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Adv."],"published-print":{"date-parts":[[2026,9,25]]},"abstract":"<jats:p>Depositional ice growth strongly influences the evolution and lifetime of ice-containing clouds on Earth and Mars, yet its physics remain uncertain because the earliest stages of ice-crystal growth cannot be directly observed and are sparsely sampled in laboratory settings. Here, we use physics-informed machine learning to learn the functional dependence of unknown physics in the depositional ice growth model from limited observations. By optimizing against mass growth time series from 290 ice crystals grown in a levitation diffusion chamber, we identify a modified capacitance growth model that more accurately captures observed early-stage growth. The resulting functional form, retrieved using symbolic regression, outperforms existing parameterizations and generalizes well to independent measurements from the AIDA Aerosol and Cloud Chamber. Our data-efficient learning framework provides an improved representation of depositional ice growth and offers a pathway to infer unresolved physical processes in other systems that are sparsely observed.<\/jats:p>","DOI":"10.1126\/sciadv.aee6174","type":"journal-article","created":{"date-parts":[[2026,9,25]],"date-time":"2026-09-25T17:58:53Z","timestamp":1790359133000},"update-policy":"https:\/\/doi.org\/10.34133\/aaas_crossmark","source":"Crossref","is-referenced-by-count":0,"title":["Discovering how ice crystals grow using neural ODEs and symbolic regression"],"prefix":"10.1126","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6883-3285","authenticated-orcid":true,"given":"Kara D.","family":"Lamb","sequence":"first","affiliation":[{"name":"Department of Earth and Environmental Engineering, Columbia University, New York, NY, USA."},{"name":"Learning the Earth with Artificial Intelligence and Physics Center, Columbia University, New York, NY, USA."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jerry Y.","family":"Harrington","sequence":"additional","affiliation":[{"name":"Department of Meteorology, Pennsylvania State University, State College, PA, USA."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Alfred M.","family":"Moyle","sequence":"additional","affiliation":[{"name":"Department of Meteorology, Pennsylvania State University, State College, PA, USA."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Gwenore F.","family":"Pokrifka","sequence":"additional","affiliation":[{"name":"High Meadows Environmental Institute, Princeton University, Princeton, NJ, USA."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Benjamin W.","family":"Clouser","sequence":"additional","affiliation":[{"name":"Department of the Geophysical Sciences, University of Chicago, Chicago, IL, USA."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1394-3097","authenticated-orcid":true,"given":"Volker","family":"Ebert","sequence":"additional","affiliation":[{"name":"Physikalisch Chemisches Institut, Heidelberg University, Heidelberg, Germany."},{"name":"Physikalisch-Technische Bundesanstalt, Braunschweig, Germany."},{"name":"Reactive Flows and Diagnostics, Technische Universit\u00e4t Darmstadt, Darmstadt, Germany."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7551-9814","authenticated-orcid":true,"given":"Ottmar","family":"M\u00f6hler","sequence":"additional","affiliation":[{"name":"Institute of Meteorology and Climate Research Atmospheric Aerosol Research, Karlsruhe Institute of Technology, Eggenstein-Leopoldshafen, Germany."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1301-8010","authenticated-orcid":true,"given":"Harald","family":"Saathoff","sequence":"additional","affiliation":[{"name":"Institute of Meteorology and Climate Research Atmospheric Aerosol Research, Karlsruhe Institute of Technology, Eggenstein-Leopoldshafen, Germany."}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"221","reference":[{"key":"e_1_3_2_2_2","doi-asserted-by":"publisher","DOI":"10.5194\/acp-23-15413-2023"},{"key":"e_1_3_2_3_2","doi-asserted-by":"publisher","DOI":"10.1126\/science.1182488"},{"key":"e_1_3_2_4_2","doi-asserted-by":"publisher","DOI":"10.1038\/s41467-018-04068-0"},{"key":"e_1_3_2_5_2","doi-asserted-by":"publisher","DOI":"10.1038\/s43247-021-00206-7"},{"key":"e_1_3_2_6_2","doi-asserted-by":"publisher","DOI":"10.5194\/acp-23-2393-2023"},{"key":"e_1_3_2_7_2","doi-asserted-by":"publisher","DOI":"10.1029\/2009JD012108"},{"key":"e_1_3_2_8_2","doi-asserted-by":"publisher","DOI":"10.1016\/0022-0248(78)90416-5"},{"key":"e_1_3_2_9_2","doi-asserted-by":"publisher","DOI":"10.2151\/jmsj1965.62.1_190"},{"key":"e_1_3_2_10_2","doi-asserted-by":"publisher","DOI":"10.1175\/JAS-D-19-0303.1"},{"key":"e_1_3_2_11_2","doi-asserted-by":"publisher","DOI":"10.1175\/JAS-D-24-0182.1"},{"key":"e_1_3_2_12_2","doi-asserted-by":"publisher","DOI":"10.1029\/2019MS001689"},{"key":"e_1_3_2_13_2","doi-asserted-by":"crossref","unstructured":"H. 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Skrotzki \u201cHigh-accuracy multiphase humidity measurements using TDLAS: Application to the investigation of ice growth in simulated cirrus clouds \u201d thesis University of Heidelberg (2012)."},{"key":"e_1_3_2_33_2","doi-asserted-by":"publisher","DOI":"10.5194\/amt-7-3177-2014"},{"key":"e_1_3_2_34_2","doi-asserted-by":"publisher","DOI":"10.1063\/1.5139244"},{"key":"e_1_3_2_35_2","doi-asserted-by":"publisher","DOI":"10.1111\/j.2153-3490.1959.tb00003.x"},{"key":"e_1_3_2_36_2","doi-asserted-by":"publisher","DOI":"10.1175\/JCLI-D-23-0250.1"},{"key":"e_1_3_2_37_2","doi-asserted-by":"publisher","DOI":"10.1029\/2025MS005341"},{"key":"e_1_3_2_38_2","doi-asserted-by":"crossref","unstructured":"D. Lamb J. Verlinde Physics and Chemistry of Clouds (Cambridge Univ. Press 2011).","DOI":"10.1017\/CBO9780511976377"},{"key":"e_1_3_2_39_2","doi-asserted-by":"publisher","DOI":"10.1256\/qj.02.04"},{"key":"e_1_3_2_40_2","unstructured":"I. Loshchilov F. 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