubMed DOI

Behler J. Chem. Rev. 2021;121:10037–10072. doi: 10.1021/acs.chemrev.0c00868. PubMed DOI

Behler J. Parrinello M. Phys. Rev. Lett. 2007;98:146401. doi: 10.1103/PhysRevLett.98.146401. PubMed DOI

Devereux C. Smith J. S. Huddleston K. K. Barros K. Zubatyuk R. Isayev O. Roitberg A. E. J. Chem. Theory Comput. 2020;16:4192–4202. doi: 10.1021/acs.jctc.0c00121. PubMed DOI

Drautz R. Phys. Rev. B. 2019;99:014104. doi: 10.1103/PhysRevB.99.014104. DOI

Batzner S. Musaelian A. Sun L. Geiger M. Mailoa J. P. Kornbluth M. Molinari N. Smidt T. E. Kozinsky B. Nat. Commun. 2022;13:2453. doi: 10.1038/s41467-022-29939-5. PubMed DOI PMC

Frank J. T. Unke O. T. Müller K.-R. Chmiela S. Nat. Commun. 2024;15:6539. doi: 10.1038/s41467-024-50620-6. PubMed DOI PMC

Schütt K. T. Kessel P. Gastegger M. Nicoli K. A. Tkatchenko A. Müller K. R. J. Chem. Theory Comput. 2019;15:448–455. doi: 10.1021/acs.jctc.8b00908. PubMed DOI

Gastegger M. McSloy A. Luya M. Schutt K. T. Maurer R. J. J. Chem. Phys. 2020;153:044123. doi: 10.1063/5.0012911. PubMed DOI

Gastegger M. Schütt K. T. Müller K.-R. Chem. Sci. 2021;12:11473–11483. doi: 10.1039/D1SC02742E. PubMed DOI PMC

Martinka J. Zhang L. Hou Y.-F. Martyka M. Pittner J. Barbatti M. Dral P. O. NPJ Comput. Mater. 2025;11(1):132. doi: 10.1038/s41524-025-01636-z. PubMed DOI PMC

Martyka M., Tong X.-Y., Jankowska J. and Dral P. O., ChemRxiv, 2025, 10.26434/chemrxiv-2025-j207x DOI

Li J. Lopez S. A. Chem.–Eur. J. 2022;28:e202200651. doi: 10.1002/chem.202200651. PubMed DOI

Zhu C. Nakamura H. J. Chem. Phys. 1995;102:7448–7461. doi: 10.1063/1.469057. DOI

Wang L. Salguero C. Lopez S. A. Li J. Chem. 2024;10:2295–2310.

Li Z. Hernández F. J. Salguero C. Lopez S. A. Crespo-Otero R. Li J. Nat. Commun. 2025;16:1194. doi: 10.1038/s41467-025-56480-y. PubMed DOI PMC

Barrett R., Dietschreit J. C. and Westermayr J., arXiv, 2025, arXiv:2502.21045, 10.48550/arXiv.2502.21045 DOI

Batatia I., Benner P., Chiang Y., Elena A. M., Kovács D. P., Riebesell J., Advincula X. R., Asta M., Avaylon M., Baldwin W. J., Berger F., Bernstein N., Bhowmik A., Blau S. M., Cărare V., Darby J. P., De S., Pia F. D., Deringer V. L., Elijošius R., El-Machachi Z., Falcioni F., Fako E., Ferrari A. C., Genreith-Schriever A., George J., Goodall R. E. A., Grey C. P., Grigorev P., Han S., Handley W., Heenen H. H., Hermansson K., Holm C., Jaafar J., Hofmann S., Jakob K. S., Jung H., Kapil V., Kaplan A. D., Karimitari N., Kermode J. R., Kroupa N., Kullgren J., Kuner M. C., Kuryla D., Liepuoniute G., Margraf J. T., Magdău I.-B., Michaelides A., Moore J. H., Naik A. A., Niblett S. P., Norwood S. W., O'Neill N., Ortner C., Persson K. A., Reuter K., Rosen A. S., Schaaf L. L., Schran C., Shi B. X., Sivonxay E., Stenczel T. K., Svahn V., Sutton C., Swinburne T. D., Tilly J., van der Oord C., Varga-Umbrich E., Vegge T., Vondrák M., Wang Y., Witt W. C., Zills F. and Csányi G., A foundation model for atomistic materials chemistry, 2024, https://arxiv.org/abs/2401.00096

Kovács D. P., Moore J. H., Browning N. J., Batatia I., Horton J. T., Pu Y., Kapil V., Witt W. C., Magdąu I.-B., Cole D. J. and Cs'anyi G., MACE-OFF: Transferable Short Range Machine Learning Force Fields for Organic Molecules, 2025, https://arxiv.org/abs/2312.15211 PubMed PMC

Cignoni E. Suman D. Nigam J. Cupellini L. Mennucci B. Ceriotti M. ACS Cent. Sci. 2024;10:637–648. doi: 10.1021/acscentsci.3c01480. PubMed DOI PMC

Westermayr J. Maurer R. J. Chem. Sci. 2021;12:10755–10764. doi: 10.1039/D1SC01542G. PubMed DOI PMC

Behler J. Chem. Rev. 2021;121:10037–10072. doi: 10.1021/acs.chemrev.0c00868. PubMed DOI

Richter M. Marquetand P. González-Vázquez J. Sola I. González L. J. Chem. Theory Comput. 2011;7:1253–1258. doi: 10.1021/ct1007394. PubMed DOI

Richings G. W. Polyak I. Spinlove K. E. Worth G. A. Burghardt I. Lasorne B. Int. Rev. Phys. Chem. 2015;34:269–308.

Spinlove K. E. Richings G. Robb M. A. Worth G. A. Faraday Discuss. 2018;212:191–215. doi: 10.1039/C8FD00090E. PubMed DOI

Błasiak B. Brey D. Koch W. Martinazzo R. Burghardt I. Chem. Phys. 2022;560:111542. doi: 10.1016/j.chemphys.2022.111542. DOI

Frankcombe T. J. Collins M. A. Worth G. A. Chem. Phys. Lett. 2010;489:242–247. doi: 10.1016/j.cplett.2010.02.068. DOI

Richings G. W. Habershon S. J. Chem. Phys. 2018;148:134116. doi: 10.1063/1.5024869. PubMed DOI

Koch W. Bonfanti M. Eisenbrandt P. Nandi A. Fu B. Bowman J. Tannor D. Burghardt I. J. Chem. Phys. 2019;151:064121. doi: 10.1063/1.5113579. DOI

Manzhos S. Carrington J. Tucker M. J. J. Chem. Phys. 2006;125:194105. doi: 10.1063/1.2387950. PubMed DOI

Westermayr J. Marquetand P. J. Chem. Phys. 2020;153:154112. doi: 10.1063/5.0021915. PubMed DOI

Zhu Y. Peng J. Xu C. Lan Z. J. Phys. Chem. Lett. 2024;15:9601–9619. doi: 10.1021/acs.jpclett.4c01751. PubMed DOI

Häse F. Galván I. F. Aspuru-Guzik A. Lindh R. Vacher M. Chem. Sci. 2019;10:2298–2307. doi: 10.1039/C8SC04516J. PubMed DOI PMC

Hare S. R. Bratholm L. A. Glowacki D. R. Carpenter B. K. Chem. Sci. 2019;10:9954–9968. doi: 10.1039/C9SC02742D. PubMed DOI PMC

Zhu Y. Peng J. Kang X. Xu C. Lan Z. Phys. Chem. Chem. Phys. 2022;24:24362–24382. doi: 10.1039/D2CP03323B. PubMed DOI

Li X. Hu D. Xie Y. Lan Z. J. Chem. Phys. 2018;149:244104. doi: 10.1063/1.5048049. PubMed DOI

Häse F. Galván I. F. Aspuru-Guzik A. Lindh R. Vacher M. Chem. Sci. 2019;10:2298–2307. doi: 10.1039/C8SC04516J. PubMed DOI PMC

Häse F. Galván I. F. Aspuru-Guzik A. Lindh R. Vacher M. J. Phys.:Conf. Ser. 2020;1412:042003. doi: 10.1088/1742-6596/1412/4/042003. DOI

Bishop C. M., Pattern Recognition and Machine Learning, Springer-Verlag, New York, 1st edn, 2006

Schölkopf B., Smola A. and Müller K.-R., Artificial Neural Networks — ICANN'97, 1997, pp. 583–588

Mead A. J. R. Stat. Soc. Ser. D. 1992;41:27–39. doi: 10.2307/2348634. DOI

Cox M. A. A. and Cox T. F., Handbook of Data Visualization, Springer Berlin Heidelberg, 2008, pp. 315–347

Xue B. X. Barbatti M. Dral P. O. J. Phys. Chem. A. 2020;124:7199–7210. doi: 10.1021/acs.jpca.0c05310. PubMed DOI PMC

Ghojogh B., Crowley M., Karray F. and Ghodsi A., in Multidimensional Scaling, Sammon Mapping, and Isomap, Springer International Publishing, Cham, 2023, pp. 185–205

Pezzotti N. Höllt T. Lelieveldt B. Eisemann E. Vilanova A. Comput. Graph. Forum. 2016;35:21–30. doi: 10.1111/cgf.12878. DOI

Ankerst M. Breunig M. M. Kriegel H.-P. Sander J. SIGMOD Rec. 1999;28:49–60. doi: 10.1145/304181.304187. DOI

Zhang T. Ramakrishnan R. Livny M. Data Min. Knowl. Discov. 1997;1:141–182. doi: 10.1023/A:1009783824328. DOI

Zhang T. Ramakrishnan R. Livny M. SIGMOD Rec. 1996;25:103–114. doi: 10.1145/235968.233324. DOI

Jin X. and Han J., Encyclopedia of Machine Learning, Springer US, 2010, pp. 563–564

Schubert E. Sander J. Ester M. Kriegel H. P. Xu X. ACM Trans. Database Syst. 2017;42:1–21. doi: 10.1145/3068335. DOI

Klem H. Hocky G. M. McCullagh M. J. Chem. Theory Comput. 2022;18:3218–3230. doi: 10.1021/acs.jctc.1c01290. PubMed DOI PMC

Rousseeuw P. J. J. Comput. Appl. Math. 1987;20:53–65. doi: 10.1016/0377-0427(87)90125-7. DOI

Ekemeyong Awong L. E. Zielinska T. Sensors. 2023;23:7925. doi: 10.3390/s23187925. PubMed DOI PMC

Shannon P. Markiel A. Ozier O. Baliga N. S. Wang J. T. Ramage D. Amin N. Schwikowski B. Ideker T. Genome Res. 2003;13:2498–2504. doi: 10.1101/gr.1239303. PubMed DOI PMC

Tavenard R. Faouzi J. Vandewiele G. Divo F. Androz G. Holtz C. Payne M. Yurchak R. Rußwurm M. Kolar K. J. Mach. Learn. Res. 2020;21:1–6.

Drautz R. Phys. Rev. B. 2019;99:014104. doi: 10.1103/PhysRevB.99.014104. DOI

Singh K. Munchmeyer J. Weber L. Leser U. Bande A. J. Chem. Theory Comput. 2022;18:4408–4417. doi: 10.1021/acs.jctc.2c00255. PubMed DOI

Curth R. Röhrkasten T. E. Müller C. Westermayr J. Sci. Data. 2025;12:1300. doi: 10.1038/s41597-025-05443-5. PubMed DOI PMC