{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,10]],"date-time":"2026-10-10T13:19:49Z","timestamp":1791638389376,"version":"4.3.4"},"reference-count":85,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"1","funder":[{"DOI":"10.13039\/100000181","name":"Air Force Office of Scientific Research","doi-asserted-by":"publisher","award":["FA9550-20-1-0358"],"award-info":[{"award-number":["FA9550-20-1-0358"]}],"id":[{"id":"10.13039\/100000181","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000185","name":"Defense Advanced Research Projects Agency","doi-asserted-by":"publisher","award":["HR00112290029"],"award-info":[{"award-number":["HR00112290029"]}],"id":[{"id":"10.13039\/100000185","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000015","name":"U.S. Department of Energy","doi-asserted-by":"publisher","award":["DE-SC0019453"],"award-info":[{"award-number":["DE-SC0019453"]}],"id":[{"id":"10.13039\/100000015","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM Rev."],"published-print":{"date-parts":[[2024,2]]},"abstract":"<jats:p>Uncertainty quantification (UQ) in machine learning is currently drawing increasing research interest, driven by the rapid deployment of deep neural networks across different fields, such as computer vision and natural language processing, and by the need for reliable tools in risk-sensitive applications. Recently, various machine learning models have also been developed to tackle problems in the field of scientific computing with applications to computational science and engineering (CSE). Physics-informed neural networks and deep operator networks are two such models for solving partial differential equations (PDEs) and learning operator mappings, respectively. In this regard, a comprehensive study of UQ methods tailored specifically for scientific machine learning (SciML) models has been provided in [A. F. Psaros et al., J. Comput. Phys., 477 (2023), art. 111902]. Nevertheless, and despite their theoretical merit, implementations of these methods are not straightforward, especially in large-scale CSE applications, hindering their broad adoption in both research and industry settings. In this paper, we present an open-source Python library (\u016dlhttps:\/\/github.com\/Crunch-UQ4MI), termed NeuralUQ and accompanied by an educational tutorial, for employing UQ methods for SciML in a convenient and structured manner. The library, designed for both educational and research purposes, supports multiple modern UQ methods and SciML models. It is based on a succinct workflow and facilitates flexible employment and easy extensions by the users. We first present a tutorial of NeuralUQ and subsequently demonstrate its applicability and efficiency in four diverse examples, involving dynamical systems and high-dimensional parametric and time-dependent PDEs.<\/jats:p>","DOI":"10.1137\/22m1518189","type":"journal-article","created":{"date-parts":[[2024,2,8]],"date-time":"2024-02-08T04:47:48Z","timestamp":1707367668000},"page":"161-190","source":"Crossref","is-referenced-by-count":66,"title":["NeuralUQ: A Comprehensive Library for Uncertainty Quantification in Neural Differential Equations and Operators"],"prefix":"10.1137","volume":"66","author":[{"given":"Zongren","family":"Zou","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Xuhui","family":"Meng","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Apostolos F.","family":"Psaros","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9713-7120","authenticated-orcid":true,"given":"George E.","family":"Karniadakis","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2024,2,8]]},"reference":[{"key":"atypb1","volume-title":"TensorFlow: Large-Scale Machine Learning on Heterogeneous Systems","author":"Abadi M.","year":"2015"},{"key":"atypb2","volume-title":"A Review of Uncertainty Quantification in Deep Learning: Techniques, Applications and Challenges, https:\/\/arxiv.org\/abs\/2011.06225","author":"Abdar M.","year":"2021"},{"key":"atypb3","doi-asserted-by":"publisher","DOI":"10.1007\/s11222-008-9110-y"},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.1029\/2019EA000954"},{"key":"atypb5","doi-asserted-by":"publisher","DOI":"10.1007\/s00332-005-0709-2"},{"key":"atypb6","volume-title":"A Conceptual Introduction to Hamiltonian Monte Carlo, preprint, https:\/\/arxiv.org\/abs\/1701.02434","author":"Betancourt M.","year":"2017"},{"key":"atypb7","first-page":"1613","volume-title":"International Conference on Machine Learning, PMLR","author":"Blundell C.","year":"2015"},{"key":"atypb8","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.1517384113"},{"key":"atypb9","doi-asserted-by":"publisher","DOI":"10.1007\/s10409-021-01148-1"},{"key":"atypb10","doi-asserted-by":"publisher","DOI":"10.1017\/jfm.2021.135"},{"key":"atypb11","doi-asserted-by":"publisher","DOI":"10.1016\/j.arcontrol.2020.10.005"},{"key":"atypb12","doi-asserted-by":"publisher","DOI":"10.21105\/joss.01931"},{"key":"atypb13","first-page":"6571","volume":"31","author":"Chen R. 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