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Instead, through minimizing our proposed local squared\n                    <jats:italic>W<\/jats:italic>\n                    <jats:sub>2<\/jats:sub>\n                    loss function, linear regression models or neural networks can be directly trained to efficiently reconstruct the distributions of the output associated with different inputs based on empirical distributions of observation data. We demonstrate the effectiveness of our proposed method across several uncertainty quantification tasks, including linear regression with coefficient uncertainty, training neural networks with weight uncertainty, and reconstructing ordinary differential equations with a latent random variable.\n                  <\/jats:p>","DOI":"10.1088\/2632-2153\/ae5c59","type":"journal-article","created":{"date-parts":[[2026,4,7]],"date-time":"2026-04-07T22:53:43Z","timestamp":1775602423000},"page":"035001","update-policy":"https:\/\/doi.org\/10.1088\/crossmark-policy","source":"Crossref","is-referenced-by-count":0,"title":["A local squared Wasserstein-2 method for efficient reconstruction of models with uncertainty"],"prefix":"10.1088","volume":"7","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2116-4712","authenticated-orcid":true,"given":"Mingtao","family":"Xia","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Qijing","family":"Shen","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"266","published-online":{"date-parts":[[2026,4,24]]},"reference":[{"key":"mlstae5c59bib1","author":"Fuller","year":"2009","type":"book"},{"key":"mlstae5c59bib2","volume":"vol 105","author":"Carroll","year":"1995","type":"book"},{"key":"mlstae5c59bib3","doi-asserted-by":"publisher","first-page":"33","DOI":"10.1111\/j.1468-0262.2004.00477.x","type":"journal-article","article-title":"Estimation of nonlinear models with measurement error","volume":"72","author":"Schennach","year":"2004","journal-title":"Econometrica"},{"key":"mlstae5c59bib4","first-page":"pp 371","type":"book","article-title":"Latent variable models","author":"Bishop","year":"1998"},{"key":"mlstae5c59bib5","doi-asserted-by":"publisher","DOI":"10.1016\/j.chaos.2020.110026","type":"journal-article","article-title":"Parameter estimation of uncertain differential equation with application to financial market","volume":"139","author":"Yang","year":"2020","journal-title":"Chaos Solitons Fractals"},{"key":"mlstae5c59bib6","doi-asserted-by":"publisher","first-page":"8264","DOI":"10.1109\/ACCESS.2016.2629987","type":"journal-article","article-title":"Heterogeneous multi-view information fusion: review of 3-D reconstruction methods and a new registration with uncertainty modeling","volume":"4","author":"Hadi Aliakbarpour","year":"2016","journal-title":"IEEE Access"},{"key":"mlstae5c59bib7","doi-asserted-by":"publisher","first-page":"666","DOI":"10.1038\/ng.3835","type":"journal-article","article-title":"Genome-to-genome analysis highlights the effect of the human innate and adaptive immune systems on the hepatitis C virus","volume":"49","author":"Ansari","year":"2017","journal-title":"Nat. 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