{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,17]],"date-time":"2026-08-17T16:53:15Z","timestamp":1786985595986,"version":"build-2736575974"},"reference-count":38,"publisher":"Elsevier BV","license":[{"start":{"date-parts":[[2021,8,1]],"date-time":"2021-08-01T00:00:00Z","timestamp":1627776000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.elsevier.com\/tdm\/userlicense\/1.0\/"},{"start":{"date-parts":[[2021,8,1]],"date-time":"2021-08-01T00:00:00Z","timestamp":1627776000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.elsevier.com\/legal\/tdmrep-license"},{"start":{"date-parts":[[2025,8,1]],"date-time":"2025-08-01T00:00:00Z","timestamp":1754006400000},"content-version":"vor","delay-in-days":1461,"URL":"http:\/\/www.elsevier.com\/open-access\/userlicense\/1.0\/"},{"start":{"date-parts":[[2021,8,1]],"date-time":"2021-08-01T00:00:00Z","timestamp":1627776000000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-017"},{"start":{"date-parts":[[2021,8,1]],"date-time":"2021-08-01T00:00:00Z","timestamp":1627776000000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-037"},{"start":{"date-parts":[[2021,8,1]],"date-time":"2021-08-01T00:00:00Z","timestamp":1627776000000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-012"},{"start":{"date-parts":[[2021,8,1]],"date-time":"2021-08-01T00:00:00Z","timestamp":1627776000000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-029"},{"start":{"date-parts":[[2021,8,1]],"date-time":"2021-08-01T00:00:00Z","timestamp":1627776000000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-004"}],"funder":[{"DOI":"10.13039\/501100000266","name":"Engineering and Physical Sciences Research Council","doi-asserted-by":"publisher","award":["EP\/T005157\/1"],"award-info":[{"award-number":["EP\/T005157\/1"]}],"id":[{"id":"10.13039\/501100000266","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100002850","name":"Fondo Nacional de Desarrollo Cient\u00edfico y Tecnol\u00f3gico","doi-asserted-by":"publisher","award":["1160774"],"award-info":[{"award-number":["1160774"]}],"id":[{"id":"10.13039\/501100002850","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100010665","name":"H2020 Marie Sk\u0142odowska-Curie Actions","doi-asserted-by":"publisher","award":["777778"],"award-info":[{"award-number":["777778"]}],"id":[{"id":"10.13039\/100010665","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100010661","name":"Horizon 2020 Framework Programme","doi-asserted-by":"publisher","id":[{"id":"10.13039\/100010661","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100007776","name":"Pontificia Universidad Cat\u00f3lica de Valpara\u00edso","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100007776","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100007601","name":"Horizon 2020","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100007601","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["elsevier.com","sciencedirect.com"],"crossmark-restriction":true},"short-container-title":["Computers &amp; Mathematics with Applications"],"published-print":{"date-parts":[[2021,8]]},"DOI":"10.1016\/j.camwa.2020.08.012","type":"journal-article","created":{"date-parts":[[2020,9,9]],"date-time":"2020-09-09T08:25:33Z","timestamp":1599639933000},"page":"186-199","update-policy":"https:\/\/doi.org\/10.1016\/elsevier_cm_policy","source":"Crossref","is-referenced-by-count":25,"special_numbering":"C","title":["A machine-learning minimal-residual (ML-MRes) framework for goal-oriented finite element discretizations"],"prefix":"10.1016","volume":"95","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1620-019X","authenticated-orcid":false,"given":"Ignacio","family":"Brevis","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4430-5167","authenticated-orcid":false,"given":"Ignacio","family":"Muga","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6830-8031","authenticated-orcid":false,"given":"Kristoffer G.","family":"van der Zee","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"78","reference":[{"issue":"1","key":"10.1016\/j.camwa.2020.08.012_b1","doi-asserted-by":"crossref","first-page":"118","DOI":"10.3934\/Mine.2018.1.118","article-title":"A machine learning framework for data driven acceleration of computations of differential equations","volume":"1","author":"Mishra","year":"2018","journal-title":"Math. Eng."},{"key":"10.1016\/j.camwa.2020.08.012_b2","series-title":"Recent Developments in Discontinuous Galerkin Finite Element Methods for Partial Differential Equations: 2012 John H Barrett Memorial Lectures","first-page":"149","article-title":"An overview of the discontinuous Petrov Galerkin method","volume":"vol. 157","author":"Demkowicz","year":"2014"},{"key":"10.1016\/j.camwa.2020.08.012_b3","series-title":"Discretization of linear problems in banach spaces: Residual minimization, nonlinear petrov\u2013galerkin, and monotone mixed methods","author":"Muga","year":"2018"},{"key":"10.1016\/j.camwa.2020.08.012_b4","doi-asserted-by":"crossref","first-page":"303","DOI":"10.1007\/BF02551274","article-title":"Approximation by superpositions of a sigmoidal function","volume":"2","author":"Cybenko","year":"1989","journal-title":"Math. Control Signals Systems"},{"key":"10.1016\/j.camwa.2020.08.012_b5","series-title":"Deep Learning","author":"Goodfellow","year":"2019"},{"key":"10.1016\/j.camwa.2020.08.012_b6","doi-asserted-by":"crossref","first-page":"436","DOI":"10.1038\/nature14539","article-title":"Deep learning","volume":"521","author":"LeCun","year":"2015","journal-title":"Nature"},{"issue":"4","key":"10.1016\/j.camwa.2020.08.012_b7","doi-asserted-by":"crossref","first-page":"860","DOI":"10.1137\/18M1165748","article-title":"Deep learning: An introduction for applied mathematicians","volume":"61","author":"Higham","year":"2019","journal-title":"SIAM Rev."},{"issue":"4","key":"10.1016\/j.camwa.2020.08.012_b8","doi-asserted-by":"crossref","DOI":"10.1126\/sciadv.1602614","article-title":"Data-driven discovery of partial differential equations","volume":"3","author":"Rudy","year":"2017","journal-title":"Sci. Adv."},{"key":"10.1016\/j.camwa.2020.08.012_b9","doi-asserted-by":"crossref","first-page":"239","DOI":"10.1016\/j.jcp.2019.01.036","article-title":"Data-driven discovery of PDEs in complex datasets","volume":"384","author":"Berg","year":"2019","journal-title":"J. Comput. Phys."},{"key":"10.1016\/j.camwa.2020.08.012_b10","doi-asserted-by":"crossref","first-page":"125","DOI":"10.1016\/j.jcp.2017.11.039","article-title":"Hidden physics models: Machine learning of nonlinear partial differential equations","volume":"357","author":"Raissi","year":"2018","journal-title":"J. Comput. Phys."},{"key":"10.1016\/j.camwa.2020.08.012_b11","doi-asserted-by":"crossref","first-page":"28","DOI":"10.1016\/j.neucom.2018.06.056","article-title":"A unified deep artificial neural network approach to partial differential equations in complex geometries","volume":"317","author":"Berg","year":"2018","journal-title":"Neurocomputing"},{"issue":"5","key":"10.1016\/j.camwa.2020.08.012_b12","doi-asserted-by":"crossref","first-page":"987","DOI":"10.1109\/72.712178","article-title":"Artificial neural networks for solving ordinary and partial differential equations","volume":"9","author":"Lagaris","year":"1998","journal-title":"IEEE Trans. Neural Netw."},{"issue":"1","key":"10.1016\/j.camwa.2020.08.012_b13","doi-asserted-by":"crossref","first-page":"110","DOI":"10.1016\/0021-9991(90)90007-N","article-title":"Neural algorithm for solving differential equations","volume":"91","author":"Lee","year":"1990","journal-title":"J. Comput. Phys."},{"key":"10.1016\/j.camwa.2020.08.012_b14","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s40304-018-0127-z","article-title":"The deep Ritz method: A deep learning-based numerical algorithm for solving variational problems","volume":"6","author":"E","year":"2018","journal-title":"Commun. Math. Stat."},{"issue":"3","key":"10.1016\/j.camwa.2020.08.012_b15","doi-asserted-by":"crossref","first-page":"A984","DOI":"10.1137\/130914619","article-title":"Data-driven parametrized model reduction in the Loewner framework","volume":"36","author":"Ionita","year":"2014","journal-title":"SIAM J. Sci. Comput."},{"key":"10.1016\/j.camwa.2020.08.012_b16","doi-asserted-by":"crossref","first-page":"55","DOI":"10.1016\/j.jcp.2018.02.037","article-title":"Non-intrusive reduced order modeling of nonlinear problems using neural networks","volume":"363","author":"Hesthaven","year":"2018","journal-title":"J. Comput. Phys."},{"key":"10.1016\/j.camwa.2020.08.012_b17","doi-asserted-by":"crossref","DOI":"10.1016\/j.jcp.2019.07.050","article-title":"Machine learning for fast and reliable solution of time-dependent differential equations","volume":"397","author":"Regazzoni","year":"2019","journal-title":"J. Comput. Phys."},{"key":"10.1016\/j.camwa.2020.08.012_b18","doi-asserted-by":"crossref","first-page":"704","DOI":"10.1016\/j.compfluid.2018.07.021","article-title":"Projection-based model reduction: Formulations for physics-based machine learning","volume":"179","author":"Swischuka","year":"2019","journal-title":"Comput. Fluids"},{"key":"10.1016\/j.camwa.2020.08.012_b19","series-title":"A theoretical analysis of deep neural networks and parametric PDEs","author":"Kutyniok","year":"2020"},{"key":"10.1016\/j.camwa.2020.08.012_b20","doi-asserted-by":"crossref","first-page":"155","DOI":"10.1017\/jfm.2016.615","article-title":"Reynolds averaged turbulence modelling using deep neural networks with embedded invariance","volume":"807","author":"Ling","year":"2016","journal-title":"J. Fluid Mech."},{"key":"10.1016\/j.camwa.2020.08.012_b21","doi-asserted-by":"crossref","first-page":"166","DOI":"10.1016\/j.jcp.2018.04.029","article-title":"An artificial neural network as a troubled-cell indicator","volume":"367","author":"Ray","year":"2018","journal-title":"J. Comput. Phys."},{"key":"10.1016\/j.camwa.2020.08.012_b22","doi-asserted-by":"crossref","DOI":"10.1016\/j.jcp.2020.109304","article-title":"Controlling oscillations in high-order Discontinuous Galerkin schemes using artificial viscosity tuned by neural networks","volume":"409","author":"Discacciati","year":"2020","journal-title":"J. Comput. Phys."},{"key":"10.1016\/j.camwa.2020.08.012_b23","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1017\/S0962492901000010","article-title":"An optimal control approach to a posteriori error estimation in finite element methods","volume":"10","author":"Becker","year":"2001","journal-title":"Acta Numer."},{"issue":"5\u20136","key":"10.1016\/j.camwa.2020.08.012_b24","doi-asserted-by":"crossref","first-page":"735","DOI":"10.1016\/S0898-1221(00)00317-5","article-title":"Goal-oriented error estimation and adaptivity for the finite element method","volume":"41","author":"Oden","year":"2001","journal-title":"Comput. Math. Appl."},{"issue":"2","key":"10.1016\/j.camwa.2020.08.012_b25","doi-asserted-by":"crossref","first-page":"861","DOI":"10.1137\/060675666","article-title":"A goal-oriented adaptive finite element method with convergence rates","volume":"47","author":"Mommer","year":"2009","journal-title":"SIAM J. Numer. Anal."},{"issue":"3","key":"10.1016\/j.camwa.2020.08.012_b26","doi-asserted-by":"crossref","first-page":"1423","DOI":"10.1137\/15M1021982","article-title":"An abstract analysis of optimal goal-oriented adaptivity","volume":"54","author":"Feischl","year":"2016","journal-title":"SIAM J. Numer. Anal."},{"key":"10.1016\/j.camwa.2020.08.012_b27","doi-asserted-by":"crossref","first-page":"256","DOI":"10.1016\/j.cma.2017.09.018","article-title":"A new goal-oriented formulation of the finite element method","volume":"327","author":"Kergrene","year":"2017","journal-title":"Comput. Methods Appl. Mech. Engrg."},{"key":"10.1016\/j.camwa.2020.08.012_b28","doi-asserted-by":"crossref","first-page":"723","DOI":"10.1016\/j.cma.2016.10.007","article-title":"Worst-case multi-objective error estimation and adaptivity","volume":"313","author":"van Brummelen","year":"2017","journal-title":"Comput. Methods Appl. Mech. Engrg."},{"issue":"4","key":"10.1016\/j.camwa.2020.08.012_b29","doi-asserted-by":"crossref","first-page":"575","DOI":"10.1515\/cmam-2017-0001","article-title":"A partition-of-unity dual-weighted residual approach for multi-objective goal functional error estimation applied to elliptic problems","volume":"17","author":"Endtmeyer","year":"2017","journal-title":"Comput. Methods Appl. Math."},{"issue":"1","key":"10.1016\/j.camwa.2020.08.012_b30","doi-asserted-by":"crossref","first-page":"1","DOI":"10.2140\/camcos.2018.13.1","article-title":"Adaptively weighted least squares finite element methods for partial differential equations with singularities","volume":"13","author":"Hayhurst","year":"2018","journal-title":"Comm. Appl. Math. Comput. Sci."},{"key":"10.1016\/j.camwa.2020.08.012_b31","doi-asserted-by":"crossref","first-page":"70","DOI":"10.1002\/num.20640","article-title":"A class of discontinuous Petrov-Galerkin methods. II. optimal test functions","volume":"27","author":"Demkowicz","year":"2010","journal-title":"Numer. Methods Partial Differential Equations"},{"issue":"11","key":"10.1016\/j.camwa.2020.08.012_b32","doi-asserted-by":"crossref","first-page":"1605","DOI":"10.1016\/j.camwa.2014.06.019","article-title":"A robust Petrov\u2013Galerkin discretisation of convection\u2013diffusion equations","volume":"68","author":"Broersen","year":"2014","journal-title":"Comput. Math. Appl."},{"key":"10.1016\/j.camwa.2020.08.012_b33","series-title":"Source Separation and Machine Learning","author":"Chien","year":"2019"},{"key":"10.1016\/j.camwa.2020.08.012_b34","series-title":"Machine learning paradigms: Advances in data analytics","author":"Tsihrintzis","year":"2019"},{"key":"10.1016\/j.camwa.2020.08.012_b35","series-title":"Optimization for Machine Learning","author":"Sra","year":"2011"},{"key":"10.1016\/j.camwa.2020.08.012_b36","series-title":"Theory and Practice of Finite Elements","author":"Ern","year":"2004"},{"key":"10.1016\/j.camwa.2020.08.012_b37","doi-asserted-by":"crossref","first-page":"537","DOI":"10.1090\/S0025-5718-2013-02721-4","article-title":"An analysis of the practical DPG method","volume":"83","author":"Gopalakrishnan","year":"2014","journal-title":"Math. Comp."},{"key":"10.1016\/j.camwa.2020.08.012_b38","doi-asserted-by":"crossref","first-page":"22","DOI":"10.1007\/BF01386205","article-title":"Estimation of iterated matrices with application to von Neumann condition","volume":"2","author":"Kato","year":"1960","journal-title":"Numer. Math."}],"container-title":["Computers &amp; Mathematics with Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.elsevier.com\/content\/article\/PII:S0898122120303199?httpAccept=text\/xml","content-type":"text\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/api.elsevier.com\/content\/article\/PII:S0898122120303199?httpAccept=text\/plain","content-type":"text\/plain","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2026,5,11]],"date-time":"2026-05-11T14:13:54Z","timestamp":1778508834000},"score":1,"resource":{"primary":{"URL":"https:\/\/linkinghub.elsevier.com\/retrieve\/pii\/S0898122120303199"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,8]]},"references-count":38,"alternative-id":["S0898122120303199"],"URL":"https:\/\/doi.org\/10.1016\/j.camwa.2020.08.012","relation":{},"ISSN":["0898-1221"],"issn-type":[{"value":"0898-1221","type":"print"}],"subject":[],"published":{"date-parts":[[2021,8]]},"assertion":[{"value":"Elsevier","name":"publisher","label":"This article is maintained by"},{"value":"A machine-learning minimal-residual (ML-MRes) framework for goal-oriented finite element discretizations","name":"articletitle","label":"Article Title"},{"value":"Computers & Mathematics with Applications","name":"journaltitle","label":"Journal Title"},{"value":"https:\/\/doi.org\/10.1016\/j.camwa.2020.08.012","name":"articlelink","label":"CrossRef DOI link to publisher maintained version"},{"value":"article","name":"content_type","label":"Content Type"},{"value":"\u00a9 2020 Elsevier Ltd.","name":"copyright","label":"Copyright"}]}}