{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,27]],"date-time":"2026-05-27T12:58:45Z","timestamp":1779886725156,"version":"3.53.1"},"reference-count":69,"publisher":"IOP Publishing","issue":"4","license":[{"start":{"date-parts":[[2024,10,29]],"date-time":"2024-10-29T00:00:00Z","timestamp":1730160000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"},{"start":{"date-parts":[[2024,10,29]],"date-time":"2024-10-29T00:00:00Z","timestamp":1730160000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/iopscience.iop.org\/info\/page\/text-and-data-mining"}],"funder":[{"name":"Danmarks Grundforskningsfond","award":["DNRF150"],"award-info":[{"award-number":["DNRF150"]}]},{"DOI":"10.13039\/100008398","name":"Villum Fonden","doi-asserted-by":"crossref","award":["16562"],"award-info":[{"award-number":["16562"]}],"id":[{"id":"10.13039\/100008398","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["iopscience.iop.org"],"crossmark-restriction":false},"short-container-title":["Mach. Learn.: Sci. Technol."],"published-print":{"date-parts":[[2024,12,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>Reliable uncertainty measures are required when using data-based machine learning interatomic potentials (MLIPs) for atomistic simulations. In this work, we propose for sparse Gaussian process regression (GPR) type MLIPs a stochastic uncertainty measure akin to the query-by-committee approach often used in conjunction with neural network based MLIPs. The uncertainty measure is coined <jats:italic>\u2018label noise\u2019<\/jats:italic> ensemble uncertainty as it emerges from adding noise to the energy labels in the training data. We find that this method of calculating an ensemble uncertainty is as well calibrated as the one obtained from the closed-form expression for the posterior variance when the sparse GPR is treated as a projected process. Comparing the two methods, our proposed ensemble uncertainty is, however, faster to evaluate than the closed-form expression. Finally, we demonstrate that the proposed uncertainty measure acts better to support a Bayesian search for optimal structure of Au<jats:sub>20<\/jats:sub> clusters.<\/jats:p>","DOI":"10.1088\/2632-2153\/ad8984","type":"journal-article","created":{"date-parts":[[2024,10,21]],"date-time":"2024-10-21T23:00:51Z","timestamp":1729551651000},"page":"045029","update-policy":"https:\/\/doi.org\/10.1088\/crossmark-policy","source":"Crossref","is-referenced-by-count":5,"title":["Efficient ensemble uncertainty estimation in Gaussian processes regression"],"prefix":"10.1088","volume":"5","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3550-8379","authenticated-orcid":true,"given":"Mads-Peter Verner","family":"Christiansen","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0009-0003-5037-4369","authenticated-orcid":true,"given":"Nikolaj","family":"R\u00f8nne","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7849-6347","authenticated-orcid":true,"given":"Bj\u00f8rk","family":"Hammer","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"266","published-online":{"date-parts":[[2024,10,29]]},"reference":[{"key":"mlstad8984bib1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.98.146401","volume":"98","author":"Behler","year":"2007","journal-title":"Phys. Rev. Lett."},{"key":"mlstad8984bib2","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.104.136403","volume":"104","author":"Bart\u00f3k","year":"2010","journal-title":"Phys. Rev. Lett."},{"key":"mlstad8984bib3","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.95.094203","volume":"95","author":"Deringer","year":"2017","journal-title":"Phys. Rev. B"},{"key":"mlstad8984bib4","doi-asserted-by":"publisher","first-page":"59","DOI":"10.1038\/s41586-020-03072-z","volume":"589","author":"Deringer","year":"2021","journal-title":"Nature"},{"key":"mlstad8984bib5","doi-asserted-by":"publisher","first-page":"5183","DOI":"10.1038\/s41467-022-32294-0","volume":"13","author":"Vandermause","year":"2022","journal-title":"Nat. Commun."},{"key":"mlstad8984bib6","doi-asserted-by":"publisher","first-page":"6924","DOI":"10.1039\/C7SC02267K","volume":"8","author":"Gastegger","year":"2017","journal-title":"Chem. Sci."},{"key":"mlstad8984bib7","doi-asserted-by":"publisher","DOI":"10.1063\/5.0150379","volume":"158","author":"Tang","year":"2023","journal-title":"J. Chem. Phys."},{"key":"mlstad8984bib8","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.97.195424","volume":"97","author":"Kolsbjerg","year":"2018","journal-title":"Phys. Rev. B"},{"key":"mlstad8984bib9","doi-asserted-by":"publisher","DOI":"10.1063\/5.0014876","volume":"153","author":"Paleico","year":"2020","journal-title":"J. Chem. Phys."},{"key":"mlstad8984bib10","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.125.206101","volume":"125","author":"Timmermann","year":"2020","journal-title":"Phys. Rev. Lett."},{"key":"mlstad8984bib11","doi-asserted-by":"publisher","DOI":"10.1088\/1361-651X\/ac4002","volume":"30","author":"Wang","year":"2021","journal-title":"Modelling Simul. Mater. Sci. Eng."},{"key":"mlstad8984bib12","doi-asserted-by":"publisher","DOI":"10.1063\/5.0071249","volume":"155","author":"Timmermann","year":"2021","journal-title":"J. Chem. Phys."},{"key":"mlstad8984bib13","doi-asserted-by":"publisher","DOI":"10.1002\/anie.202204244","volume":"61","author":"Merte","year":"2022","journal-title":"Angew. Chem. Int. Ed."},{"key":"mlstad8984bib14","doi-asserted-by":"publisher","DOI":"10.1021\/acs.jpcc.3c04049","volume":"127","author":"Lee","year":"2023","journal-title":"J. Phys. Chem. C"},{"key":"mlstad8984bib15","doi-asserted-by":"publisher","DOI":"10.1039\/C5NR03903G","volume":"7","author":"Ouyang","year":"2015","journal-title":"Nanoscale"},{"key":"mlstad8984bib16","doi-asserted-by":"publisher","first-page":"31","DOI":"10.1039\/C8FD00055G","volume":"211","author":"Tong","year":"2018","journal-title":"Faraday Discuss."},{"key":"mlstad8984bib17","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1038\/s41524-021-00537-1","volume":"7","author":"Arrigoni","year":"2021","journal-title":"npj Comput. Mater."},{"key":"mlstad8984bib18","doi-asserted-by":"publisher","first-page":"nwad128","DOI":"10.1093\/nsr\/nwad128","volume":"10","author":"Wang","year":"2023","journal-title":"Nat. Sci. Rev."},{"key":"mlstad8984bib19","doi-asserted-by":"publisher","first-page":"507","DOI":"10.1107\/S0108767310026395","volume":"66","author":"Valle","year":"2010","journal-title":"Acta Cryst. A"},{"key":"mlstad8984bib20","doi-asserted-by":"publisher","DOI":"10.1063\/1.3553717","volume":"134","author":"Behler","year":"2011","journal-title":"J. Chem. Phys."},{"key":"mlstad8984bib21","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.108.058301","volume":"108","author":"Rupp","year":"2012","journal-title":"Phys. Rev. Lett."},{"key":"mlstad8984bib22","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.87.184115","volume":"87","author":"Bart\u00f3k","year":"2013","journal-title":"Phys. Rev. B"},{"key":"mlstad8984bib23","doi-asserted-by":"publisher","DOI":"10.1063\/1.5020710","volume":"148","author":"Faber","year":"2018","journal-title":"J. Chem. Phys."},{"key":"mlstad8984bib24","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.99.014104","volume":"99","author":"Drautz","year":"2019","journal-title":"Phys. Rev. B"},{"key":"mlstad8984bib25","doi-asserted-by":"publisher","DOI":"10.1088\/2632-2153\/aca005","volume":"3","author":"Huo","year":"2022","journal-title":"Mach. Learn.: Sci. Technol."},{"key":"mlstad8984bib26","volume":"vol 70","author":"Gilmer","year":"2017"},{"key":"mlstad8984bib27","volume":"vol 30","author":"Sch\u00fctt","year":"2017"},{"key":"mlstad8984bib28","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.120.145301","volume":"120","author":"Xie","year":"2018","journal-title":"Phys. Rev. Lett."},{"key":"mlstad8984bib29","author":"Gasteiger","year":"2019"},{"key":"mlstad8984bib30","article-title":"Tensor field networks: rotation- and translation-equivariant neural networks for 3D point clouds","author":"Thomas","year":"2018"},{"key":"mlstad8984bib31","article-title":"Cormorant: covariant molecular neural networks","author":"Anderson","year":"2019"},{"key":"mlstad8984bib32","first-page":"pp 9377","author":"Sch\u00fctt","year":"2021"},{"key":"mlstad8984bib33","doi-asserted-by":"publisher","first-page":"2453","DOI":"10.1038\/s41467-022-29939-5","volume":"13","author":"Batzner","year":"2022","journal-title":"Nat. Commun."},{"key":"mlstad8984bib34","first-page":"pp 11423","volume":"vol 35","author":"Batatia","year":"2022"},{"key":"mlstad8984bib35","doi-asserted-by":"publisher","first-page":"3678","DOI":"10.1021\/acs.jctc.9b00181","volume":"15","author":"Unke","year":"2019","journal-title":"J. Chem. Theory Comput."},{"key":"mlstad8984bib36","doi-asserted-by":"publisher","first-page":"142","DOI":"10.1140\/epjb\/s10051-021-00156-1","volume":"94","author":"Behler","year":"2021","journal-title":"Eur. Phys. J. B"},{"key":"mlstad8984bib37","doi-asserted-by":"publisher","first-page":"2417","DOI":"10.1021\/acs.jpca.2c06778","volume":"127","author":"Anstine","year":"2023","journal-title":"J. Phys. Chem. A"},{"key":"mlstad8984bib38","doi-asserted-by":"publisher","first-page":"511","DOI":"10.1021\/acs.jpcc.6b10908","volume":"121","author":"Botu","year":"2017","journal-title":"J. Phys. Chem. C"},{"key":"mlstad8984bib39","doi-asserted-by":"publisher","DOI":"10.1039\/C7CP00375G","volume":"19","author":"Peterson","year":"2017","journal-title":"Phys. Chem. Chem. Phys."},{"key":"mlstad8984bib40","doi-asserted-by":"publisher","DOI":"10.1088\/2632-2153\/ab7e1a","volume":"1","author":"Tran","year":"2020","journal-title":"Mach. Learn.: Sci. Technol."},{"key":"mlstad8984bib41","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1038\/s41524-020-00390-8","volume":"6","author":"Wen","year":"2020","journal-title":"npj Comput. Mater."},{"key":"mlstad8984bib42","doi-asserted-by":"publisher","DOI":"10.1088\/2632-2153\/aca7b1","volume":"3","author":"Hu","year":"2022","journal-title":"Mach. Learn.: Sci. Technol."},{"key":"mlstad8984bib43","doi-asserted-by":"publisher","first-page":"121","DOI":"10.1186\/s13321-023-00790-0","volume":"15","author":"Rasmussen","year":"2023","journal-title":"J. Cheminf."},{"key":"mlstad8984bib44","doi-asserted-by":"publisher","DOI":"10.1063\/5.0146905","volume":"158","author":"Carrete","year":"2023","journal-title":"J. Chem. Phys."},{"key":"mlstad8984bib45","article-title":"Coherent energy and force uncertainty in deep learning force fields","author":"J\u00f8rgensen","year":"2023"},{"key":"mlstad8984bib46","doi-asserted-by":"publisher","DOI":"10.1063\/5.0136574","volume":"158","author":"Zhu","year":"2023","journal-title":"J. Chem. Phys."},{"key":"mlstad8984bib47","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1038\/s41524-023-01180-8","volume":"9","author":"Tan","year":"2023","journal-title":"npj Comput. Mater."},{"key":"mlstad8984bib48","doi-asserted-by":"publisher","first-page":"1032","DOI":"10.1002\/qua.24890","volume":"115","author":"Behler","year":"2015","journal-title":"Int. J. Quantum Chem."},{"key":"mlstad8984bib49","doi-asserted-by":"publisher","first-page":"605","DOI":"10.1039\/D2DD00034B","volume":"1","author":"Zaverkin","year":"2022","journal-title":"Digit. Discovery"},{"key":"mlstad8984bib50","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1038\/s41524-023-01104-6","volume":"9","author":"van der Oord","year":"2023","journal-title":"npj Comput. Mater."},{"key":"mlstad8984bib51","doi-asserted-by":"publisher","first-page":"230","DOI":"10.1038\/s43588-023-00406-5","volume":"3","author":"Kulichenko","year":"2023","journal-title":"Nat. Comput. Sci."},{"key":"mlstad8984bib52","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1038\/s41524-024-01254-1","volume":"10","author":"Zaverkin","year":"2024","journal-title":"npj Comput. Mater."},{"key":"mlstad8984bib53","first-page":"pp 1470","author":"Hern\u00e1ndez-Lobato","year":"2017"},{"key":"mlstad8984bib54","doi-asserted-by":"publisher","first-page":"35","DOI":"10.1038\/s41524-019-0175-2","volume":"5","author":"Todorovi\u0107","year":"2019","journal-title":"npj Comput. Mater."},{"key":"mlstad8984bib55","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.124.086102","volume":"124","author":"Bisbo","year":"2020","journal-title":"Phys. Rev. Lett."},{"key":"mlstad8984bib56","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.103.174114","volume":"103","author":"Kaappa","year":"2021","journal-title":"Phys. Rev. B"},{"key":"mlstad8984bib57","doi-asserted-by":"publisher","DOI":"10.1063\/5.0016004","volume":"153","author":"Schran","year":"2020","journal-title":"J. Chem. Phys."},{"key":"mlstad8984bib58","doi-asserted-by":"publisher","first-page":"88","DOI":"10.1021\/acs.jcim.1c01380","volume":"62","author":"Montes-Campos","year":"2022","journal-title":"J. Chem. Inf. Model."},{"key":"mlstad8984bib59","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.105.015311","volume":"105","author":"Kahle","year":"2022","journal-title":"Phys. Rev. E"},{"key":"mlstad8984bib60","doi-asserted-by":"publisher","DOI":"10.1039\/D3CP02143B","volume":"25","author":"Busk","year":"2023","journal-title":"Phys. Chem. Chem. Phys."},{"key":"mlstad8984bib61","doi-asserted-by":"publisher","DOI":"10.1021\/acs.chemrev.1c00022","volume":"121","author":"Deringer","year":"2021","journal-title":"Chem. Rev."},{"key":"mlstad8984bib62","doi-asserted-by":"publisher","DOI":"10.1063\/5.0121748","volume":"157","author":"R\u00f8nne","year":"2022","journal-title":"J. Chem. Phys."},{"key":"mlstad8984bib63","author":"Rasmussen","year":"2006"},{"key":"mlstad8984bib64","first-page":"pp 2796","author":"Kuleshov","year":"2018"},{"key":"mlstad8984bib65","doi-asserted-by":"publisher","first-page":"5540","DOI":"10.3390\/s22155540","volume":"22","author":"Levi","year":"2022","journal-title":"Sensors"},{"key":"mlstad8984bib66","doi-asserted-by":"publisher","first-page":"457","DOI":"10.1007\/s10994-021-05946-3","volume":"110","author":"H\u00fcllermeier","year":"2021","journal-title":"Mach. Learn."},{"key":"mlstad8984bib67","doi-asserted-by":"publisher","first-page":"1513","DOI":"10.1007\/s10462-023-10562-9","volume":"56","author":"Gawlikowski","year":"2023","journal-title":"Artif. Intell. Rev."},{"key":"mlstad8984bib68","doi-asserted-by":"publisher","DOI":"10.1063\/5.0094165","volume":"157","author":"Christiansen","year":"2022","journal-title":"J. Chem. Phys."},{"key":"mlstad8984bib69","doi-asserted-by":"publisher","DOI":"10.1063\/5.0207801","volume":"161","author":"Slavensky","year":"2024","journal-title":"J. Chem. Phys."}],"container-title":["Machine Learning: Science and Technology"],"original-title":[],"link":[{"URL":"https:\/\/iopscience.iop.org\/article\/10.1088\/2632-2153\/ad8984","content-type":"text\/html","content-version":"am","intended-application":"text-mining"},{"URL":"https:\/\/iopscience.iop.org\/article\/10.1088\/2632-2153\/ad8984\/pdf","content-type":"application\/pdf","content-version":"am","intended-application":"text-mining"},{"URL":"https:\/\/iopscience.iop.org\/article\/10.1088\/2632-2153\/ad8984","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/iopscience.iop.org\/article\/10.1088\/2632-2153\/ad8984\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/iopscience.iop.org\/article\/10.1088\/2632-2153\/ad8984\/pdf","content-type":"application\/pdf","content-version":"am","intended-application":"syndication"},{"URL":"https:\/\/iopscience.iop.org\/article\/10.1088\/2632-2153\/ad8984\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https:\/\/iopscience.iop.org\/article\/10.1088\/2632-2153\/ad8984\/pdf","content-type":"application\/pdf","content-version":"am","intended-application":"similarity-checking"},{"URL":"https:\/\/iopscience.iop.org\/article\/10.1088\/2632-2153\/ad8984\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,10,29]],"date-time":"2024-10-29T08:56:56Z","timestamp":1730192216000},"score":1,"resource":{"primary":{"URL":"https:\/\/iopscience.iop.org\/article\/10.1088\/2632-2153\/ad8984"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,10,29]]},"references-count":69,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2024,10,29]]},"published-print":{"date-parts":[[2024,12,1]]}},"URL":"https:\/\/doi.org\/10.1088\/2632-2153\/ad8984","relation":{},"ISSN":["2632-2153"],"issn-type":[{"value":"2632-2153","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,10,29]]},"assertion":[{"value":"Efficient ensemble uncertainty estimation in Gaussian processes regression","name":"article_title","label":"Article Title"},{"value":"Machine Learning: Science and Technology","name":"journal_title","label":"Journal Title"},{"value":"paper","name":"article_type","label":"Article Type"},{"value":"\u00a9 2024 The Author(s). Published by IOP Publishing Ltd","name":"copyright_information","label":"Copyright Information"},{"value":"2024-07-17","name":"date_received","label":"Date Received","group":{"name":"publication_dates","label":"Publication dates"}},{"value":"2024-10-21","name":"date_accepted","label":"Date Accepted","group":{"name":"publication_dates","label":"Publication dates"}},{"value":"2024-10-29","name":"date_epub","label":"Online publication date","group":{"name":"publication_dates","label":"Publication dates"}}]}}