{"id":148308,"date":"2026-08-22T19:37:38","date_gmt":"2026-08-22T19:37:38","guid":{"rendered":"https:\/\/www.europesays.com\/ai\/148308\/"},"modified":"2026-08-22T19:37:38","modified_gmt":"2026-08-22T19:37:38","slug":"explainable-ai-learning-from-the-learners","status":"publish","type":"post","link":"https:\/\/www.europesays.com\/ai\/148308\/","title":{"rendered":"Explainable AI: learning from the learners"},"content":{"rendered":"<p class=\"c-article-references__text\" id=\"ref-CR1\">Jumper, J. et al. Highly accurate protein structure prediction with AlphaFold. Nature 596, 583\u2013589 (2021).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"10.1038\/s41586-021-03819-2\" data-track-item_id=\"10.1038\/s41586-021-03819-2\" data-track-value=\"article reference\" data-track-action=\"article reference\" href=\"https:\/\/doi.org\/10.1038%2Fs41586-021-03819-2\" aria-label=\"Article reference 1\" data-doi=\"10.1038\/s41586-021-03819-2\" target=\"_blank\">Article<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2021Natur.596..583J\" aria-label=\"ADS reference 1\" target=\"_blank\">ADS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"cas reference\" data-track-action=\"cas reference\" href=\"https:\/\/www.nature.com\/articles\/cas-redirect\/1:CAS:528:DC%2BB3MXhvVaktrrL\" aria-label=\"CAS reference 1\" target=\"_blank\">CAS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"pubmed reference\" data-track-action=\"pubmed reference\" href=\"http:\/\/www.ncbi.nlm.nih.gov\/entrez\/query.fcgi?cmd=Retrieve&amp;db=PubMed&amp;dopt=Abstract&amp;list_uids=34265844\" aria-label=\"PubMed reference 1\" target=\"_blank\">PubMed<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"pubmed central reference\" data-track-action=\"pubmed central reference\" href=\"http:\/\/www.ncbi.nlm.nih.gov\/pmc\/articles\/PMC8371605\" aria-label=\"PubMed Central reference 1\" target=\"_blank\">PubMed Central<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 1\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Highly%20accurate%20protein%20structure%20prediction%20with%20AlphaFold&amp;journal=Nature&amp;doi=10.1038%2Fs41586-021-03819-2&amp;volume=596&amp;pages=583-589&amp;publication_year=2021&amp;author=Jumper%2CJ\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<p class=\"c-article-references__text\" id=\"ref-CR2\">Guastoni, L., Rabault, J., Schlatter, P., Azizpour, H. &amp; Vinuesa, R. Deep reinforcement learning for turbulent drag reduction in channel flows. Eur. Phys. J. E 46, 27 (2023).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"10.1140\/epje\/s10189-023-00285-8\" data-track-item_id=\"10.1140\/epje\/s10189-023-00285-8\" data-track-value=\"article reference\" data-track-action=\"article reference\" href=\"https:\/\/doi.org\/10.1140%2Fepje%2Fs10189-023-00285-8\" aria-label=\"Article reference 2\" data-doi=\"10.1140\/epje\/s10189-023-00285-8\" target=\"_blank\">Article<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"cas reference\" data-track-action=\"cas reference\" href=\"https:\/\/www.nature.com\/articles\/cas-redirect\/1:CAS:528:DC%2BB3sXns1Wgtrg%3D\" aria-label=\"CAS reference 2\" target=\"_blank\">CAS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"pubmed reference\" data-track-action=\"pubmed reference\" href=\"http:\/\/www.ncbi.nlm.nih.gov\/entrez\/query.fcgi?cmd=Retrieve&amp;db=PubMed&amp;dopt=Abstract&amp;list_uids=37039923\" aria-label=\"PubMed reference 2\" target=\"_blank\">PubMed<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"pubmed central reference\" data-track-action=\"pubmed central reference\" href=\"http:\/\/www.ncbi.nlm.nih.gov\/pmc\/articles\/PMC10090012\" aria-label=\"PubMed Central reference 2\" target=\"_blank\">PubMed Central<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 2\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Deep%20reinforcement%20learning%20for%20turbulent%20drag%20reduction%20in%20channel%20flows&amp;journal=Eur.%20Phys.%20J.%20E&amp;doi=10.1140%2Fepje%2Fs10189-023-00285-8&amp;volume=46&amp;publication_year=2023&amp;author=Guastoni%2CL&amp;author=Rabault%2CJ&amp;author=Schlatter%2CP&amp;author=Azizpour%2CH&amp;author=Vinuesa%2CR\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<p class=\"c-article-references__text\" id=\"ref-CR3\">Lundberg, S. M. &amp; Lee, S.-I. A unified approach to interpreting model predictions. Advances in Neural Information Processing Systems 30, (2017).<\/p>\n<p class=\"c-article-references__text\" id=\"ref-CR4\">Rudin, C. Stop explaining black box machine learning models for high-stakes decisions and use interpretable models instead. Nat. Mach. Intell. 1, 206\u2013215 (2019).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"10.1038\/s42256-019-0048-x\" data-track-item_id=\"10.1038\/s42256-019-0048-x\" data-track-value=\"article reference\" data-track-action=\"article reference\" href=\"https:\/\/doi.org\/10.1038%2Fs42256-019-0048-x\" aria-label=\"Article reference 4\" data-doi=\"10.1038\/s42256-019-0048-x\" target=\"_blank\">Article<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"pubmed reference\" data-track-action=\"pubmed reference\" href=\"http:\/\/www.ncbi.nlm.nih.gov\/entrez\/query.fcgi?cmd=Retrieve&amp;db=PubMed&amp;dopt=Abstract&amp;list_uids=35603010\" aria-label=\"PubMed reference 4\" target=\"_blank\">PubMed<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"pubmed central reference\" data-track-action=\"pubmed central reference\" href=\"http:\/\/www.ncbi.nlm.nih.gov\/pmc\/articles\/PMC9122117\" aria-label=\"PubMed Central reference 4\" target=\"_blank\">PubMed Central<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 4\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Stop%20explaining%20black%20box%20machine%20learning%20models%20for%20high-stakes%20decisions%20and%20use%20interpretable%20models%20instead&amp;journal=Nat.%20Mach.%20Intell.&amp;doi=10.1038%2Fs42256-019-0048-x&amp;volume=1&amp;pages=206-215&amp;publication_year=2019&amp;author=Rudin%2CC\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<p class=\"c-article-references__text\" id=\"ref-CR5\">Camps-Valls, G. et al. Discovering causal relations and equations from data. Phys. Rep. 1044, 1\u201368 (2023).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"10.1016\/j.physrep.2023.10.005\" data-track-item_id=\"10.1016\/j.physrep.2023.10.005\" data-track-value=\"article reference\" data-track-action=\"article reference\" href=\"https:\/\/doi.org\/10.1016%2Fj.physrep.2023.10.005\" aria-label=\"Article reference 5\" data-doi=\"10.1016\/j.physrep.2023.10.005\" target=\"_blank\">Article<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2023PhR..1044....1C\" aria-label=\"ADS reference 5\" target=\"_blank\">ADS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"mathscinet reference\" data-track-action=\"mathscinet reference\" href=\"http:\/\/www.ams.org\/mathscinet-getitem?mr=4665016\" aria-label=\"MathSciNet reference 5\" target=\"_blank\">MathSciNet<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 5\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Discovering%20causal%20relations%20and%20equations%20from%20data&amp;journal=Phys.%20Rep.&amp;doi=10.1016%2Fj.physrep.2023.10.005&amp;volume=1044&amp;pages=1-68&amp;publication_year=2023&amp;author=Camps-Valls%2CG\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<p class=\"c-article-references__text\" id=\"ref-CR6\">Carloni, G., Berti, A. &amp; Colantonio, S. The role of causality in explainable artificial intelligence. WIREs Data Mining Knowl Discov. 15, e70015 (2025).<\/p>\n<p class=\"c-article-references__text\" id=\"ref-CR7\">Cremades, A., Hoyas, S. &amp; Vinuesa, R. Classically studied coherent structures only paint a partial picture of wall-bounded turbulence. Nat. Commun. (2025).<\/p>\n<p class=\"c-article-references__text\" id=\"ref-CR8\">Pearl, J. Causality (Cambridge University Press, 2009).<\/p>\n<p class=\"c-article-references__text\" id=\"ref-CR9\">Peters, J., Janzing, D. &amp; Scholkopf, B. Elements of Causal Inference: Foundations and Learning Algorithms. (MIT Press, 2017).<\/p>\n<p class=\"c-article-references__text\" id=\"ref-CR10\">Mart\u00ednez-S\u00e1nchez, A., Arranz, G. &amp; Lozano-Dur\u00e1n, A. Decomposing causality into its synergistic, unique, and redundant components. Nat. Commun. 15, 9296 (2024).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"10.1038\/s41467-024-53373-4\" data-track-item_id=\"10.1038\/s41467-024-53373-4\" data-track-value=\"article reference\" data-track-action=\"article reference\" href=\"https:\/\/doi.org\/10.1038%2Fs41467-024-53373-4\" aria-label=\"Article reference 10\" data-doi=\"10.1038\/s41467-024-53373-4\" target=\"_blank\">Article<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2024NatCo..15.9296M\" aria-label=\"ADS reference 10\" target=\"_blank\">ADS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"pubmed reference\" data-track-action=\"pubmed reference\" href=\"http:\/\/www.ncbi.nlm.nih.gov\/entrez\/query.fcgi?cmd=Retrieve&amp;db=PubMed&amp;dopt=Abstract&amp;list_uids=39487116\" aria-label=\"PubMed reference 10\" target=\"_blank\">PubMed<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"pubmed central reference\" data-track-action=\"pubmed central reference\" href=\"http:\/\/www.ncbi.nlm.nih.gov\/pmc\/articles\/PMC11530654\" aria-label=\"PubMed Central reference 10\" target=\"_blank\">PubMed Central<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 10\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Decomposing%20causality%20into%20its%20synergistic%2C%20unique%2C%20and%20redundant%20components&amp;journal=Nat.%20Commun.&amp;doi=10.1038%2Fs41467-024-53373-4&amp;volume=15&amp;publication_year=2024&amp;author=Mart%C3%ADnez-S%C3%A1nchez%2CA&amp;author=Arranz%2CG&amp;author=Lozano-Dur%C3%A1n%2CA\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<p class=\"c-article-references__text\" id=\"ref-CR11\">Sundararajan, M., Taly, A. &amp; Yan, Q. Axiomatic attribution for deep networks. Proc. 34th Int. Conf. Mach. Learn. 70, 3319\u20133328 (2017).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 11\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Axiomatic%20attribution%20for%20deep%20networks&amp;journal=Proc.%2034th%20Int.%20Conf.%20Mach.%20Learn.&amp;volume=70&amp;pages=3319-3328&amp;publication_year=2017&amp;author=Sundararajan%2CM&amp;author=Taly%2CA&amp;author=Yan%2CQ\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<p class=\"c-article-references__text\" id=\"ref-CR12\">Sch\u00f6lkopf, B. et al. On causal and anticausal learning. CML&#8217;12: Proceedings of the 29th International Coference on International Conference on Machine Learning. 459\u2013466 (2012).<\/p>\n<p class=\"c-article-references__text\" id=\"ref-CR13\">Peters, J., B\u00fchlmann, P. &amp; Meinshausen, N. Causal inference by using invariant prediction: identification and confidence intervals. J. R. Stat. Soc. Ser. B: Stat. Methodol. 78, 947\u20131012 (2016).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"10.1111\/rssb.12167\" data-track-item_id=\"10.1111\/rssb.12167\" data-track-value=\"article reference\" data-track-action=\"article reference\" href=\"https:\/\/doi.org\/10.1111%2Frssb.12167\" aria-label=\"Article reference 13\" data-doi=\"10.1111\/rssb.12167\" target=\"_blank\">Article<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"mathscinet reference\" data-track-action=\"mathscinet reference\" href=\"http:\/\/www.ams.org\/mathscinet-getitem?mr=3557186\" aria-label=\"MathSciNet reference 13\" target=\"_blank\">MathSciNet<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 13\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Causal%20inference%20by%20using%20invariant%20prediction%3A%20identification%20and%20confidence%20intervals&amp;journal=J.%20R.%20Stat.%20Soc.%20Ser.%20B%3A%20Stat.%20Methodol.&amp;doi=10.1111%2Frssb.12167&amp;volume=78&amp;pages=947-1012&amp;publication_year=2016&amp;author=Peters%2CJ&amp;author=B%C3%BChlmann%2CP&amp;author=Meinshausen%2CN\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<p class=\"c-article-references__text\" id=\"ref-CR14\">B\u00fchlmann, P. Invariance, causality and robustness. Stat. Sci. 35, 404\u2013426 (2020).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 14\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Invariance%2C%20causality%20and%20robustness&amp;journal=Stat.%20Sci.&amp;volume=35&amp;pages=404-426&amp;publication_year=2020&amp;author=B%C3%BChlmann%2CP\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<p class=\"c-article-references__text\" id=\"ref-CR15\">Sch\u00f6lkopf, B. Causality for machine learning. Probabilistic and causal inference: The works of Judea Pearl. 765\u2013804 (2022).<\/p>\n<p class=\"c-article-references__text\" id=\"ref-CR16\">Sch\u00f6lkopf, B. et al. Toward causal representation learning. Proc. IEEE 109, 612\u2013634 (2021).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"10.1109\/JPROC.2021.3058954\" data-track-item_id=\"10.1109\/JPROC.2021.3058954\" data-track-value=\"article reference\" data-track-action=\"article reference\" href=\"https:\/\/doi.org\/10.1109%2FJPROC.2021.3058954\" aria-label=\"Article reference 16\" data-doi=\"10.1109\/JPROC.2021.3058954\" target=\"_blank\">Article<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2021IEEEP.109..612S\" aria-label=\"ADS reference 16\" target=\"_blank\">ADS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 16\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Toward%20causal%20representation%20learning&amp;journal=Proc.%20IEEE&amp;doi=10.1109%2FJPROC.2021.3058954&amp;volume=109&amp;pages=612-634&amp;publication_year=2021&amp;author=Sch%C3%B6lkopf%2CB\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<p class=\"c-article-references__text\" id=\"ref-CR17\">Binkyte, R. et al. Trustworthy AI suffers from invariance conflicts, and causality is the solution. Preprint at arXiv <a href=\"https:\/\/doi.org\/10.48550\/arXiv.2605.02640\" data-track=\"click_references\" data-track-action=\"external reference\" data-track-value=\"external reference\" data-track-label=\"10.48550\/arXiv.2605.02640\" rel=\"nofollow noopener\" target=\"_blank\">https:\/\/doi.org\/10.48550\/arXiv.2605.02640<\/a> (2026).<\/p>\n<p class=\"c-article-references__text\" id=\"ref-CR18\">Wei, J. et al. Xai4extremes: An interpretable machine learning framework for understanding extreme-weather precursors under climate change. Tackling Climate Change with Machine Learning (ICLR. 2025).<\/p>\n<p class=\"c-article-references__text\" id=\"ref-CR19\">Camps-Valls, G. et al. Artificial intelligence for modeling and understanding extreme weather and climate events. Nat. Commun. 16, 1919 (2025).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"10.1038\/s41467-025-56573-8\" data-track-item_id=\"10.1038\/s41467-025-56573-8\" data-track-value=\"article reference\" data-track-action=\"article reference\" href=\"https:\/\/doi.org\/10.1038%2Fs41467-025-56573-8\" aria-label=\"Article reference 19\" data-doi=\"10.1038\/s41467-025-56573-8\" target=\"_blank\">Article<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2025NatCo..16.1919C\" aria-label=\"ADS reference 19\" target=\"_blank\">ADS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"cas reference\" data-track-action=\"cas reference\" href=\"https:\/\/www.nature.com\/articles\/cas-redirect\/1:CAS:528:DC%2BB2MXkvFylt70%3D\" aria-label=\"CAS reference 19\" target=\"_blank\">CAS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"pubmed reference\" data-track-action=\"pubmed reference\" href=\"http:\/\/www.ncbi.nlm.nih.gov\/entrez\/query.fcgi?cmd=Retrieve&amp;db=PubMed&amp;dopt=Abstract&amp;list_uids=39994190\" aria-label=\"PubMed reference 19\" target=\"_blank\">PubMed<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"pubmed central reference\" data-track-action=\"pubmed central reference\" href=\"http:\/\/www.ncbi.nlm.nih.gov\/pmc\/articles\/PMC11850610\" aria-label=\"PubMed Central reference 19\" target=\"_blank\">PubMed Central<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 19\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Artificial%20intelligence%20for%20modeling%20and%20understanding%20extreme%20weather%20and%20climate%20events&amp;journal=Nat.%20Commun.&amp;doi=10.1038%2Fs41467-025-56573-8&amp;volume=16&amp;publication_year=2025&amp;author=Camps-Valls%2CG\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<p class=\"c-article-references__text\" id=\"ref-CR20\">Yang, L. et al. Diffusion models: a comprehensive survey of methods and applications. ACM Comput. Surv. 56, 1\u201339 (2023).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"10.1145\/3554729\" data-track-item_id=\"10.1145\/3554729\" data-track-value=\"article reference\" data-track-action=\"article reference\" href=\"https:\/\/doi.org\/10.1145%2F3554729\" aria-label=\"Article reference 20\" data-doi=\"10.1145\/3554729\" target=\"_blank\">Article<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 20\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Diffusion%20models%3A%20a%20comprehensive%20survey%20of%20methods%20and%20applications&amp;journal=ACM%20Comput.%20Surv.&amp;doi=10.1145%2F3554729&amp;volume=56&amp;pages=1-39&amp;publication_year=2023&amp;author=Yang%2CL\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<p class=\"c-article-references__text\" id=\"ref-CR21\">Ameisen, E. et al. Circuit tracing: revealing computational graphs in language models. Transformer Circuits Thread (2025).<\/p>\n<p class=\"c-article-references__text\" id=\"ref-CR22\">Mengaldo, G. Explain the black box for the sake of science: the scientific method in the era of generative artificial intelligence. Preprint at arXiv <a href=\"https:\/\/doi.org\/10.48550\/arXiv.2406.10557\" data-track=\"click_references\" data-track-action=\"external reference\" data-track-value=\"external reference\" data-track-label=\"10.48550\/arXiv.2406.10557\" rel=\"nofollow noopener\" target=\"_blank\">https:\/\/doi.org\/10.48550\/arXiv.2406.10557<\/a> (2024).<\/p>\n<p class=\"c-article-references__text\" id=\"ref-CR23\">Turb\u00e9, H., Bjelogrlic, M., Lovis, C. &amp; Mengaldo, G. Evaluation of post-hoc interpretability methods in time-series classification. Nat. Mach. Intell. 5, 250\u2013260 (2023).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"10.1038\/s42256-023-00620-w\" data-track-item_id=\"10.1038\/s42256-023-00620-w\" data-track-value=\"article reference\" data-track-action=\"article reference\" href=\"https:\/\/doi.org\/10.1038%2Fs42256-023-00620-w\" aria-label=\"Article reference 23\" data-doi=\"10.1038\/s42256-023-00620-w\" target=\"_blank\">Article<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 23\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Evaluation%20of%20post-hoc%20interpretability%20methods%20in%20time-series%20classification&amp;journal=Nat.%20Mach.%20Intell.&amp;doi=10.1038%2Fs42256-023-00620-w&amp;volume=5&amp;pages=250-260&amp;publication_year=2023&amp;author=Turb%C3%A9%2CH&amp;author=Bjelogrlic%2CM&amp;author=Lovis%2CC&amp;author=Mengaldo%2CG\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<p class=\"c-article-references__text\" id=\"ref-CR24\">Wei, J., Turb\u00e9, H. &amp; Mengaldo, G. Revisiting the robustness of post-hoc interpretability methods. Preprint at arXiv <a href=\"https:\/\/doi.org\/10.48550\/arXiv:2407.19683\" data-track=\"click_references\" data-track-action=\"external reference\" data-track-value=\"external reference\" data-track-label=\"10.48550\/arXiv:2407.19683\" rel=\"nofollow noopener\" target=\"_blank\">https:\/\/doi.org\/10.48550\/arXiv:2407.19683<\/a> (2024).<\/p>\n<p class=\"c-article-references__text\" id=\"ref-CR25\">Bommer, P. L., Kretschmer, M., Hedstr\u00f6m, A., Bareeva, D. &amp; H\u00f6hne, M. M.-C. Finding the right XAI method\u2014a guide for the evaluation and ranking of explainable AI methods in climate science. Artif. Intell. Earth Syst. 3, e230074 (2024).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 25\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Finding%20the%20right%20XAI%20method%E2%80%94a%20guide%20for%20the%20evaluation%20and%20ranking%20of%20explainable%20AI%20methods%20in%20climate%20science&amp;journal=Artif.%20Intell.%20Earth%20Syst.&amp;volume=3&amp;publication_year=2024&amp;author=Bommer%2CPL&amp;author=Kretschmer%2CM&amp;author=Hedstr%C3%B6m%2CA&amp;author=Bareeva%2CD&amp;author=H%C3%B6hne%2CMM-C\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<p class=\"c-article-references__text\" id=\"ref-CR26\">Mamalakis, A., Barnes, E. A. &amp; Ebert-Uphoff, I. Investigation of the fidelity of explainable artificial intelligence methods in applications of convolutional neural networks in geoscience. Artif. Intell. Earth Syst. 1, e220012 (2022).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 26\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Investigation%20of%20the%20fidelity%20of%20explainable%20artificial%20intelligence%20methods%20in%20applications%20of%20convolutional%20neural%20networks%20in%20geoscience&amp;journal=Artif.%20Intell.%20Earth%20Syst.&amp;volume=1&amp;publication_year=2022&amp;author=Mamalakis%2CA&amp;author=Barnes%2CEA&amp;author=Ebert-Uphoff%2CI\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<p class=\"c-article-references__text\" id=\"ref-CR27\">Mamalakis, A., Ebert-Uphoff, I. &amp; Barnes, E. A. Neural network attribution methods for problems in geoscience: a novel synthetic benchmark dataset. Environ. Data Sci. <a href=\"https:\/\/doi.org\/10.1017\/eds.2022.7\" data-track=\"click_references\" data-track-action=\"external reference\" data-track-value=\"external reference\" data-track-label=\"10.1017\/eds.2022.7\" rel=\"nofollow noopener\" target=\"_blank\">https:\/\/doi.org\/10.1017\/eds.2022.7<\/a> (2022).<\/p>\n<p class=\"c-article-references__text\" id=\"ref-CR28\">Mamalakis, A., Ebert-Uphoff, I. &amp; Barnes, E. A. in Explainable artificial intelligence in meteorology and climate science: Model fine-tuning, calibrating trust, and learning new science. (eds Holzinger, A. et al.) xxAI\u2014Beyond Explainable AI: International Workshop, Held in Conjunction with ICML 2020, July 18, 2020, Vienna, Austria, Revised and Extended Papers Lecture Notes in Computer Science (Springer, Cham, 2022).<\/p>\n<p class=\"c-article-references__text\" id=\"ref-CR29\">Brunton, S. L., Proctor, J. L. &amp; Kutz, J. N. Discovering governing equations from data by sparse identification of nonlinear dynamical systems. Proc. Natl. Acad. Sci. 113, 3932\u20133937 (2016).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"10.1073\/pnas.1517384113\" data-track-item_id=\"10.1073\/pnas.1517384113\" data-track-value=\"article reference\" data-track-action=\"article reference\" href=\"https:\/\/doi.org\/10.1073%2Fpnas.1517384113\" aria-label=\"Article reference 29\" data-doi=\"10.1073\/pnas.1517384113\" target=\"_blank\">Article<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2016PNAS..113.3932B\" aria-label=\"ADS reference 29\" target=\"_blank\">ADS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"mathscinet reference\" data-track-action=\"mathscinet reference\" href=\"http:\/\/www.ams.org\/mathscinet-getitem?mr=3494081\" aria-label=\"MathSciNet reference 29\" target=\"_blank\">MathSciNet<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"cas reference\" data-track-action=\"cas reference\" href=\"https:\/\/www.nature.com\/articles\/cas-redirect\/1:CAS:528:DC%2BC28XkvFGis7w%3D\" aria-label=\"CAS reference 29\" target=\"_blank\">CAS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"pubmed reference\" data-track-action=\"pubmed reference\" href=\"http:\/\/www.ncbi.nlm.nih.gov\/entrez\/query.fcgi?cmd=Retrieve&amp;db=PubMed&amp;dopt=Abstract&amp;list_uids=27035946\" aria-label=\"PubMed reference 29\" target=\"_blank\">PubMed<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"pubmed central reference\" data-track-action=\"pubmed central reference\" href=\"http:\/\/www.ncbi.nlm.nih.gov\/pmc\/articles\/PMC4839439\" aria-label=\"PubMed Central reference 29\" target=\"_blank\">PubMed Central<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 29\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Discovering%20governing%20equations%20from%20data%20by%20sparse%20identification%20of%20nonlinear%20dynamical%20systems&amp;journal=Proc.%20Natl.%20Acad.%20Sci.&amp;doi=10.1073%2Fpnas.1517384113&amp;volume=113&amp;pages=3932-3937&amp;publication_year=2016&amp;author=Brunton%2CSL&amp;author=Proctor%2CJL&amp;author=Kutz%2CJN\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<p class=\"c-article-references__text\" id=\"ref-CR30\">Cranmer, M. Interpretable machine learning for science with PySR and symbolic regression. Preprint at arXiv <a href=\"https:\/\/doi.org\/10.48550\/arXiv.2305.01582\" data-track=\"click_references\" data-track-action=\"external reference\" data-track-value=\"external reference\" data-track-label=\"10.48550\/arXiv.2305.01582\" rel=\"nofollow noopener\" target=\"_blank\">https:\/\/doi.org\/10.48550\/arXiv.2305.01582<\/a> (2023).<\/p>\n<p class=\"c-article-references__text\" id=\"ref-CR31\">Vinuesa, R. et al. Decoding complexity: how machine learning is redefining scientific discovery. Preprint at Commun. Phys. 9, 168 (2026).<\/p>\n<p class=\"c-article-references__text\" id=\"ref-CR32\">No\u00e9, F., Tkatchenko, A., M\u00fcller, K.-R. &amp; Clementi, C. Machine learning for molecular simulation. Annu. Rev. Phys. Chem. 71, 361\u2013390 (2020).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"10.1146\/annurev-physchem-042018-052331\" data-track-item_id=\"10.1146\/annurev-physchem-042018-052331\" data-track-value=\"article reference\" data-track-action=\"article reference\" href=\"https:\/\/doi.org\/10.1146%2Fannurev-physchem-042018-052331\" aria-label=\"Article reference 32\" data-doi=\"10.1146\/annurev-physchem-042018-052331\" target=\"_blank\">Article<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2020ARPC...71..361N\" aria-label=\"ADS reference 32\" target=\"_blank\">ADS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"pubmed reference\" data-track-action=\"pubmed reference\" href=\"http:\/\/www.ncbi.nlm.nih.gov\/entrez\/query.fcgi?cmd=Retrieve&amp;db=PubMed&amp;dopt=Abstract&amp;list_uids=32092281\" aria-label=\"PubMed reference 32\" target=\"_blank\">PubMed<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 32\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Machine%20learning%20for%20molecular%20simulation&amp;journal=Annu.%20Rev.%20Phys.%20Chem.&amp;doi=10.1146%2Fannurev-physchem-042018-052331&amp;volume=71&amp;pages=361-390&amp;publication_year=2020&amp;author=No%C3%A9%2CF&amp;author=Tkatchenko%2CA&amp;author=M%C3%BCller%2CK-R&amp;author=Clementi%2CC\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<p class=\"c-article-references__text\" id=\"ref-CR33\">Baek, M. &amp; Baker, D. Deep learning and protein structure modeling. Nat. Methods 19, 13\u201314 (2022).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"10.1038\/s41592-021-01360-8\" data-track-item_id=\"10.1038\/s41592-021-01360-8\" data-track-value=\"article reference\" data-track-action=\"article reference\" href=\"https:\/\/doi.org\/10.1038%2Fs41592-021-01360-8\" aria-label=\"Article reference 33\" data-doi=\"10.1038\/s41592-021-01360-8\" target=\"_blank\">Article<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"cas reference\" data-track-action=\"cas reference\" href=\"https:\/\/www.nature.com\/articles\/cas-redirect\/1:CAS:528:DC%2BB38XpvFaqtA%3D%3D\" aria-label=\"CAS reference 33\" target=\"_blank\">CAS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"pubmed reference\" data-track-action=\"pubmed reference\" href=\"http:\/\/www.ncbi.nlm.nih.gov\/entrez\/query.fcgi?cmd=Retrieve&amp;db=PubMed&amp;dopt=Abstract&amp;list_uids=35017724\" aria-label=\"PubMed reference 33\" target=\"_blank\">PubMed<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 33\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Deep%20learning%20and%20protein%20structure%20modeling&amp;journal=Nat.%20Methods&amp;doi=10.1038%2Fs41592-021-01360-8&amp;volume=19&amp;pages=13-14&amp;publication_year=2022&amp;author=Baek%2CM&amp;author=Baker%2CD\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<p class=\"c-article-references__text\" id=\"ref-CR34\">Degrave, J. et al. Magnetic control of tokamak plasmas through deep reinforcement learning. Nature 602, 414\u2013419 (2022).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"10.1038\/s41586-021-04301-9\" data-track-item_id=\"10.1038\/s41586-021-04301-9\" data-track-value=\"article reference\" data-track-action=\"article reference\" href=\"https:\/\/doi.org\/10.1038%2Fs41586-021-04301-9\" aria-label=\"Article reference 34\" data-doi=\"10.1038\/s41586-021-04301-9\" target=\"_blank\">Article<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2022Natur.602..414D\" aria-label=\"ADS reference 34\" target=\"_blank\">ADS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"cas reference\" data-track-action=\"cas reference\" href=\"https:\/\/www.nature.com\/articles\/cas-redirect\/1:CAS:528:DC%2BB38XjvFOkur8%3D\" aria-label=\"CAS reference 34\" target=\"_blank\">CAS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"pubmed reference\" data-track-action=\"pubmed reference\" href=\"http:\/\/www.ncbi.nlm.nih.gov\/entrez\/query.fcgi?cmd=Retrieve&amp;db=PubMed&amp;dopt=Abstract&amp;list_uids=35173339\" aria-label=\"PubMed reference 34\" target=\"_blank\">PubMed<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"pubmed central reference\" data-track-action=\"pubmed central reference\" href=\"http:\/\/www.ncbi.nlm.nih.gov\/pmc\/articles\/PMC8850200\" aria-label=\"PubMed Central reference 34\" target=\"_blank\">PubMed Central<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 34\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Magnetic%20control%20of%20tokamak%20plasmas%20through%20deep%20reinforcement%20learning&amp;journal=Nature&amp;doi=10.1038%2Fs41586-021-04301-9&amp;volume=602&amp;pages=414-419&amp;publication_year=2022&amp;author=Degrave%2CJ\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<p class=\"c-article-references__text\" id=\"ref-CR35\">Tamayo, D. et al. Predicting the long-term stability of compact multiplanet systems. Proc. Natl. Acad. Sci. 117, 18194\u201318205 (2020).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"10.1073\/pnas.2001258117\" data-track-item_id=\"10.1073\/pnas.2001258117\" data-track-value=\"article reference\" data-track-action=\"article reference\" href=\"https:\/\/doi.org\/10.1073%2Fpnas.2001258117\" aria-label=\"Article reference 35\" data-doi=\"10.1073\/pnas.2001258117\" target=\"_blank\">Article<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2020PNAS..11718194T\" aria-label=\"ADS reference 35\" target=\"_blank\">ADS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"mathscinet reference\" data-track-action=\"mathscinet reference\" href=\"http:\/\/www.ams.org\/mathscinet-getitem?mr=4242696\" aria-label=\"MathSciNet reference 35\" target=\"_blank\">MathSciNet<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"cas reference\" data-track-action=\"cas reference\" href=\"https:\/\/www.nature.com\/articles\/cas-redirect\/1:CAS:528:DC%2BB3cXhs1SlurrO\" aria-label=\"CAS reference 35\" target=\"_blank\">CAS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"pubmed reference\" data-track-action=\"pubmed reference\" href=\"http:\/\/www.ncbi.nlm.nih.gov\/entrez\/query.fcgi?cmd=Retrieve&amp;db=PubMed&amp;dopt=Abstract&amp;list_uids=32675234\" aria-label=\"PubMed reference 35\" target=\"_blank\">PubMed<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"pubmed central reference\" data-track-action=\"pubmed central reference\" href=\"http:\/\/www.ncbi.nlm.nih.gov\/pmc\/articles\/PMC7414196\" aria-label=\"PubMed Central reference 35\" target=\"_blank\">PubMed Central<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 35\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Predicting%20the%20long-term%20stability%20of%20compact%20multiplanet%20systems&amp;journal=Proc.%20Natl.%20Acad.%20Sci.&amp;doi=10.1073%2Fpnas.2001258117&amp;volume=117&amp;pages=18194-18205&amp;publication_year=2020&amp;author=Tamayo%2CD\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<p class=\"c-article-references__text\" id=\"ref-CR36\">Parker, L. et al. Astroclip: a cross-modal foundation model for galaxies. Mon. Not. R. Astron. Soc. 531, 4990\u20135011 (2024).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"10.1093\/mnras\/stae1450\" data-track-item_id=\"10.1093\/mnras\/stae1450\" data-track-value=\"article reference\" data-track-action=\"article reference\" href=\"https:\/\/doi.org\/10.1093%2Fmnras%2Fstae1450\" aria-label=\"Article reference 36\" data-doi=\"10.1093\/mnras\/stae1450\" target=\"_blank\">Article<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2024MNRAS.531.4990P\" aria-label=\"ADS reference 36\" target=\"_blank\">ADS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"cas reference\" data-track-action=\"cas reference\" href=\"https:\/\/www.nature.com\/articles\/cas-redirect\/1:CAS:528:DC%2BB2MXhvVansrY%3D\" aria-label=\"CAS reference 36\" target=\"_blank\">CAS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 36\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Astroclip%3A%20a%20cross-modal%20foundation%20model%20for%20galaxies&amp;journal=Mon.%20Not.%20R.%20Astron.%20Soc.&amp;doi=10.1093%2Fmnras%2Fstae1450&amp;volume=531&amp;pages=4990-5011&amp;publication_year=2024&amp;author=Parker%2CL\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<p class=\"c-article-references__text\" id=\"ref-CR37\">Angeloudi, E. et al. The multimodal universe: enabling large-scale machine learning with 100 tb of astronomical scientific data. Adv. Neural Inf. Process. Syst. 37, 57841\u201357913 (2024).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"10.52202\/079017-1845\" data-track-item_id=\"10.52202\/079017-1845\" data-track-value=\"article reference\" data-track-action=\"article reference\" href=\"https:\/\/doi.org\/10.52202%2F079017-1845\" aria-label=\"Article reference 37\" data-doi=\"10.52202\/079017-1845\" target=\"_blank\">Article<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 37\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=The%20multimodal%20universe%3A%20enabling%20large-scale%20machine%20learning%20with%20100%20tb%20of%20astronomical%20scientific%20data&amp;journal=Adv.%20Neural%20Inf.%20Process.%20Syst.&amp;doi=10.52202%2F079017-1845&amp;volume=37&amp;pages=57841-57913&amp;publication_year=2024&amp;author=Angeloudi%2CE\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<p class=\"c-article-references__text\" id=\"ref-CR38\">Ling, J., Kurzawski, A. &amp; Templeton, J. Reynolds averaged turbulence modelling using deep neural networks with embedded invariance. J. Fluid Mech. 807, 155\u2013166 (2016).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"10.1017\/jfm.2016.615\" data-track-item_id=\"10.1017\/jfm.2016.615\" data-track-value=\"article reference\" data-track-action=\"article reference\" href=\"https:\/\/doi.org\/10.1017%2Fjfm.2016.615\" aria-label=\"Article reference 38\" data-doi=\"10.1017\/jfm.2016.615\" target=\"_blank\">Article<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2016JFM...807..155L\" aria-label=\"ADS reference 38\" target=\"_blank\">ADS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"mathscinet reference\" data-track-action=\"mathscinet reference\" href=\"http:\/\/www.ams.org\/mathscinet-getitem?mr=3569308\" aria-label=\"MathSciNet reference 38\" target=\"_blank\">MathSciNet<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"cas reference\" data-track-action=\"cas reference\" href=\"https:\/\/www.nature.com\/articles\/cas-redirect\/1:CAS:528:DC%2BC2sXitVSrtbrM\" aria-label=\"CAS reference 38\" target=\"_blank\">CAS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 38\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Reynolds%20averaged%20turbulence%20modelling%20using%20deep%20neural%20networks%20with%20embedded%20invariance&amp;journal=J.%20Fluid%20Mech.&amp;doi=10.1017%2Fjfm.2016.615&amp;volume=807&amp;pages=155-166&amp;publication_year=2016&amp;author=Ling%2CJ&amp;author=Kurzawski%2CA&amp;author=Templeton%2CJ\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<p class=\"c-article-references__text\" id=\"ref-CR39\">Loiseau, J.-C. &amp; Brunton, S. L. Constrained sparse Galerkin regression. J. Fluid Mech. 838, 42\u201367 (2018).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"10.1017\/jfm.2017.823\" data-track-item_id=\"10.1017\/jfm.2017.823\" data-track-value=\"article reference\" data-track-action=\"article reference\" href=\"https:\/\/doi.org\/10.1017%2Fjfm.2017.823\" aria-label=\"Article reference 39\" data-doi=\"10.1017\/jfm.2017.823\" target=\"_blank\">Article<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2018JFM...838...42L\" aria-label=\"ADS reference 39\" target=\"_blank\">ADS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"mathscinet reference\" data-track-action=\"mathscinet reference\" href=\"http:\/\/www.ams.org\/mathscinet-getitem?mr=3757034\" aria-label=\"MathSciNet reference 39\" target=\"_blank\">MathSciNet<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 39\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Constrained%20sparse%20Galerkin%20regression&amp;journal=J.%20Fluid%20Mech.&amp;doi=10.1017%2Fjfm.2017.823&amp;volume=838&amp;pages=42-67&amp;publication_year=2018&amp;author=Loiseau%2CJ-C&amp;author=Brunton%2CSL\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<p class=\"c-article-references__text\" id=\"ref-CR40\">Fukami, K., Goto, S. &amp; Taira, K. Data-driven nonlinear turbulent flow scaling with Buckingham Pi variables. J. Fluid Mech. 984, R4 (2024).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"10.1017\/jfm.2024.211\" data-track-item_id=\"10.1017\/jfm.2024.211\" data-track-value=\"article reference\" data-track-action=\"article reference\" href=\"https:\/\/doi.org\/10.1017%2Fjfm.2024.211\" aria-label=\"Article reference 40\" data-doi=\"10.1017\/jfm.2024.211\" target=\"_blank\">Article<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2024JFM...984R...4F\" aria-label=\"ADS reference 40\" target=\"_blank\">ADS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"mathscinet reference\" data-track-action=\"mathscinet reference\" href=\"http:\/\/www.ams.org\/mathscinet-getitem?mr=4725640\" aria-label=\"MathSciNet reference 40\" target=\"_blank\">MathSciNet<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"cas reference\" data-track-action=\"cas reference\" href=\"https:\/\/www.nature.com\/articles\/cas-redirect\/1:CAS:528:DC%2BB2cXpt1ehu74%3D\" aria-label=\"CAS reference 40\" target=\"_blank\">CAS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 40\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Data-driven%20nonlinear%20turbulent%20flow%20scaling%20with%20Buckingham%20Pi%20variables&amp;journal=J.%20Fluid%20Mech.&amp;doi=10.1017%2Fjfm.2024.211&amp;volume=984&amp;publication_year=2024&amp;author=Fukami%2CK&amp;author=Goto%2CS&amp;author=Taira%2CK\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<p class=\"c-article-references__text\" id=\"ref-CR41\">Brunton, S. L., Zolman, N., Kutz, J. N. &amp; Fasel, U. Machine learning for sparse nonlinear modeling and control. Annu. Rev. Control Robot. Auton. Syst. 8, 127\u2013152 (2025).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"10.1146\/annurev-control-030123-015238\" data-track-item_id=\"10.1146\/annurev-control-030123-015238\" data-track-value=\"article reference\" data-track-action=\"article reference\" href=\"https:\/\/doi.org\/10.1146%2Fannurev-control-030123-015238\" aria-label=\"Article reference 41\" data-doi=\"10.1146\/annurev-control-030123-015238\" target=\"_blank\">Article<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 41\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Machine%20learning%20for%20sparse%20nonlinear%20modeling%20and%20control.&amp;journal=Annu.%20Rev.%20Control%20Robot.%20Auton.%20Syst.&amp;doi=10.1146%2Fannurev-control-030123-015238&amp;volume=8&amp;pages=127-152&amp;publication_year=2025&amp;author=Brunton%2CSL&amp;author=Zolman%2CN&amp;author=Kutz%2CJN&amp;author=Fasel%2CU\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<p class=\"c-article-references__text\" id=\"ref-CR42\">Bongard, J. &amp; Lipson, H. Automated reverse engineering of nonlinear dynamical systems. Proc. Natl. Acad. Sci. 104, 9943\u20139948 (2007).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"10.1073\/pnas.0609476104\" data-track-item_id=\"10.1073\/pnas.0609476104\" data-track-value=\"article reference\" data-track-action=\"article reference\" href=\"https:\/\/doi.org\/10.1073%2Fpnas.0609476104\" aria-label=\"Article reference 42\" data-doi=\"10.1073\/pnas.0609476104\" target=\"_blank\">Article<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2007PNAS..104.9943B\" aria-label=\"ADS reference 42\" target=\"_blank\">ADS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"cas reference\" data-track-action=\"cas reference\" href=\"https:\/\/www.nature.com\/articles\/cas-redirect\/1:CAS:528:DC%2BD2sXmvVChtrk%3D\" aria-label=\"CAS reference 42\" target=\"_blank\">CAS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"pubmed reference\" data-track-action=\"pubmed reference\" href=\"http:\/\/www.ncbi.nlm.nih.gov\/entrez\/query.fcgi?cmd=Retrieve&amp;db=PubMed&amp;dopt=Abstract&amp;list_uids=17553966\" aria-label=\"PubMed reference 42\" target=\"_blank\">PubMed<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"pubmed central reference\" data-track-action=\"pubmed central reference\" href=\"http:\/\/www.ncbi.nlm.nih.gov\/pmc\/articles\/PMC1891254\" aria-label=\"PubMed Central reference 42\" target=\"_blank\">PubMed Central<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 42\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Automated%20reverse%20engineering%20of%20nonlinear%20dynamical%20systems&amp;journal=Proc.%20Natl.%20Acad.%20Sci.&amp;doi=10.1073%2Fpnas.0609476104&amp;volume=104&amp;pages=9943-9948&amp;publication_year=2007&amp;author=Bongard%2CJ&amp;author=Lipson%2CH\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<p class=\"c-article-references__text\" id=\"ref-CR43\">Schmidt, M. &amp; Lipson, H. Distilling free-form natural laws from experimental data. Science 324, 81\u201385 (2009).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"10.1126\/science.1165893\" data-track-item_id=\"10.1126\/science.1165893\" data-track-value=\"article reference\" data-track-action=\"article reference\" href=\"https:\/\/doi.org\/10.1126%2Fscience.1165893\" aria-label=\"Article reference 43\" data-doi=\"10.1126\/science.1165893\" target=\"_blank\">Article<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2009Sci...324...81S\" aria-label=\"ADS reference 43\" target=\"_blank\">ADS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"cas reference\" data-track-action=\"cas reference\" href=\"https:\/\/www.nature.com\/articles\/cas-redirect\/1:CAS:528:DC%2BD1MXjvVajsbw%3D\" aria-label=\"CAS reference 43\" target=\"_blank\">CAS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"pubmed reference\" data-track-action=\"pubmed reference\" href=\"http:\/\/www.ncbi.nlm.nih.gov\/entrez\/query.fcgi?cmd=Retrieve&amp;db=PubMed&amp;dopt=Abstract&amp;list_uids=19342586\" aria-label=\"PubMed reference 43\" target=\"_blank\">PubMed<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 43\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Distilling%20free-form%20natural%20laws%20from%20experimental%20data&amp;journal=Science&amp;doi=10.1126%2Fscience.1165893&amp;volume=324&amp;pages=81-85&amp;publication_year=2009&amp;author=Schmidt%2CM&amp;author=Lipson%2CH\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<p class=\"c-article-references__text\" id=\"ref-CR44\">Petersen, B. K. Deep symbolic regression: Recovering mathematical expressions from data via policy gradients. Preprint at arXiv <a href=\"https:\/\/doi.org\/10.48550\/arXiv.1912.04871\" data-track=\"click_references\" data-track-action=\"external reference\" data-track-value=\"external reference\" data-track-label=\"10.48550\/arXiv.1912.04871\" rel=\"nofollow noopener\" target=\"_blank\">https:\/\/doi.org\/10.48550\/arXiv.1912.04871<\/a> (2019).<\/p>\n<p class=\"c-article-references__text\" id=\"ref-CR45\">Vaddireddy, H. &amp; San, O. Equation discovery using fast function extraction: a deterministic symbolic regression approach. Fluids 4, 111 (2019).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"10.3390\/fluids4020111\" data-track-item_id=\"10.3390\/fluids4020111\" data-track-value=\"article reference\" data-track-action=\"article reference\" href=\"https:\/\/doi.org\/10.3390%2Ffluids4020111\" aria-label=\"Article reference 45\" data-doi=\"10.3390\/fluids4020111\" target=\"_blank\">Article<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2019Fluid...4..111V\" aria-label=\"ADS reference 45\" target=\"_blank\">ADS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 45\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Equation%20discovery%20using%20fast%20function%20extraction%3A%20a%20deterministic%20symbolic%20regression%20approach&amp;journal=Fluids&amp;doi=10.3390%2Ffluids4020111&amp;volume=4&amp;publication_year=2019&amp;author=Vaddireddy%2CH&amp;author=San%2CO\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<p class=\"c-article-references__text\" id=\"ref-CR46\">Udrescu, S.-M. &amp; Tegmark, M. AI Feynman: a physics-inspired method for symbolic regression. Sci. Adv. 6, eaay2631 (2020).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"10.1126\/sciadv.aay2631\" data-track-item_id=\"10.1126\/sciadv.aay2631\" data-track-value=\"article reference\" data-track-action=\"article reference\" href=\"https:\/\/doi.org\/10.1126%2Fsciadv.aay2631\" aria-label=\"Article reference 46\" data-doi=\"10.1126\/sciadv.aay2631\" target=\"_blank\">Article<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2020SciA....6.2631U\" aria-label=\"ADS reference 46\" target=\"_blank\">ADS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"pubmed reference\" data-track-action=\"pubmed reference\" href=\"http:\/\/www.ncbi.nlm.nih.gov\/entrez\/query.fcgi?cmd=Retrieve&amp;db=PubMed&amp;dopt=Abstract&amp;list_uids=32426452\" aria-label=\"PubMed reference 46\" target=\"_blank\">PubMed<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"pubmed central reference\" data-track-action=\"pubmed central reference\" href=\"http:\/\/www.ncbi.nlm.nih.gov\/pmc\/articles\/PMC7159912\" aria-label=\"PubMed Central reference 46\" target=\"_blank\">PubMed Central<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 46\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=AI%20Feynman%3A%20a%20physics-inspired%20method%20for%20symbolic%20regression&amp;journal=Sci.%20Adv.&amp;doi=10.1126%2Fsciadv.aay2631&amp;volume=6&amp;publication_year=2020&amp;author=Udrescu%2CS-M&amp;author=Tegmark%2CM\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<p class=\"c-article-references__text\" id=\"ref-CR47\">Zolman, N., Fasel, U., Kutz, J. N. &amp; Brunton, S. L. SINDy-RL: interpretable and efficient model-based reinforcement learning. Nat. Commun. 16, 10714 (2025).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"10.1038\/s41467-025-65738-4\" data-track-item_id=\"10.1038\/s41467-025-65738-4\" data-track-value=\"article reference\" data-track-action=\"article reference\" href=\"https:\/\/doi.org\/10.1038%2Fs41467-025-65738-4\" aria-label=\"Article reference 47\" data-doi=\"10.1038\/s41467-025-65738-4\" target=\"_blank\">Article<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2025NatCo..1610714Z\" aria-label=\"ADS reference 47\" target=\"_blank\">ADS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"cas reference\" data-track-action=\"cas reference\" href=\"https:\/\/www.nature.com\/articles\/cas-redirect\/1:CAS:528:DC%2BB2MXivFGltLfI\" aria-label=\"CAS reference 47\" target=\"_blank\">CAS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"pubmed reference\" data-track-action=\"pubmed reference\" href=\"http:\/\/www.ncbi.nlm.nih.gov\/entrez\/query.fcgi?cmd=Retrieve&amp;db=PubMed&amp;dopt=Abstract&amp;list_uids=41315255\" aria-label=\"PubMed reference 47\" target=\"_blank\">PubMed<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"pubmed central reference\" data-track-action=\"pubmed central reference\" href=\"http:\/\/www.ncbi.nlm.nih.gov\/pmc\/articles\/PMC12663201\" aria-label=\"PubMed Central reference 47\" target=\"_blank\">PubMed Central<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 47\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=SINDy-RL%3A%20interpretable%20and%20efficient%20model-based%20reinforcement%20learning&amp;journal=Nat.%20Commun.&amp;doi=10.1038%2Fs41467-025-65738-4&amp;volume=16&amp;publication_year=2025&amp;author=Zolman%2CN&amp;author=Fasel%2CU&amp;author=Kutz%2CJN&amp;author=Brunton%2CSL\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<p class=\"c-article-references__text\" id=\"ref-CR48\">Kaptanoglu, A. A., Callaham, J. L., Hansen, C. J., Aravkin, A. &amp; Brunton, S. L. Promoting global stability in data-driven models of quadratic nonlinear dynamics. Phys. Rev. Fluids 6, 094401 (2021).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"10.1103\/PhysRevFluids.6.094401\" data-track-item_id=\"10.1103\/PhysRevFluids.6.094401\" data-track-value=\"article reference\" data-track-action=\"article reference\" href=\"https:\/\/doi.org\/10.1103%2FPhysRevFluids.6.094401\" aria-label=\"Article reference 48\" data-doi=\"10.1103\/PhysRevFluids.6.094401\" target=\"_blank\">Article<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2021PhRvF...6i4401K\" aria-label=\"ADS reference 48\" target=\"_blank\">ADS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 48\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Promoting%20global%20stability%20in%20data-driven%20models%20of%20quadratic%20nonlinear%20dynamics&amp;journal=Phys.%20Rev.%20Fluids&amp;doi=10.1103%2FPhysRevFluids.6.094401&amp;volume=6&amp;publication_year=2021&amp;author=Kaptanoglu%2CAA&amp;author=Callaham%2CJL&amp;author=Hansen%2CCJ&amp;author=Aravkin%2CA&amp;author=Brunton%2CSL\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<p class=\"c-article-references__text\" id=\"ref-CR49\">Otto, S. E., Zolman, N., Kutz, J. N. &amp; Brunton, S. L. A unified framework to enforce, discover, and promote symmetry in machine learning. J. Mach. Learn. Res. 26, 1\u201383 (2025).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"mathscinet reference\" data-track-action=\"mathscinet reference\" href=\"http:\/\/www.ams.org\/mathscinet-getitem?mr=5017190\" aria-label=\"MathSciNet reference 49\" target=\"_blank\">MathSciNet<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 49\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=A%20unified%20framework%20to%20enforce%2C%20discover%2C%20and%20promote%20symmetry%20in%20machine%20learning&amp;journal=J.%20Mach.%20Learn.%20Res.&amp;volume=26&amp;pages=1-83&amp;publication_year=2025&amp;author=Otto%2CSE&amp;author=Zolman%2CN&amp;author=Kutz%2CJN&amp;author=Brunton%2CSL\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<p class=\"c-article-references__text\" id=\"ref-CR50\">Rudy, S. H., Brunton, S. L., Proctor, J. L. &amp; Kutz, J. N. Data-driven discovery of partial differential equations. Sci. Adv. 3, e1602614 (2017).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"10.1126\/sciadv.1602614\" data-track-item_id=\"10.1126\/sciadv.1602614\" data-track-value=\"article reference\" data-track-action=\"article reference\" href=\"https:\/\/doi.org\/10.1126%2Fsciadv.1602614\" aria-label=\"Article reference 50\" data-doi=\"10.1126\/sciadv.1602614\" target=\"_blank\">Article<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2017SciA....3E2614R\" aria-label=\"ADS reference 50\" target=\"_blank\">ADS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"pubmed reference\" data-track-action=\"pubmed reference\" href=\"http:\/\/www.ncbi.nlm.nih.gov\/entrez\/query.fcgi?cmd=Retrieve&amp;db=PubMed&amp;dopt=Abstract&amp;list_uids=28508044\" aria-label=\"PubMed reference 50\" target=\"_blank\">PubMed<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"pubmed central reference\" data-track-action=\"pubmed central reference\" href=\"http:\/\/www.ncbi.nlm.nih.gov\/pmc\/articles\/PMC5406137\" aria-label=\"PubMed Central reference 50\" target=\"_blank\">PubMed Central<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 50\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Data-driven%20discovery%20of%20partial%20differential%20equations.&amp;journal=Sci.%20Adv.&amp;doi=10.1126%2Fsciadv.1602614&amp;volume=3&amp;publication_year=2017&amp;author=Rudy%2CSH&amp;author=Brunton%2CSL&amp;author=Proctor%2CJL&amp;author=Kutz%2CJN\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<p class=\"c-article-references__text\" id=\"ref-CR51\">Schaeffer, H. Learning partial differential equations via data discovery and sparse optimization. Proc. R. Soc. A 473, 20160446 (2017).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"10.1098\/rspa.2016.0446\" data-track-item_id=\"10.1098\/rspa.2016.0446\" data-track-value=\"article reference\" data-track-action=\"article reference\" href=\"https:\/\/doi.org\/10.1098%2Frspa.2016.0446\" aria-label=\"Article reference 51\" data-doi=\"10.1098\/rspa.2016.0446\" target=\"_blank\">Article<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2017RSPSA.47360446S\" aria-label=\"ADS reference 51\" target=\"_blank\">ADS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"mathscinet reference\" data-track-action=\"mathscinet reference\" href=\"http:\/\/www.ams.org\/mathscinet-getitem?mr=3615300\" aria-label=\"MathSciNet reference 51\" target=\"_blank\">MathSciNet<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"pubmed reference\" data-track-action=\"pubmed reference\" href=\"http:\/\/www.ncbi.nlm.nih.gov\/entrez\/query.fcgi?cmd=Retrieve&amp;db=PubMed&amp;dopt=Abstract&amp;list_uids=28265183\" aria-label=\"PubMed reference 51\" target=\"_blank\">PubMed<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"pubmed central reference\" data-track-action=\"pubmed central reference\" href=\"http:\/\/www.ncbi.nlm.nih.gov\/pmc\/articles\/PMC5312119\" aria-label=\"PubMed Central reference 51\" target=\"_blank\">PubMed Central<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 51\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Learning%20partial%20differential%20equations%20via%20data%20discovery%20and%20sparse%20optimization&amp;journal=Proc.%20R.%20Soc.%20A&amp;doi=10.1098%2Frspa.2016.0446&amp;volume=473&amp;publication_year=2017&amp;author=Schaeffer%2CH\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<p class=\"c-article-references__text\" id=\"ref-CR52\">Messenger, D. A. &amp; Bortz, D. M. Weak sindy for partial differential equations. J. Comput. Phys. <a href=\"https:\/\/doi.org\/10.1016\/j.jcp.2021.110525\" data-track=\"click_references\" data-track-action=\"external reference\" data-track-value=\"external reference\" data-track-label=\"10.1016\/j.jcp.2021.110525\" rel=\"nofollow noopener\" target=\"_blank\">https:\/\/doi.org\/10.1016\/j.jcp.2021.110525<\/a> (2021).<\/p>\n<p class=\"c-article-references__text\" id=\"ref-CR53\">Reinbold, P. A., Kageorge, L. M., Schatz, M. F. &amp; Grigoriev, R. O. Robust learning from noisy, incomplete, high-dimensional experimental data via physically constrained symbolic regression. Nat. Commun. 12, 1\u20138 (2021).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"10.1038\/s41467-021-23479-0\" data-track-item_id=\"10.1038\/s41467-021-23479-0\" data-track-value=\"article reference\" data-track-action=\"article reference\" href=\"https:\/\/doi.org\/10.1038%2Fs41467-021-23479-0\" aria-label=\"Article reference 53\" data-doi=\"10.1038\/s41467-021-23479-0\" target=\"_blank\">Article<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 53\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Robust%20learning%20from%20noisy%2C%20incomplete%2C%20high-dimensional%20experimental%20data%20via%20physically%20constrained%20symbolic%20regression&amp;journal=Nat.%20Commun.&amp;doi=10.1038%2Fs41467-021-23479-0&amp;volume=12&amp;pages=1-8&amp;publication_year=2021&amp;author=Reinbold%2CPA&amp;author=Kageorge%2CLM&amp;author=Schatz%2CMF&amp;author=Grigoriev%2CRO\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<p class=\"c-article-references__text\" id=\"ref-CR54\">Fasel, U., Kutz, J. N., Brunton, B. W. &amp; Brunton, S. L. Ensemble-sindy: Robust sparse model discovery in the low-data, high-noise limit, with active learning and control. Proc. R. Soc. A. 478, 20210904 (2022).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"10.1098\/rspa.2021.0904\" data-track-item_id=\"10.1098\/rspa.2021.0904\" data-track-value=\"article reference\" data-track-action=\"article reference\" href=\"https:\/\/doi.org\/10.1098%2Frspa.2021.0904\" aria-label=\"Article reference 54\" data-doi=\"10.1098\/rspa.2021.0904\" target=\"_blank\">Article<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2022RSPSA.47810904F\" aria-label=\"ADS reference 54\" target=\"_blank\">ADS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"mathscinet reference\" data-track-action=\"mathscinet reference\" href=\"http:\/\/www.ams.org\/mathscinet-getitem?mr=4416372\" aria-label=\"MathSciNet reference 54\" target=\"_blank\">MathSciNet<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"cas reference\" data-track-action=\"cas reference\" href=\"https:\/\/www.nature.com\/articles\/cas-redirect\/1:STN:280:DC%2BB2MrgtFWqtw%3D%3D\" aria-label=\"CAS reference 54\" target=\"_blank\">CAS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"pubmed reference\" data-track-action=\"pubmed reference\" href=\"http:\/\/www.ncbi.nlm.nih.gov\/entrez\/query.fcgi?cmd=Retrieve&amp;db=PubMed&amp;dopt=Abstract&amp;list_uids=35450025\" aria-label=\"PubMed reference 54\" target=\"_blank\">PubMed<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"pubmed central reference\" data-track-action=\"pubmed central reference\" href=\"http:\/\/www.ncbi.nlm.nih.gov\/pmc\/articles\/PMC9006119\" aria-label=\"PubMed Central reference 54\" target=\"_blank\">PubMed Central<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 54\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Ensemble-sindy%3A%20Robust%20sparse%20model%20discovery%20in%20the%20low-data%2C%20high-noise%20limit%2C%20with%20active%20learning%20and%20control&amp;journal=Proc.%20R.%20Soc.%20A&amp;doi=10.1098%2Frspa.2021.0904&amp;volume=478&amp;publication_year=2022&amp;author=Fasel%2CU&amp;author=Kutz%2CJN&amp;author=Brunton%2CBW&amp;author=Brunton%2CSL\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<p class=\"c-article-references__text\" id=\"ref-CR55\">Beetham, S. &amp; Capecelatro, J. Formulating turbulence closures using sparse regression with embedded form invariance. Phys. Rev. Fluids 5, 084611 (2020).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"10.1103\/PhysRevFluids.5.084611\" data-track-item_id=\"10.1103\/PhysRevFluids.5.084611\" data-track-value=\"article reference\" data-track-action=\"article reference\" href=\"https:\/\/doi.org\/10.1103%2FPhysRevFluids.5.084611\" aria-label=\"Article reference 55\" data-doi=\"10.1103\/PhysRevFluids.5.084611\" target=\"_blank\">Article<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2020PhRvF...5h4611B\" aria-label=\"ADS reference 55\" target=\"_blank\">ADS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 55\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Formulating%20turbulence%20closures%20using%20sparse%20regression%20with%20embedded%20form%20invariance&amp;journal=Phys.%20Rev.%20Fluids&amp;doi=10.1103%2FPhysRevFluids.5.084611&amp;volume=5&amp;publication_year=2020&amp;author=Beetham%2CS&amp;author=Capecelatro%2CJ\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<p class=\"c-article-references__text\" id=\"ref-CR56\">Beetham, S., Fox, R. O. &amp; Capecelatro, J. Sparse identification of multiphase turbulence closures for coupled fluid\u2013particle flows. Journal of Fluid Mechanics <a href=\"https:\/\/doi.org\/10.1017\/jfm.2021.53\" data-track=\"click_references\" data-track-action=\"external reference\" data-track-value=\"external reference\" data-track-label=\"10.1017\/jfm.2021.53\" rel=\"nofollow noopener\" target=\"_blank\">https:\/\/doi.org\/10.1017\/jfm.2021.53<\/a> (2021).<\/p>\n<p class=\"c-article-references__text\" id=\"ref-CR57\">Schmelzer, M., Dwight, R. P. &amp; Cinnella, P. Discovery of algebraic Reynolds-stress models using sparse symbolic regression. Flow, Turbulence Combust. 104, 579\u2013603 (2020).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" rel=\"noopener nofollow\" data-track-label=\"10.1007\/s10494-019-00089-x\" data-track-item_id=\"10.1007\/s10494-019-00089-x\" data-track-value=\"article reference\" data-track-action=\"article reference\" href=\"https:\/\/link.springer.com\/doi\/10.1007\/s10494-019-00089-x\" aria-label=\"Article reference 57\" data-doi=\"10.1007\/s10494-019-00089-x\" target=\"_blank\">Article<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2020FTC...104..579S\" aria-label=\"ADS reference 57\" target=\"_blank\">ADS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 57\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Discovery%20of%20algebraic%20Reynolds-stress%20models%20using%20sparse%20symbolic%20regression&amp;journal=Flow%2C%20Turbulence%20Combust.&amp;doi=10.1007%2Fs10494-019-00089-x&amp;volume=104&amp;pages=579-603&amp;publication_year=2020&amp;author=Schmelzer%2CM&amp;author=Dwight%2CRP&amp;author=Cinnella%2CP\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<p class=\"c-article-references__text\" id=\"ref-CR58\">Alves, E. P. &amp; Fiuza, F. Data-driven discovery of reduced plasma physics models from fully-kinetic simulations. Phys. Rev. Research 4, 033192 (2022).<\/p>\n<p class=\"c-article-references__text\" id=\"ref-CR59\">Supekar, R. et al. Learning hydrodynamic equations for active matter from particle simulations and experiments. Proc. Natl. Acad. Sci. 120, e2206994120 (2023).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"10.1073\/pnas.2206994120\" data-track-item_id=\"10.1073\/pnas.2206994120\" data-track-value=\"article reference\" data-track-action=\"article reference\" href=\"https:\/\/doi.org\/10.1073%2Fpnas.2206994120\" aria-label=\"Article reference 59\" data-doi=\"10.1073\/pnas.2206994120\" target=\"_blank\">Article<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"mathscinet reference\" data-track-action=\"mathscinet reference\" href=\"http:\/\/www.ams.org\/mathscinet-getitem?mr=4575284\" aria-label=\"MathSciNet reference 59\" target=\"_blank\">MathSciNet<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"cas reference\" data-track-action=\"cas reference\" href=\"https:\/\/www.nature.com\/articles\/cas-redirect\/1:CAS:528:DC%2BB3sXkt1Cjt70%3D\" aria-label=\"CAS reference 59\" target=\"_blank\">CAS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"pubmed reference\" data-track-action=\"pubmed reference\" href=\"http:\/\/www.ncbi.nlm.nih.gov\/entrez\/query.fcgi?cmd=Retrieve&amp;db=PubMed&amp;dopt=Abstract&amp;list_uids=36763535\" aria-label=\"PubMed reference 59\" target=\"_blank\">PubMed<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"pubmed central reference\" data-track-action=\"pubmed central reference\" href=\"http:\/\/www.ncbi.nlm.nih.gov\/pmc\/articles\/PMC9963139\" aria-label=\"PubMed Central reference 59\" target=\"_blank\">PubMed Central<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 59\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Learning%20hydrodynamic%20equations%20for%20active%20matter%20from%20particle%20simulations%20and%20experiments&amp;journal=Proc.%20Natl.%20Acad.%20Sci.&amp;doi=10.1073%2Fpnas.2206994120&amp;volume=120&amp;publication_year=2023&amp;author=Supekar%2CR\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<p class=\"c-article-references__text\" id=\"ref-CR60\">Zanna, L. &amp; Bolton, T. Data-driven equation discovery of ocean mesoscale closures. Geophys. Res. Lett. 47, e2020GL088376 (2020).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"10.1029\/2020GL088376\" data-track-item_id=\"10.1029\/2020GL088376\" data-track-value=\"article reference\" data-track-action=\"article reference\" href=\"https:\/\/doi.org\/10.1029%2F2020GL088376\" aria-label=\"Article reference 60\" data-doi=\"10.1029\/2020GL088376\" target=\"_blank\">Article<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2020GeoRL..4788376Z\" aria-label=\"ADS reference 60\" target=\"_blank\">ADS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 60\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Data-driven%20equation%20discovery%20of%20ocean%20mesoscale%20closures&amp;journal=Geophys.%20Res.%20Lett.&amp;doi=10.1029%2F2020GL088376&amp;volume=47&amp;publication_year=2020&amp;author=Zanna%2CL&amp;author=Bolton%2CT\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<p class=\"c-article-references__text\" id=\"ref-CR61\">Long, Z., Lu, Y. &amp; Dong, B. Pde-net 2.0: Learning PDEs from data with a numeric-symbolic hybrid deep network. J. Comput. Phys. 399, 108925 (2019).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"10.1016\/j.jcp.2019.108925\" data-track-item_id=\"10.1016\/j.jcp.2019.108925\" data-track-value=\"article reference\" data-track-action=\"article reference\" href=\"https:\/\/doi.org\/10.1016%2Fj.jcp.2019.108925\" aria-label=\"Article reference 61\" data-doi=\"10.1016\/j.jcp.2019.108925\" target=\"_blank\">Article<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"mathscinet reference\" data-track-action=\"mathscinet reference\" href=\"http:\/\/www.ams.org\/mathscinet-getitem?mr=4013148\" aria-label=\"MathSciNet reference 61\" target=\"_blank\">MathSciNet<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 61\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Pde-net%202.0%3A%20Learning%20PDEs%20from%20data%20with%20a%20numeric-symbolic%20hybrid%20deep%20network&amp;journal=J.%20Comput.%20Phys.&amp;doi=10.1016%2Fj.jcp.2019.108925&amp;volume=399&amp;publication_year=2019&amp;author=Long%2CZ&amp;author=Lu%2CY&amp;author=Dong%2CB\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<p class=\"c-article-references__text\" id=\"ref-CR62\">Kingma, D. P. &amp; Welling, M. Auto-encoding variational Bayes. Preprint at arXiv <a href=\"https:\/\/doi.org\/10.48550\/arXiv.1312.6114\" data-track=\"click_references\" data-track-action=\"external reference\" data-track-value=\"external reference\" data-track-label=\"10.48550\/arXiv.1312.6114\" rel=\"nofollow noopener\" target=\"_blank\">https:\/\/doi.org\/10.48550\/arXiv.1312.6114<\/a> (2013).<\/p>\n<p class=\"c-article-references__text\" id=\"ref-CR63\">Lusch, B., Kutz, J. N. &amp; Brunton, S. L. Deep learning for universal linear embeddings of nonlinear dynamics. Nat. Commun. 9, 4950 (2018).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"10.1038\/s41467-018-07210-0\" data-track-item_id=\"10.1038\/s41467-018-07210-0\" data-track-value=\"article reference\" data-track-action=\"article reference\" href=\"https:\/\/doi.org\/10.1038%2Fs41467-018-07210-0\" aria-label=\"Article reference 63\" data-doi=\"10.1038\/s41467-018-07210-0\" target=\"_blank\">Article<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2018NatCo...9.4950L\" aria-label=\"ADS reference 63\" target=\"_blank\">ADS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"pubmed reference\" data-track-action=\"pubmed reference\" href=\"http:\/\/www.ncbi.nlm.nih.gov\/entrez\/query.fcgi?cmd=Retrieve&amp;db=PubMed&amp;dopt=Abstract&amp;list_uids=30470743\" aria-label=\"PubMed reference 63\" target=\"_blank\">PubMed<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"pubmed central reference\" data-track-action=\"pubmed central reference\" href=\"http:\/\/www.ncbi.nlm.nih.gov\/pmc\/articles\/PMC6251871\" aria-label=\"PubMed Central reference 63\" target=\"_blank\">PubMed Central<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 63\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Deep%20learning%20for%20universal%20linear%20embeddings%20of%20nonlinear%20dynamics&amp;journal=Nat.%20Commun.&amp;doi=10.1038%2Fs41467-018-07210-0&amp;volume=9&amp;publication_year=2018&amp;author=Lusch%2CB&amp;author=Kutz%2CJN&amp;author=Brunton%2CSL\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<p class=\"c-article-references__text\" id=\"ref-CR64\">Champion, K., Lusch, B., Kutz, J. N. &amp; Brunton, S. L. Data-driven discovery of coordinates and governing equations. Proc. Natl. Acad. Sci. 116, 22445\u201322451 (2019).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"10.1073\/pnas.1906995116\" data-track-item_id=\"10.1073\/pnas.1906995116\" data-track-value=\"article reference\" data-track-action=\"article reference\" href=\"https:\/\/doi.org\/10.1073%2Fpnas.1906995116\" aria-label=\"Article reference 64\" data-doi=\"10.1073\/pnas.1906995116\" target=\"_blank\">Article<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2019PNAS..11622445C\" aria-label=\"ADS reference 64\" target=\"_blank\">ADS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"mathscinet reference\" data-track-action=\"mathscinet reference\" href=\"http:\/\/www.ams.org\/mathscinet-getitem?mr=4032517\" aria-label=\"MathSciNet reference 64\" target=\"_blank\">MathSciNet<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"cas reference\" data-track-action=\"cas reference\" href=\"https:\/\/www.nature.com\/articles\/cas-redirect\/1:CAS:528:DC%2BC1MXitFWgu7vK\" aria-label=\"CAS reference 64\" target=\"_blank\">CAS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"pubmed reference\" data-track-action=\"pubmed reference\" href=\"http:\/\/www.ncbi.nlm.nih.gov\/entrez\/query.fcgi?cmd=Retrieve&amp;db=PubMed&amp;dopt=Abstract&amp;list_uids=31636218\" aria-label=\"PubMed reference 64\" target=\"_blank\">PubMed<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"pubmed central reference\" data-track-action=\"pubmed central reference\" href=\"http:\/\/www.ncbi.nlm.nih.gov\/pmc\/articles\/PMC6842598\" aria-label=\"PubMed Central reference 64\" target=\"_blank\">PubMed Central<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 64\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Data-driven%20discovery%20of%20coordinates%20and%20governing%20equations&amp;journal=Proc.%20Natl.%20Acad.%20Sci.&amp;doi=10.1073%2Fpnas.1906995116&amp;volume=116&amp;pages=22445-22451&amp;publication_year=2019&amp;author=Champion%2CK&amp;author=Lusch%2CB&amp;author=Kutz%2CJN&amp;author=Brunton%2CSL\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<p class=\"c-article-references__text\" id=\"ref-CR65\">Lee, K. &amp; Carlberg, K. T. Model reduction of dynamical systems on nonlinear manifolds using deep convolutional autoencoders. J. Comput. Phys. 404, 108973 (2020).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"10.1016\/j.jcp.2019.108973\" data-track-item_id=\"10.1016\/j.jcp.2019.108973\" data-track-value=\"article reference\" data-track-action=\"article reference\" href=\"https:\/\/doi.org\/10.1016%2Fj.jcp.2019.108973\" aria-label=\"Article reference 65\" data-doi=\"10.1016\/j.jcp.2019.108973\" target=\"_blank\">Article<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"mathscinet reference\" data-track-action=\"mathscinet reference\" href=\"http:\/\/www.ams.org\/mathscinet-getitem?mr=4043884\" aria-label=\"MathSciNet reference 65\" target=\"_blank\">MathSciNet<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 65\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Model%20reduction%20of%20dynamical%20systems%20on%20nonlinear%20manifolds%20using%20deep%20convolutional%20autoencoders&amp;journal=J.%20Comput.%20Phys.&amp;doi=10.1016%2Fj.jcp.2019.108973&amp;volume=404&amp;publication_year=2020&amp;author=Lee%2CK&amp;author=Carlberg%2CKT\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<p class=\"c-article-references__text\" id=\"ref-CR66\">Conti, P., Gobat, G., Fresca, S., Manzoni, A. &amp; Frangi, A. Reduced order modeling of parametrized systems through autoencoders and sindy approach: continuation of periodic solutions. Comput. Methods Appl. Mech. Eng. 411, 116072 (2023).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"10.1016\/j.cma.2023.116072\" data-track-item_id=\"10.1016\/j.cma.2023.116072\" data-track-value=\"article reference\" data-track-action=\"article reference\" href=\"https:\/\/doi.org\/10.1016%2Fj.cma.2023.116072\" aria-label=\"Article reference 66\" data-doi=\"10.1016\/j.cma.2023.116072\" target=\"_blank\">Article<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"mathscinet reference\" data-track-action=\"mathscinet reference\" href=\"http:\/\/www.ams.org\/mathscinet-getitem?mr=4583709\" aria-label=\"MathSciNet reference 66\" target=\"_blank\">MathSciNet<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 66\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Reduced%20order%20modeling%20of%20parametrized%20systems%20through%20autoencoders%20and%20sindy%20approach%3A%20continuation%20of%20periodic%20solutions&amp;journal=Comput.%20Methods%20Appl.%20Mech.%20Eng.&amp;doi=10.1016%2Fj.cma.2023.116072&amp;volume=411&amp;publication_year=2023&amp;author=Conti%2CP&amp;author=Gobat%2CG&amp;author=Fresca%2CS&amp;author=Manzoni%2CA&amp;author=Frangi%2CA\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<p class=\"c-article-references__text\" id=\"ref-CR67\">Mounayer, J., Rodriguez, S., Ghnatios, C., Farhat, C. &amp; Chinesta, F. Rank reduction autoencoders\u2013enhancing interpolation on nonlinear manifolds. Preprint at arXiv <a href=\"https:\/\/doi.org\/10.48550\/arXiv.2405.13980\" data-track=\"click_references\" data-track-action=\"external reference\" data-track-value=\"external reference\" data-track-label=\"10.48550\/arXiv.2405.13980\" rel=\"nofollow noopener\" target=\"_blank\">https:\/\/doi.org\/10.48550\/arXiv.2405.13980<\/a> (2024).<\/p>\n<p class=\"c-article-references__text\" id=\"ref-CR68\">Brunton, S. L. &amp; Kutz, J. N.Data-Driven Science and Engineering: Machine Learning, Dynamical Systems, and Control 2nd edn (Cambridge University Press, 2022).<\/p>\n<p class=\"c-article-references__text\" id=\"ref-CR69\">Bakarji, J., Champion, K., Nathan Kutz, J. &amp; Brunton, S. L. Discovering governing equations from partial measurements with deep delay autoencoders. Proc. R. Soc. A 479, 20230422 (2023).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"10.1098\/rspa.2023.0422\" data-track-item_id=\"10.1098\/rspa.2023.0422\" data-track-value=\"article reference\" data-track-action=\"article reference\" href=\"https:\/\/doi.org\/10.1098%2Frspa.2023.0422\" aria-label=\"Article reference 69\" data-doi=\"10.1098\/rspa.2023.0422\" target=\"_blank\">Article<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2023RSPSA.47930422B\" aria-label=\"ADS reference 69\" target=\"_blank\">ADS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"mathscinet reference\" data-track-action=\"mathscinet reference\" href=\"http:\/\/www.ams.org\/mathscinet-getitem?mr=4636877\" aria-label=\"MathSciNet reference 69\" target=\"_blank\">MathSciNet<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 69\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Discovering%20governing%20equations%20from%20partial%20measurements%20with%20deep%20delay%20autoencoders&amp;journal=Proc.%20R.%20Soc.%20A&amp;doi=10.1098%2Frspa.2023.0422&amp;volume=479&amp;publication_year=2023&amp;author=Bakarji%2CJ&amp;author=Champion%2CK&amp;author=Nathan%20Kutz%2CJ&amp;author=Brunton%2CSL\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<p class=\"c-article-references__text\" id=\"ref-CR70\">Takeishi, N., Kawahara, Y. &amp; Yairi, T. Learning Koopman invariant subspaces for dynamic mode decomposition (2017).<\/p>\n<p class=\"c-article-references__text\" id=\"ref-CR71\">Yeung, E., Kundu, S. &amp; Hodas, N. Learning deep neural network representations for Koopman operators of nonlinear dynamical systems. In Proccedings of American Control Conference (ACC) (Philadelphia, PA, USA, 2019).<\/p>\n<p class=\"c-article-references__text\" id=\"ref-CR72\">Wehmeyer, C. &amp; No\u00e9, F. Time-lagged autoencoders: Deep learning of slow collective variables for molecular kinetics.  J. Chem. Phys. 148, 1\u20139 (2018).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"10.1063\/1.5011399\" data-track-item_id=\"10.1063\/1.5011399\" data-track-value=\"article reference\" data-track-action=\"article reference\" href=\"https:\/\/doi.org\/10.1063%2F1.5011399\" aria-label=\"Article reference 72\" data-doi=\"10.1063\/1.5011399\" target=\"_blank\">Article<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 72\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Time-lagged%20autoencoders%3A%20Deep%20learning%20of%20slow%20collective%20variables%20for%20molecular%20kinetics&amp;journal=J.%20Chem.%20Phys.&amp;doi=10.1063%2F1.5011399&amp;volume=148&amp;pages=1-9&amp;publication_year=2018&amp;author=Wehmeyer%2CC&amp;author=No%C3%A9%2CF\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<p class=\"c-article-references__text\" id=\"ref-CR73\">Mardt, A., Pasquali, L., Wu, H. &amp; No\u00e9, F. VAMPnets: deep learning of molecular kinetics. Nat. Commun. <a href=\"https:\/\/doi.org\/10.1038\/s41467-017-02388-1\" data-track=\"click_references\" data-track-action=\"external reference\" data-track-value=\"external reference\" data-track-label=\"10.1038\/s41467-017-02388-1\" rel=\"nofollow noopener\" target=\"_blank\">https:\/\/doi.org\/10.1038\/s41467-017-02388-1<\/a> (2018).<\/p>\n<p class=\"c-article-references__text\" id=\"ref-CR74\">Otto, S. E. &amp; Rowley, C. W. Linearly-recurrent autoencoder networks for learning dynamics. SIAM J. Appl. Dyn. Syst. 18, 558\u2013593 (2019).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"10.1137\/18M1177846\" data-track-item_id=\"10.1137\/18M1177846\" data-track-value=\"article reference\" data-track-action=\"article reference\" href=\"https:\/\/doi.org\/10.1137%2F18M1177846\" aria-label=\"Article reference 74\" data-doi=\"10.1137\/18M1177846\" target=\"_blank\">Article<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"mathscinet reference\" data-track-action=\"mathscinet reference\" href=\"http:\/\/www.ams.org\/mathscinet-getitem?mr=3932614\" aria-label=\"MathSciNet reference 74\" target=\"_blank\">MathSciNet<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 74\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Linearly-recurrent%20autoencoder%20networks%20for%20learning%20dynamics&amp;journal=SIAM%20J.%20Appl.%20Dyn.%20Syst.&amp;doi=10.1137%2F18M1177846&amp;volume=18&amp;pages=558-593&amp;publication_year=2019&amp;author=Otto%2CSE&amp;author=Rowley%2CCW\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<p class=\"c-article-references__text\" id=\"ref-CR75\">Locatello, F. et al. Challenging common assumptions in the unsupervised learning of disentangled representations, 4114\u20134124 (PMLR, 2019).<\/p>\n<p class=\"c-article-references__text\" id=\"ref-CR76\">Lachapelle, S. et al. Nonparametric partial disentanglement via mechanism sparsity: Sparse actions, interventions and sparse temporal dependencies. J. Mach. Learn. Res. 27, 1\u201390 (2026).<\/p>\n<p class=\"c-article-references__text\" id=\"ref-CR77\">Lippe, P. et al. Citris: Causal identifiability from temporal intervened sequences, 13557\u201313603 (PMLR, 2022).<\/p>\n<p class=\"c-article-references__text\" id=\"ref-CR78\">Cranmer, M. et al. Disentangled sparsity networks for explainable AI. (2021).<\/p>\n<p class=\"c-article-references__text\" id=\"ref-CR79\">No\u00e9, F., Olsson, S., K\u00f6hler, J. &amp; Wu, H. Boltzmann generators: Sampling equilibrium states of many-body systems with deep learning. Science 365, eaaw1147 (2019).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"10.1126\/science.aaw1147\" data-track-item_id=\"10.1126\/science.aaw1147\" data-track-value=\"article reference\" data-track-action=\"article reference\" href=\"https:\/\/doi.org\/10.1126%2Fscience.aaw1147\" aria-label=\"Article reference 79\" data-doi=\"10.1126\/science.aaw1147\" target=\"_blank\">Article<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2019Sci...365.1147N\" aria-label=\"ADS reference 79\" target=\"_blank\">ADS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"pubmed reference\" data-track-action=\"pubmed reference\" href=\"http:\/\/www.ncbi.nlm.nih.gov\/entrez\/query.fcgi?cmd=Retrieve&amp;db=PubMed&amp;dopt=Abstract&amp;list_uids=31488660\" aria-label=\"PubMed reference 79\" target=\"_blank\">PubMed<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 79\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Boltzmann%20generators%3A%20Sampling%20equilibrium%20states%20of%20many-body%20systems%20with%20deep%20learning&amp;journal=Science&amp;doi=10.1126%2Fscience.aaw1147&amp;volume=365&amp;publication_year=2019&amp;author=No%C3%A9%2CF&amp;author=Olsson%2CS&amp;author=K%C3%B6hler%2CJ&amp;author=Wu%2CH\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<p class=\"c-article-references__text\" id=\"ref-CR80\">Greydanus, S., Dzamba, M. &amp; Yosinski, J. Hamiltonian neural networks. Adv. Neural Inf. Process. Syst. 32, 15379\u201315389 (2019).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 80\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Hamiltonian%20neural%20networks&amp;journal=Adv.%20Neural%20Inf.%20Process.%20Syst.&amp;volume=32&amp;pages=15379-15389&amp;publication_year=2019&amp;author=Greydanus%2CS&amp;author=Dzamba%2CM&amp;author=Yosinski%2CJ\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<p class=\"c-article-references__text\" id=\"ref-CR81\">Cranmer, M. et al. Lagrangian neural networks. Preprint at arXiv <a href=\"https:\/\/doi.org\/10.48550\/arXiv.2003.04630\" data-track=\"click_references\" data-track-action=\"external reference\" data-track-value=\"external reference\" data-track-label=\"10.48550\/arXiv.2003.04630\" rel=\"nofollow noopener\" target=\"_blank\">https:\/\/doi.org\/10.48550\/arXiv.2003.04630<\/a> (2020).<\/p>\n<p class=\"c-article-references__text\" id=\"ref-CR82\">Miller, B. K., Geiger, M., Smidt, T. E. &amp; No\u00e9, F. Relevance of rotationally equivariant convolutions for predicting molecular properties. Preprint at arXiv <a href=\"https:\/\/doi.org\/10.48550\/arXiv.2008.08461\" data-track=\"click_references\" data-track-action=\"external reference\" data-track-value=\"external reference\" data-track-label=\"10.48550\/arXiv.2008.08461\" rel=\"nofollow noopener\" target=\"_blank\">https:\/\/doi.org\/10.48550\/arXiv.2008.08461<\/a> (2020).<\/p>\n<p class=\"c-article-references__text\" id=\"ref-CR83\">Batzner, S. et al. E (3)-equivariant graph neural networks for data-efficient and accurate interatomic potentials. Nat. Commun. 13, 2453 (2022).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"10.1038\/s41467-022-29939-5\" data-track-item_id=\"10.1038\/s41467-022-29939-5\" data-track-value=\"article reference\" data-track-action=\"article reference\" href=\"https:\/\/doi.org\/10.1038%2Fs41467-022-29939-5\" aria-label=\"Article reference 83\" data-doi=\"10.1038\/s41467-022-29939-5\" target=\"_blank\">Article<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2022NatCo..13.2453B\" aria-label=\"ADS reference 83\" target=\"_blank\">ADS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"cas reference\" data-track-action=\"cas reference\" href=\"https:\/\/www.nature.com\/articles\/cas-redirect\/1:CAS:528:DC%2BB38Xht1ShtLrE\" aria-label=\"CAS reference 83\" target=\"_blank\">CAS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"pubmed reference\" data-track-action=\"pubmed reference\" href=\"http:\/\/www.ncbi.nlm.nih.gov\/entrez\/query.fcgi?cmd=Retrieve&amp;db=PubMed&amp;dopt=Abstract&amp;list_uids=35508450\" aria-label=\"PubMed reference 83\" target=\"_blank\">PubMed<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"pubmed central reference\" data-track-action=\"pubmed central reference\" href=\"http:\/\/www.ncbi.nlm.nih.gov\/pmc\/articles\/PMC9068614\" aria-label=\"PubMed Central reference 83\" target=\"_blank\">PubMed Central<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 83\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=E%20%283%29-equivariant%20graph%20neural%20networks%20for%20data-efficient%20and%20accurate%20interatomic%20potentials&amp;journal=Nat.%20Commun.&amp;doi=10.1038%2Fs41467-022-29939-5&amp;volume=13&amp;publication_year=2022&amp;author=Batzner%2CS\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<p class=\"c-article-references__text\" id=\"ref-CR84\">Brandstetter, J., Berg, R. v. d., Welling, M. &amp; Gupta, J. K. Clifford neural layers for PDE modeling. In International Conference on Learning Representations <a href=\"https:\/\/mlanthology.org\/iclr\/2023\/brandstetter2023iclr-clifford\/\" data-track=\"click_references\" data-track-action=\"external reference\" data-track-value=\"external reference\" data-track-label=\"https:\/\/mlanthology.org\/iclr\/2023\/brandstetter2023iclr-clifford\/\" rel=\"nofollow noopener\" target=\"_blank\">https:\/\/mlanthology.org\/iclr\/2023\/brandstetter2023iclr-clifford\/<\/a> (2023).<\/p>\n<p class=\"c-article-references__text\" id=\"ref-CR85\">Brandstetter, J., Welling, M. &amp; Worrall, D. E. Lie point symmetry data augmentation for neural PDE solvers. International Conference on Machine Learning 2241\u20132256 (2022).<\/p>\n<p class=\"c-article-references__text\" id=\"ref-CR86\">Du, P., Parikh, M. H., Fan, X., Liu, X. &amp; Wang, J. Conditional neural field latent diffusion model for generating spatiotemporal turbulence. Nat. Commun. 15, 10416 (2024).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"10.1038\/s41467-024-54712-1\" data-track-item_id=\"10.1038\/s41467-024-54712-1\" data-track-value=\"article reference\" data-track-action=\"article reference\" href=\"https:\/\/doi.org\/10.1038%2Fs41467-024-54712-1\" aria-label=\"Article reference 86\" data-doi=\"10.1038\/s41467-024-54712-1\" target=\"_blank\">Article<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2024NatCo..1510416D\" aria-label=\"ADS reference 86\" target=\"_blank\">ADS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"cas reference\" data-track-action=\"cas reference\" href=\"https:\/\/www.nature.com\/articles\/cas-redirect\/1:CAS:528:DC%2BB2cXis1GhsL3M\" aria-label=\"CAS reference 86\" target=\"_blank\">CAS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"pubmed reference\" data-track-action=\"pubmed reference\" href=\"http:\/\/www.ncbi.nlm.nih.gov\/entrez\/query.fcgi?cmd=Retrieve&amp;db=PubMed&amp;dopt=Abstract&amp;list_uids=39613755\" aria-label=\"PubMed reference 86\" target=\"_blank\">PubMed<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"pubmed central reference\" data-track-action=\"pubmed central reference\" href=\"http:\/\/www.ncbi.nlm.nih.gov\/pmc\/articles\/PMC11607081\" aria-label=\"PubMed Central reference 86\" target=\"_blank\">PubMed Central<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 86\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Conditional%20neural%20field%20latent%20diffusion%20model%20for%20generating%20spatiotemporal%20turbulence.&amp;journal=Nat.%20Commun.&amp;doi=10.1038%2Fs41467-024-54712-1&amp;volume=15&amp;publication_year=2024&amp;author=Du%2CP&amp;author=Parikh%2CMH&amp;author=Fan%2CX&amp;author=Liu%2CX&amp;author=Wang%2CJ\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<p class=\"c-article-references__text\" id=\"ref-CR87\">Solera-Rico, A. et al. \u03b2-Variational autoencoders and transformers for reduced-order modelling of fluid flows. Nat. Commun. 15, 1361 (2024).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"10.1038\/s41467-024-45578-4\" data-track-item_id=\"10.1038\/s41467-024-45578-4\" data-track-value=\"article reference\" data-track-action=\"article reference\" href=\"https:\/\/doi.org\/10.1038%2Fs41467-024-45578-4\" aria-label=\"Article reference 87\" data-doi=\"10.1038\/s41467-024-45578-4\" target=\"_blank\">Article<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2024NatCo..15.1361S\" aria-label=\"ADS reference 87\" target=\"_blank\">ADS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"cas reference\" data-track-action=\"cas reference\" href=\"https:\/\/www.nature.com\/articles\/cas-redirect\/1:CAS:528:DC%2BB2cXksVWmtLw%3D\" aria-label=\"CAS reference 87\" target=\"_blank\">CAS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"pubmed reference\" data-track-action=\"pubmed reference\" href=\"http:\/\/www.ncbi.nlm.nih.gov\/entrez\/query.fcgi?cmd=Retrieve&amp;db=PubMed&amp;dopt=Abstract&amp;list_uids=38355720\" aria-label=\"PubMed reference 87\" target=\"_blank\">PubMed<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"pubmed central reference\" data-track-action=\"pubmed central reference\" href=\"http:\/\/www.ncbi.nlm.nih.gov\/pmc\/articles\/PMC10866995\" aria-label=\"PubMed Central reference 87\" target=\"_blank\">PubMed Central<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 87\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=%CE%B2-Variational%20autoencoders%20and%20transformers%20for%20reduced-order%20modelling%20of%20fluid%20flows&amp;journal=Nat.%20Commun.&amp;doi=10.1038%2Fs41467-024-45578-4&amp;volume=15&amp;publication_year=2024&amp;author=Solera-Rico%2CA\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<p class=\"c-article-references__text\" id=\"ref-CR88\">Mnih, V. et al. Human-level control through deep reinforcement learning. Nature 518, 529\u2013533 (2015).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"10.1038\/nature14236\" data-track-item_id=\"10.1038\/nature14236\" data-track-value=\"article reference\" data-track-action=\"article reference\" href=\"https:\/\/doi.org\/10.1038%2Fnature14236\" aria-label=\"Article reference 88\" data-doi=\"10.1038\/nature14236\" target=\"_blank\">Article<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2015Natur.518..529M\" aria-label=\"ADS reference 88\" target=\"_blank\">ADS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"cas reference\" data-track-action=\"cas reference\" href=\"https:\/\/www.nature.com\/articles\/cas-redirect\/1:CAS:528:DC%2BC2MXjsVagur0%3D\" aria-label=\"CAS reference 88\" target=\"_blank\">CAS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"pubmed reference\" data-track-action=\"pubmed reference\" href=\"http:\/\/www.ncbi.nlm.nih.gov\/entrez\/query.fcgi?cmd=Retrieve&amp;db=PubMed&amp;dopt=Abstract&amp;list_uids=25719670\" aria-label=\"PubMed reference 88\" target=\"_blank\">PubMed<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 88\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Human-level%20control%20through%20deep%20reinforcement%20learning&amp;journal=Nature&amp;doi=10.1038%2Fnature14236&amp;volume=518&amp;pages=529-533&amp;publication_year=2015&amp;author=Mnih%2CV\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<p class=\"c-article-references__text\" id=\"ref-CR89\">Lillicrap, T. P. et al. Continuous control with deep reinforcement learning. Preprint at arXiv <a href=\"https:\/\/doi.org\/10.48550\/arXiv.1509.02971\" data-track=\"click_references\" data-track-action=\"external reference\" data-track-value=\"external reference\" data-track-label=\"10.48550\/arXiv.1509.02971\" rel=\"nofollow noopener\" target=\"_blank\">https:\/\/doi.org\/10.48550\/arXiv.1509.02971<\/a> (2015).<\/p>\n<p class=\"c-article-references__text\" id=\"ref-CR90\">Erion, G., Janizek, J. D., Sturmfels, P., Lundberg, S. M. &amp; Lee, S.-I. Improving performance of deep learning models with axiomatic attribution priors and expected gradients. Nat. Mach. Intell. 3, 620\u2013631 (2021).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"10.1038\/s42256-021-00343-w\" data-track-item_id=\"10.1038\/s42256-021-00343-w\" data-track-value=\"article reference\" data-track-action=\"article reference\" href=\"https:\/\/doi.org\/10.1038%2Fs42256-021-00343-w\" aria-label=\"Article reference 90\" data-doi=\"10.1038\/s42256-021-00343-w\" target=\"_blank\">Article<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 90\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Improving%20performance%20of%20deep%20learning%20models%20with%20axiomatic%20attribution%20priors%20and%20expected%20gradients&amp;journal=Nat.%20Mach.%20Intell.&amp;doi=10.1038%2Fs42256-021-00343-w&amp;volume=3&amp;pages=620-631&amp;publication_year=2021&amp;author=Erion%2CG&amp;author=Janizek%2CJD&amp;author=Sturmfels%2CP&amp;author=Lundberg%2CSM&amp;author=Lee%2CS-I\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<p class=\"c-article-references__text\" id=\"ref-CR91\">Martins, J. R. R. A. &amp; Lambe, A. B. Multidisciplinary design optimization: a survey of architectures. AIAA J. 51, 2049\u20132075 (2013).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"10.2514\/1.J051895\" data-track-item_id=\"10.2514\/1.J051895\" data-track-value=\"article reference\" data-track-action=\"article reference\" href=\"https:\/\/doi.org\/10.2514%2F1.J051895\" aria-label=\"Article reference 91\" data-doi=\"10.2514\/1.J051895\" target=\"_blank\">Article<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2013AIAAJ..51.2049M\" aria-label=\"ADS reference 91\" target=\"_blank\">ADS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 91\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Multidisciplinary%20design%20optimization%3A%20a%20survey%20of%20architectures&amp;journal=AIAA%20J.&amp;doi=10.2514%2F1.J051895&amp;volume=51&amp;pages=2049-2075&amp;publication_year=2013&amp;author=Martins%2CJRRA&amp;author=Lambe%2CAB\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<p class=\"c-article-references__text\" id=\"ref-CR92\">Jameson, A. Aerodynamic design via control theory. J. Sci. Comput. 3, 233\u2013260 (1988).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" rel=\"noopener nofollow\" data-track-label=\"10.1007\/BF01061285\" data-track-item_id=\"10.1007\/BF01061285\" data-track-value=\"article reference\" data-track-action=\"article reference\" href=\"https:\/\/link.springer.com\/doi\/10.1007\/BF01061285\" aria-label=\"Article reference 92\" data-doi=\"10.1007\/BF01061285\" target=\"_blank\">Article<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 92\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Aerodynamic%20design%20via%20control%20theory&amp;journal=J.%20Sci.%20Comput.&amp;doi=10.1007%2FBF01061285&amp;volume=3&amp;pages=233-260&amp;publication_year=1988&amp;author=Jameson%2CA\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<p class=\"c-article-references__text\" id=\"ref-CR93\">Shapley, L. S. et al. A value for n-person games (1953).<\/p>\n<p class=\"c-article-references__text\" id=\"ref-CR94\">Simonyan, K., Vedaldi, A. &amp; Zisserman, A. Deep inside convolutional networks: Visualizing image classification models and saliency maps. Preprint at arXiv <a href=\"https:\/\/doi.org\/10.48550\/arXiv.1312.6034\" data-track=\"click_references\" data-track-action=\"external reference\" data-track-value=\"external reference\" data-track-label=\"10.48550\/arXiv.1312.6034\" rel=\"nofollow noopener\" target=\"_blank\">https:\/\/doi.org\/10.48550\/arXiv.1312.6034<\/a> (2013).<\/p>\n<p class=\"c-article-references__text\" id=\"ref-CR95\">Cremades, A. et al. Identifying regions of importance in wall-bounded turbulence through explainable deep learning. Nat. Commun. 15, 3864 (2024).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"10.1038\/s41467-024-47954-6\" data-track-item_id=\"10.1038\/s41467-024-47954-6\" data-track-value=\"article reference\" data-track-action=\"article reference\" href=\"https:\/\/doi.org\/10.1038%2Fs41467-024-47954-6\" aria-label=\"Article reference 95\" data-doi=\"10.1038\/s41467-024-47954-6\" target=\"_blank\">Article<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2024NatCo..15.3864C\" aria-label=\"ADS reference 95\" target=\"_blank\">ADS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"cas reference\" data-track-action=\"cas reference\" href=\"https:\/\/www.nature.com\/articles\/cas-redirect\/1:CAS:528:DC%2BB2cXht1Clsr3P\" aria-label=\"CAS reference 95\" target=\"_blank\">CAS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"pubmed reference\" data-track-action=\"pubmed reference\" href=\"http:\/\/www.ncbi.nlm.nih.gov\/entrez\/query.fcgi?cmd=Retrieve&amp;db=PubMed&amp;dopt=Abstract&amp;list_uids=38740802\" aria-label=\"PubMed reference 95\" target=\"_blank\">PubMed<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"pubmed central reference\" data-track-action=\"pubmed central reference\" href=\"http:\/\/www.ncbi.nlm.nih.gov\/pmc\/articles\/PMC11091079\" aria-label=\"PubMed Central reference 95\" target=\"_blank\">PubMed Central<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 95\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Identifying%20regions%20of%20importance%20in%20wall-bounded%20turbulence%20through%20explainable%20deep%20learning&amp;journal=Nat.%20Commun.&amp;doi=10.1038%2Fs41467-024-47954-6&amp;volume=15&amp;publication_year=2024&amp;author=Cremades%2CA\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<p class=\"c-article-references__text\" id=\"ref-CR96\">Larra\u00f1aga, A., Sandberg, R. D., Mart\u00ednez, J. &amp; Porteiro, J. On the machine learning-assisted identification of the fundamental parameters of nonstandard microfin arrays to assess their heat transfer performance. Eng. Appl. Artif. Intell. 136, 108945 (2024).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"10.1016\/j.engappai.2024.108945\" data-track-item_id=\"10.1016\/j.engappai.2024.108945\" data-track-value=\"article reference\" data-track-action=\"article reference\" href=\"https:\/\/doi.org\/10.1016%2Fj.engappai.2024.108945\" aria-label=\"Article reference 96\" data-doi=\"10.1016\/j.engappai.2024.108945\" target=\"_blank\">Article<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 96\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=On%20the%20machine%20learning-assisted%20identification%20of%20the%20fundamental%20parameters%20of%20nonstandard%20microfin%20arrays%20to%20assess%20their%20heat%20transfer%20performance&amp;journal=Eng.%20Appl.%20Artif.%20Intell.&amp;doi=10.1016%2Fj.engappai.2024.108945&amp;volume=136&amp;publication_year=2024&amp;author=Larra%C3%B1aga%2CA&amp;author=Sandberg%2CRD&amp;author=Mart%C3%ADnez%2CJ&amp;author=Porteiro%2CJ\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<p class=\"c-article-references__text\" id=\"ref-CR97\">Zhao, P. et al. A machine learning and CFD modeling hybrid approach for predicting real-time heat transfer during cokemaking processes. Fuel 373, 132273 (2024).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"10.1016\/j.fuel.2024.132273\" data-track-item_id=\"10.1016\/j.fuel.2024.132273\" data-track-value=\"article reference\" data-track-action=\"article reference\" href=\"https:\/\/doi.org\/10.1016%2Fj.fuel.2024.132273\" aria-label=\"Article reference 97\" data-doi=\"10.1016\/j.fuel.2024.132273\" target=\"_blank\">Article<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"cas reference\" data-track-action=\"cas reference\" href=\"https:\/\/www.nature.com\/articles\/cas-redirect\/1:CAS:528:DC%2BB2cXhtlyqur7I\" aria-label=\"CAS reference 97\" target=\"_blank\">CAS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 97\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=A%20machine%20learning%20and%20CFD%20modeling%20hybrid%20approach%20for%20predicting%20real-time%20heat%20transfer%20during%20cokemaking%20processes&amp;journal=Fuel&amp;doi=10.1016%2Fj.fuel.2024.132273&amp;volume=373&amp;publication_year=2024&amp;author=Zhao%2CP\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<p class=\"c-article-references__text\" id=\"ref-CR98\">Singh, S., Suman, S., Kumar, M. &amp; Mitra, S. Ann model for prediction of thermo-hydraulic performance of a solar air heater with vertical cylindrical ribs. Energy Rep. 8, 585\u2013592 (2022).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"10.1016\/j.egyr.2022.10.263\" data-track-item_id=\"10.1016\/j.egyr.2022.10.263\" data-track-value=\"article reference\" data-track-action=\"article reference\" href=\"https:\/\/doi.org\/10.1016%2Fj.egyr.2022.10.263\" aria-label=\"Article reference 98\" data-doi=\"10.1016\/j.egyr.2022.10.263\" target=\"_blank\">Article<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 98\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Ann%20model%20for%20prediction%20of%20thermo-hydraulic%20performance%20of%20a%20solar%20air%20heater%20with%20vertical%20cylindrical%20ribs&amp;journal=Energy%20Rep.&amp;doi=10.1016%2Fj.egyr.2022.10.263&amp;volume=8&amp;pages=585-592&amp;publication_year=2022&amp;author=Singh%2CS&amp;author=Suman%2CS&amp;author=Kumar%2CM&amp;author=Mitra%2CS\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<p class=\"c-article-references__text\" id=\"ref-CR99\">Verma, S., Novati, G. &amp; Koumoutsakos, P. Efficient collective swimming by harnessing vortices through deep reinforcement learning. Proc. Natl. Acad. Sci. USA. 115, 5849\u20135854 (2018).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"10.1073\/pnas.1800923115\" data-track-item_id=\"10.1073\/pnas.1800923115\" data-track-value=\"article reference\" data-track-action=\"article reference\" href=\"https:\/\/doi.org\/10.1073%2Fpnas.1800923115\" aria-label=\"Article reference 99\" data-doi=\"10.1073\/pnas.1800923115\" target=\"_blank\">Article<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2018PNAS..115.5849V\" aria-label=\"ADS reference 99\" target=\"_blank\">ADS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"cas reference\" data-track-action=\"cas reference\" href=\"https:\/\/www.nature.com\/articles\/cas-redirect\/1:CAS:528:DC%2BC1cXitlWrtbjN\" aria-label=\"CAS reference 99\" target=\"_blank\">CAS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"pubmed reference\" data-track-action=\"pubmed reference\" href=\"http:\/\/www.ncbi.nlm.nih.gov\/entrez\/query.fcgi?cmd=Retrieve&amp;db=PubMed&amp;dopt=Abstract&amp;list_uids=29784820\" aria-label=\"PubMed reference 99\" target=\"_blank\">PubMed<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"pubmed central reference\" data-track-action=\"pubmed central reference\" href=\"http:\/\/www.ncbi.nlm.nih.gov\/pmc\/articles\/PMC6003313\" aria-label=\"PubMed Central reference 99\" target=\"_blank\">PubMed Central<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 99\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Efficient%20collective%20swimming%20by%20harnessing%20vortices%20through%20deep%20reinforcement%20learning&amp;journal=Proc.%20Natl.%20Acad.%20Sci.%20USA&amp;doi=10.1073%2Fpnas.1800923115&amp;volume=115&amp;pages=5849-5854&amp;publication_year=2018&amp;author=Verma%2CS&amp;author=Novati%2CG&amp;author=Koumoutsakos%2CP\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<p class=\"c-article-references__text\" id=\"ref-CR100\">Vishwasrao, A. et al. Diff-sport: Diffusion-based sensor placement optimization and reconstruction of turbulent flows in urban environments. Preprint at arXiv <a href=\"https:\/\/doi.org\/10.48550\/arXiv.2506.00214\" data-track=\"click_references\" data-track-action=\"external reference\" data-track-value=\"external reference\" data-track-label=\"10.48550\/arXiv.2506.00214\" rel=\"nofollow noopener\" target=\"_blank\">https:\/\/doi.org\/10.48550\/arXiv.2506.00214<\/a> (2025).<\/p>\n<p class=\"c-article-references__text\" id=\"ref-CR101\">Beneitez, M., Cremades, A., Guastoni, L. &amp; Vinuesa, R. Improving turbulence control through explainable deep learning. Preprint at arXiv <a href=\"https:\/\/doi.org\/10.48550\/arXiv.2504.02354\" data-track=\"click_references\" data-track-action=\"external reference\" data-track-value=\"external reference\" data-track-label=\"10.48550\/arXiv.2504.02354\" rel=\"nofollow noopener\" target=\"_blank\">https:\/\/doi.org\/10.48550\/arXiv.2504.02354<\/a> (2025).<\/p>\n<p class=\"c-article-references__text\" id=\"ref-CR102\">Heaton, H. &amp; Wu Fung, S. Explainable AI via learning to optimize. Sci. Rep. 13, 10103 (2023).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"10.1038\/s41598-023-36249-3\" data-track-item_id=\"10.1038\/s41598-023-36249-3\" data-track-value=\"article reference\" data-track-action=\"article reference\" href=\"https:\/\/doi.org\/10.1038%2Fs41598-023-36249-3\" aria-label=\"Article reference 102\" data-doi=\"10.1038\/s41598-023-36249-3\" target=\"_blank\">Article<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2023NatSR..1310103H\" aria-label=\"ADS reference 102\" target=\"_blank\">ADS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"cas reference\" data-track-action=\"cas reference\" href=\"https:\/\/www.nature.com\/articles\/cas-redirect\/1:CAS:528:DC%2BB3sXhtlSrur3M\" aria-label=\"CAS reference 102\" target=\"_blank\">CAS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"pubmed reference\" data-track-action=\"pubmed reference\" href=\"http:\/\/www.ncbi.nlm.nih.gov\/entrez\/query.fcgi?cmd=Retrieve&amp;db=PubMed&amp;dopt=Abstract&amp;list_uids=37344533\" aria-label=\"PubMed reference 102\" target=\"_blank\">PubMed<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"pubmed central reference\" data-track-action=\"pubmed central reference\" href=\"http:\/\/www.ncbi.nlm.nih.gov\/pmc\/articles\/PMC10284861\" aria-label=\"PubMed Central reference 102\" target=\"_blank\">PubMed Central<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 102\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Explainable%20AI%20via%20learning%20to%20optimize&amp;journal=Sci.%20Rep.&amp;doi=10.1038%2Fs41598-023-36249-3&amp;volume=13&amp;publication_year=2023&amp;author=Heaton%2CH&amp;author=Wu%20Fung%2CS\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<p class=\"c-article-references__text\" id=\"ref-CR103\">Murray, B. et al. Explainable AI for understanding decisions and data-driven optimization of the Choquet integral. 2018 IEEE International Conference on Fuzzy Systems. (FUZZ-IEEE) (2018).<\/p>\n<p class=\"c-article-references__text\" id=\"ref-CR104\">Shen, Y., Huang, W., Wang, Z. &amp; Xu, D. F. An automatic visible explainer of geometric knowledge for aeroshape design optimization based on SHAP. Aerosp. Sci. Technol. 127, 107800 (2022).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 104\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=An%20automatic%20visible%20explainer%20of%20geometric%20knowledge%20for%20aeroshape%20design%20optimization%20based%20on%20SHAP&amp;journal=Aerosp.%20Sci.%20Technol.&amp;volume=127&amp;publication_year=2022&amp;author=Shen%2CY&amp;author=Huang%2CW&amp;author=Wang%2CZ&amp;author=Xu%2CDF\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<p class=\"c-article-references__text\" id=\"ref-CR105\">Hinton, G. E. &amp; Salakhutdinov, R. R. Reducing the dimensionality of data with neural networks. Science 313, 504\u2013507 (2006).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"10.1126\/science.1127647\" data-track-item_id=\"10.1126\/science.1127647\" data-track-value=\"article reference\" data-track-action=\"article reference\" href=\"https:\/\/doi.org\/10.1126%2Fscience.1127647\" aria-label=\"Article reference 105\" data-doi=\"10.1126\/science.1127647\" target=\"_blank\">Article<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2006Sci...313..504H\" aria-label=\"ADS reference 105\" target=\"_blank\">ADS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"mathscinet reference\" data-track-action=\"mathscinet reference\" href=\"http:\/\/www.ams.org\/mathscinet-getitem?mr=2242509\" aria-label=\"MathSciNet reference 105\" target=\"_blank\">MathSciNet<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"cas reference\" data-track-action=\"cas reference\" href=\"https:\/\/www.nature.com\/articles\/cas-redirect\/1:CAS:528:DC%2BD28Xnt1KntrY%3D\" aria-label=\"CAS reference 105\" target=\"_blank\">CAS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"pubmed reference\" data-track-action=\"pubmed reference\" href=\"http:\/\/www.ncbi.nlm.nih.gov\/entrez\/query.fcgi?cmd=Retrieve&amp;db=PubMed&amp;dopt=Abstract&amp;list_uids=16873662\" aria-label=\"PubMed reference 105\" target=\"_blank\">PubMed<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 105\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Reducing%20the%20dimensionality%20of%20data%20with%20neural%20networks&amp;journal=Science&amp;doi=10.1126%2Fscience.1127647&amp;volume=313&amp;pages=504-507&amp;publication_year=2006&amp;author=Hinton%2CGE&amp;author=Salakhutdinov%2CRR\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<p class=\"c-article-references__text\" id=\"ref-CR106\">Higgins, I. et al. beta-VAE: learning basic visual concepts with a constrained variational framework. International Conference on Learning Representations (ICLR) (2017).<\/p>\n<p class=\"c-article-references__text\" id=\"ref-CR107\">Arranz, G. &amp; Lozano-Dur\u00e1n, A. Informative and non-informative decomposition of turbulent flow fields. J. Fluid Mech. 1000, A95 (2024).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"10.1017\/jfm.2024.1007\" data-track-item_id=\"10.1017\/jfm.2024.1007\" data-track-value=\"article reference\" data-track-action=\"article reference\" href=\"https:\/\/doi.org\/10.1017%2Fjfm.2024.1007\" aria-label=\"Article reference 107\" data-doi=\"10.1017\/jfm.2024.1007\" target=\"_blank\">Article<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"ads reference\" data-track-action=\"ads reference\" href=\"http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-data_query?link_type=ABSTRACT&amp;bibcode=2024JFM..1000A..95A\" aria-label=\"ADS reference 107\" target=\"_blank\">ADS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"mathscinet reference\" data-track-action=\"mathscinet reference\" href=\"http:\/\/www.ams.org\/mathscinet-getitem?mr=4834852\" aria-label=\"MathSciNet reference 107\" target=\"_blank\">MathSciNet<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"cas reference\" data-track-action=\"cas reference\" href=\"https:\/\/www.nature.com\/articles\/cas-redirect\/1:CAS:528:DC%2BB2MXht12gt7c%3D\" aria-label=\"CAS reference 107\" target=\"_blank\">CAS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 107\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Informative%20and%20non-informative%20decomposition%20of%20turbulent%20flow%20fields&amp;journal=J.%20Fluid%20Mech.&amp;doi=10.1017%2Fjfm.2024.1007&amp;volume=1000&amp;publication_year=2024&amp;author=Arranz%2CG&amp;author=Lozano-Dur%C3%A1n%2CA\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<p class=\"c-article-references__text\" id=\"ref-CR108\">Chen, M. et al. Explainable machine learning model for load-deformation correlation in long-span suspension bridges using XGBoost-SHAP. Dev. Built Environ. 20, 100569 (2024).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"10.1016\/j.dibe.2024.100569\" data-track-item_id=\"10.1016\/j.dibe.2024.100569\" data-track-value=\"article reference\" data-track-action=\"article reference\" href=\"https:\/\/doi.org\/10.1016%2Fj.dibe.2024.100569\" aria-label=\"Article reference 108\" data-doi=\"10.1016\/j.dibe.2024.100569\" target=\"_blank\">Article<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 108\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Explainable%20machine%20learning%20model%20for%20load-deformation%20correlation%20in%20long-span%20suspension%20bridges%20using%20XGBoost-SHAP&amp;journal=Dev.%20Built%20Environ.&amp;doi=10.1016%2Fj.dibe.2024.100569&amp;volume=20&amp;publication_year=2024&amp;author=Chen%2CM\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<p class=\"c-article-references__text\" id=\"ref-CR109\">Chung, W. T., Mishra, A. A. &amp; Ihme, M. Interpretable data-driven methods for subgrid-scale closure in les for transcritical LOX\/GCH4 combustion. Combust. Flame 239, 111758 (2022).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"10.1016\/j.combustflame.2021.111758\" data-track-item_id=\"10.1016\/j.combustflame.2021.111758\" data-track-value=\"article reference\" data-track-action=\"article reference\" href=\"https:\/\/doi.org\/10.1016%2Fj.combustflame.2021.111758\" aria-label=\"Article reference 109\" data-doi=\"10.1016\/j.combustflame.2021.111758\" target=\"_blank\">Article<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"cas reference\" data-track-action=\"cas reference\" href=\"https:\/\/www.nature.com\/articles\/cas-redirect\/1:CAS:528:DC%2BB3MXitFSgu7nN\" aria-label=\"CAS reference 109\" target=\"_blank\">CAS<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 109\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Interpretable%20data-driven%20methods%20for%20subgrid-scale%20closure%20in%20les%20for%20transcritical%20LOX%2FGCH4%20combustion&amp;journal=Combust.%20Flame&amp;doi=10.1016%2Fj.combustflame.2021.111758&amp;volume=239&amp;publication_year=2022&amp;author=Chung%2CWT&amp;author=Mishra%2CAA&amp;author=Ihme%2CM\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<p class=\"c-article-references__text\" id=\"ref-CR110\">Sanchis-Agudo, M. et al. Easy attention: A simple attention mechanism for temporal predictions with transformers. APL Comput. Phys. 1, 016104 (2025).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"10.1063\/5.0284085\" data-track-item_id=\"10.1063\/5.0284085\" data-track-value=\"article reference\" data-track-action=\"article reference\" href=\"https:\/\/doi.org\/10.1063%2F5.0284085\" aria-label=\"Article reference 110\" data-doi=\"10.1063\/5.0284085\" target=\"_blank\">Article<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 110\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Easy%20attention%3A%20A%20simple%20attention%20mechanism%20for%20temporal%20predictions%20with%20transformers&amp;journal=APL%20Comput.%20Phys.&amp;doi=10.1063%2F5.0284085&amp;volume=1&amp;publication_year=2025&amp;author=Sanchis-Agudo%2CM\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<p class=\"c-article-references__text\" id=\"ref-CR111\">Bommasani, R. et al. On the opportunities and risks of foundation models. Preprint at arXiv <a href=\"https:\/\/doi.org\/10.48550\/arXiv.2108.07258\" data-track=\"click_references\" data-track-action=\"external reference\" data-track-value=\"external reference\" data-track-label=\"10.48550\/arXiv.2108.07258\" rel=\"nofollow noopener\" target=\"_blank\">https:\/\/doi.org\/10.48550\/arXiv.2108.07258<\/a> (2022).<\/p>\n<p class=\"c-article-references__text\" id=\"ref-CR112\">Gottweis, J. et al. Towards an AI co-scientist. Nature 655, 487\u2013496 (2026).<\/p>\n<p class=\"c-article-references__text\" id=\"ref-CR113\">Vinuesa, R., Manch\u00f3n, P., Hoyas, S. &amp; Garc\u00eda-Mart\u00ednez, J. Balancing ai and human insights in scientific discovery: Challenges and guidelines. Innovation 7, 101144 (2026).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"pubmed reference\" data-track-action=\"pubmed reference\" href=\"http:\/\/www.ncbi.nlm.nih.gov\/entrez\/query.fcgi?cmd=Retrieve&amp;db=PubMed&amp;dopt=Abstract&amp;list_uids=41658493\" aria-label=\"PubMed reference 113\" target=\"_blank\">PubMed<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 113\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Balancing%20ai%20and%20human%20insights%20in%20scientific%20discovery%3A%20Challenges%20and%20guidelines&amp;journal=Innovation&amp;volume=7&amp;publication_year=2026&amp;author=Vinuesa%2CR&amp;author=Manch%C3%B3n%2CP&amp;author=Hoyas%2CS&amp;author=Garc%C3%ADa-Mart%C3%ADnez%2CJ\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<p class=\"c-article-references__text\" id=\"ref-CR114\">Wadell, A. et al. Foundation models for discovery and exploration in chemical space. Preprint at arXiv <a href=\"https:\/\/doi.org\/10.48550\/arXiv.2510.18900\" data-track=\"click_references\" data-track-action=\"external reference\" data-track-value=\"external reference\" data-track-label=\"10.48550\/arXiv.2510.18900\" rel=\"nofollow noopener\" target=\"_blank\">https:\/\/doi.org\/10.48550\/arXiv.2510.18900<\/a> (2025).<\/p>\n<p class=\"c-article-references__text\" id=\"ref-CR115\">Carreon, A., Sharma, V. &amp; Raman, V. Automated design optimization via strategic search with large language models. Preprint at arXiv <a href=\"https:\/\/doi.org\/10.48550\/arXiv.2511.22651\" data-track=\"click_references\" data-track-action=\"external reference\" data-track-value=\"external reference\" data-track-label=\"10.48550\/arXiv.2511.22651\" rel=\"nofollow noopener\" target=\"_blank\">https:\/\/doi.org\/10.48550\/arXiv.2511.22651<\/a> (2025).<\/p>\n<p class=\"c-article-references__text\" id=\"ref-CR116\">Administration, F. A. Summary of the FAA\u2019s Review of the Boeing 737MAX. Tech. Rep. <a href=\"https:\/\/www.faa.gov\/sites\/faa.gov\/files\/2022-08\/737_RTS_Summary.pdf\" data-track=\"click_references\" data-track-action=\"external reference\" data-track-value=\"external reference\" data-track-label=\"https:\/\/www.faa.gov\/sites\/faa.gov\/files\/2022-08\/737_RTS_Summary.pdf\" rel=\"nofollow noopener\" target=\"_blank\">https:\/\/www.faa.gov\/sites\/faa.gov\/files\/2022-08\/737_RTS_Summary.pdf<\/a> (2020).<\/p>\n<p class=\"c-article-references__text\" id=\"ref-CR117\">Wang, X. et al. Condensnet: Enabling stable long-term climate simulations via hybrid deep learning models with adaptive physical constraints. npj Climate and Atmospheric Science (2025).<\/p>\n<p class=\"c-article-references__text\" id=\"ref-CR118\">Bostrom, N.Superintelligence: Paths, Dangers, Strategies (Oxford University Press, 2014).<\/p>\n<p class=\"c-article-references__text\" id=\"ref-CR119\">Wei, J., Verona, E., Bertolini, A. &amp; Mengaldo, G. Explainability matters: The effect of liability rules on the healthcare sector. Preprint at arXiv <a href=\"https:\/\/doi.org\/10.48550\/arXiv.2509.17334\" data-track=\"click_references\" data-track-action=\"external reference\" data-track-value=\"external reference\" data-track-label=\"10.48550\/arXiv.2509.17334\" rel=\"nofollow noopener\" target=\"_blank\">https:\/\/doi.org\/10.48550\/arXiv.2509.17334<\/a> (2025).<\/p>\n<p class=\"c-article-references__text\" id=\"ref-CR120\">Vinuesa, R. &amp; Sirmacek, B. Interpretable deep-learning models to help achieve the Sustainable Development Goals. Nat. Mach. Intell. 3, 926 (2021).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"10.1038\/s42256-021-00414-y\" data-track-item_id=\"10.1038\/s42256-021-00414-y\" data-track-value=\"article reference\" data-track-action=\"article reference\" href=\"https:\/\/doi.org\/10.1038%2Fs42256-021-00414-y\" aria-label=\"Article reference 120\" data-doi=\"10.1038\/s42256-021-00414-y\" target=\"_blank\">Article<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 120\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Interpretable%20deep-learning%20models%20to%20help%20achieve%20the%20Sustainable%20Development%20Goals&amp;journal=Nat.%20Mach.%20Intell.&amp;doi=10.1038%2Fs42256-021-00414-y&amp;volume=3&amp;publication_year=2021&amp;author=Vinuesa%2CR&amp;author=Sirmacek%2CB\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<p class=\"c-article-references__text\" id=\"ref-CR121\">Selbst, A. D. &amp; Barocas, S. The intuitive appeal of explainable machines. Fordham L. Rev. 87, 1085\u20131139 (2018).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 121\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=The%20intuitive%20appeal%20of%20explainable%20machines&amp;journal=Fordham%20L.%20Rev.&amp;volume=87&amp;pages=1085-1139&amp;publication_year=2018&amp;author=Selbst%2CAD&amp;author=Barocas%2CS\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<p class=\"c-article-references__text\" id=\"ref-CR122\">Krenn, M. et al. On scientific understanding with artificial intelligence. Nat. Rev. Phys. 4, 761\u2013769 (2022).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"10.1038\/s42254-022-00518-3\" data-track-item_id=\"10.1038\/s42254-022-00518-3\" data-track-value=\"article reference\" data-track-action=\"article reference\" href=\"https:\/\/doi.org\/10.1038%2Fs42254-022-00518-3\" aria-label=\"Article reference 122\" data-doi=\"10.1038\/s42254-022-00518-3\" target=\"_blank\">Article<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"pubmed reference\" data-track-action=\"pubmed reference\" href=\"http:\/\/www.ncbi.nlm.nih.gov\/entrez\/query.fcgi?cmd=Retrieve&amp;db=PubMed&amp;dopt=Abstract&amp;list_uids=36247217\" aria-label=\"PubMed reference 122\" target=\"_blank\">PubMed<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"link\" data-track-item_id=\"link\" data-track-value=\"pubmed central reference\" data-track-action=\"pubmed central reference\" href=\"http:\/\/www.ncbi.nlm.nih.gov\/pmc\/articles\/PMC9552145\" aria-label=\"PubMed Central reference 122\" target=\"_blank\">PubMed Central<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 122\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=On%20scientific%20understanding%20with%20artificial%20intelligence&amp;journal=Nat.%20Rev.%20Phys.&amp;doi=10.1038%2Fs42254-022-00518-3&amp;volume=4&amp;pages=761-769&amp;publication_year=2022&amp;author=Krenn%2CM\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n<p class=\"c-article-references__text\" id=\"ref-CR123\">Gamella, J. L., Peters, J. &amp; B\u00fchlmann, P. Causal chambers as a real-world physical testbed for AI methodology. Nat. Mach. Intell. 7, 107\u2013118 (2025).<\/p>\n<p class=\"c-article-references__links u-hide-print\"><a data-track=\"click_references\" rel=\"nofollow noopener\" data-track-label=\"10.1038\/s42256-024-00964-x\" data-track-item_id=\"10.1038\/s42256-024-00964-x\" data-track-value=\"article reference\" data-track-action=\"article reference\" href=\"https:\/\/doi.org\/10.1038%2Fs42256-024-00964-x\" aria-label=\"Article reference 123\" data-doi=\"10.1038\/s42256-024-00964-x\" target=\"_blank\">Article<\/a>\u00a0<br \/>\n    <a data-track=\"click_references\" data-track-action=\"google scholar reference\" data-track-value=\"google scholar reference\" data-track-label=\"link\" data-track-item_id=\"link\" rel=\"nofollow noopener\" aria-label=\"Google Scholar reference 123\" href=\"http:\/\/scholar.google.com\/scholar_lookup?&amp;title=Causal%20chambers%20as%20a%20real-world%20physical%20testbed%20for%20AI%20methodology&amp;journal=Nat.%20Mach.%20Intell.&amp;doi=10.1038%2Fs42256-024-00964-x&amp;volume=7&amp;pages=107-118&amp;publication_year=2025&amp;author=Gamella%2CJL&amp;author=Peters%2CJ&amp;author=B%C3%BChlmann%2CP\" target=\"_blank\"><br \/>\n                    Google Scholar<\/a>\u00a0\n                <\/p>\n","protected":false},"excerpt":{"rendered":"Jumper, J. et al. Highly accurate protein structure prediction with AlphaFold. Nature 596, 583\u2013589 (2021). Article\u00a0 ADS\u00a0 CAS\u00a0&hellip;\n","protected":false},"author":2,"featured_media":148309,"comment_status":"","ping_status":"","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[2],"tags":[24,25,1632,1633,1743,25409,1744,160],"class_list":["post-148308","post","type-post","status-publish","format-standard","has-post-thumbnail","category-ai","tag-ai","tag-artificial-intelligence","tag-computational-science","tag-computer-science","tag-humanities-and-social-sciences","tag-mechanical-engineering","tag-multidisciplinary","tag-science"],"_links":{"self":[{"href":"https:\/\/www.europesays.com\/ai\/wp-json\/wp\/v2\/posts\/148308","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.europesays.com\/ai\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.europesays.com\/ai\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.europesays.com\/ai\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/www.europesays.com\/ai\/wp-json\/wp\/v2\/comments?post=148308"}],"version-history":[{"count":0,"href":"https:\/\/www.europesays.com\/ai\/wp-json\/wp\/v2\/posts\/148308\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.europesays.com\/ai\/wp-json\/wp\/v2\/media\/148309"}],"wp:attachment":[{"href":"https:\/\/www.europesays.com\/ai\/wp-json\/wp\/v2\/media?parent=148308"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.europesays.com\/ai\/wp-json\/wp\/v2\/categories?post=148308"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.europesays.com\/ai\/wp-json\/wp\/v2\/tags?post=148308"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}