{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,15]],"date-time":"2026-10-15T03:51:04Z","timestamp":1792036264506,"version":"4.4.0"},"reference-count":80,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"4","funder":[{"name":"DOD MURI","award":["FA9550-21-1-0084"],"award-info":[{"award-number":["FA9550-21-1-0084"]}]},{"name":"Texas Advanced Computing Center (TACC) at The University of Texas at Austin"},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DMS 2012453"],"award-info":[{"award-number":["DMS 2012453"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000015","name":"U.S. Department of Energy","doi-asserted-by":"publisher","award":["DE-SC0019303"],"award-info":[{"award-number":["DE-SC0019303"]}],"id":[{"id":"10.13039\/100000015","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000015","name":"U.S. Department of Energy","doi-asserted-by":"publisher","award":["DE-SC0021239"],"award-info":[{"award-number":["DE-SC0021239"]}],"id":[{"id":"10.13039\/100000015","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000015","name":"U.S. Department of Energy","doi-asserted-by":"publisher","award":["DE-SC0023171"],"award-info":[{"award-number":["DE-SC0023171"]}],"id":[{"id":"10.13039\/100000015","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Sci. Comput."],"published-print":{"date-parts":[[2025,8,31]]},"abstract":"<jats:p>Abstract.<\/jats:p>\n                  <jats:p>We propose a novel machine learning framework for solving optimization problems governed by large-scale partial differential equations (PDEs) with high-dimensional random parameters. Such optimization under uncertainty (OUU) problems may be computationally prohibitive using classical methods, as the estimation of risk measures may require a large number of solutions of an expensive-to-solve PDE at every iteration of an optimization algorithm. To address this challenge, we propose the multi-input reduced basis derivative-informed neural operator (MR-DINO), as an approximation to the PDE solution operator. The MR-DINO has the combined merits of (1) being able to accurately approximate not only the mapping from the joint inputs of random parameters and optimization variables to the PDE state, but also its derivative with respect to the optimization variables, (2) using a reduced basis architecture that can be efficiently constructed and is scalable to high-dimensional OUU problems, and (3) requiring only a limited amount of training data to achieve high accuracy for both the PDE solution and the OUU solution. We demonstrate the capability of MR-DINOs through several numerical experiments, including the risk-averse control of a semilinear elliptic PDE and the steady-state Navier\u2013Stokes equations in two and three spatial dimensions. Across the examples, MR-DINOs offer [Formula: see text]\u2013[Formula: see text] reductions in execution times and produce OUU solutions of comparable accuracies to PDE-based OUU solutions using standard quasi-Newton methods (BFGS\/L-BFGS-B) while being one to two orders of magnitude more cost-efficient after accounting for construction costs. Moreover, the construction cost of MR-DINOs can be amortized over many optimization runs with differing objectives and constraints.<\/jats:p>","DOI":"10.1137\/23m157956x","type":"journal-article","created":{"date-parts":[[2025,7,24]],"date-time":"2025-07-24T07:58:20Z","timestamp":1753343900000},"page":"C899-C931","source":"Crossref","is-referenced-by-count":10,"title":["Efficient PDE-Constrained Optimization Under High-Dimensional Uncertainty Using Derivative-Informed Neural Operators"],"prefix":"10.1137","volume":"47","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0466-6902","authenticated-orcid":true,"given":"Dingcheng","family":"Luo","sequence":"first","affiliation":[{"name":"Oden Institute for Computational Engineering and Sciences, The University of Texas at Austin, Austin, TX 78712 USA."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Thomas","family":"O\u2019Leary-Roseberry","sequence":"additional","affiliation":[{"name":"Oden Institute for Computational Engineering and Sciences, The University of Texas at Austin, Austin, TX 78712 USA."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Peng","family":"Chen","sequence":"additional","affiliation":[{"name":"School of Computational Science and Engineering, Georgia Institute of Technology, Atlanta, GA 30332 USA."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Omar","family":"Ghattas","sequence":"additional","affiliation":[{"name":"Oden Institute for Computational Engineering and Sciences, The University of Texas at Austin, Austin, TX 78712 USA, and Walker Department of Mechanical Engineering, The University of Texas at Austin, Austin, TX 78712 USA."}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2025,7,24]]},"reference":[{"key":"ref1","doi-asserted-by":"publisher","DOI":"10.1137\/16M106306X"},{"key":"ref2","doi-asserted-by":"publisher","DOI":"10.1137\/16M109870X"},{"key":"ref3","doi-asserted-by":"publisher","DOI":"10.1007\/s10444-018-9653-1"},{"key":"ref4","doi-asserted-by":"publisher","DOI":"10.1137\/S0895479899358194"},{"key":"ref5","doi-asserted-by":"publisher","DOI":"10.5802\/smai-jcm.74"},{"key":"ref6","doi-asserted-by":"publisher","DOI":"10.1137\/100795772"},{"key":"ref7","doi-asserted-by":"publisher","DOI":"10.1007\/s00791-010-0134-4"},{"key":"ref8","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2023.112104"},{"key":"ref9","doi-asserted-by":"publisher","DOI":"10.2514\/1.J060539"},{"key":"ref10","doi-asserted-by":"publisher","DOI":"10.1137\/20M1381381"},{"key":"ref11","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2021.110114"},{"key":"ref12","doi-asserted-by":"publisher","DOI":"10.1137\/130940517"},{"key":"ref13","doi-asserted-by":"publisher","DOI":"10.1007\/s00211-015-0743-4"},{"key":"ref14","doi-asserted-by":"publisher","DOI":"10.1137\/151004550"},{"key":"ref15","doi-asserted-by":"crossref","unstructured":"P. Chen and J. O. Royset, Performance Bounds for PDE-Constrained Optimization Under Uncertainty, SIAM J. Optim., 33 (2023), pp. 1828\u20131854, https:\/\/doi.org\/10.1137\/21M1457916.","DOI":"10.1137\/21M1457916"},{"key":"ref16","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2015.08.006"},{"key":"ref17","doi-asserted-by":"crossref","unstructured":"P. Chen and C. Schwab, Model order reduction methods in computational uncertainty quantification, in Handbook of Uncertainty Quantification, Springer, Cham, 2016, pp. 1\u201353.","DOI":"10.1007\/978-3-319-11259-6_70-1"},{"key":"ref18","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2016.02.055"},{"key":"ref19","doi-asserted-by":"crossref","unstructured":"P. Chen, U. Villa, and O. Ghattas, Taylor approximation and variance reduction for PDE-constrained optimal control under uncertainty, J. Comput. Phys., 385 (2019), 163\u2013186, https:\/\/doi.org\/10.1016\/j.jcp.2019.01.047.","DOI":"10.1016\/j.jcp.2019.01.047"},{"key":"ref20","doi-asserted-by":"publisher","DOI":"10.1017\/S0962492915000033"},{"key":"ref21","doi-asserted-by":"crossref","unstructured":"Y. Daon and G. Stadler, Mitigating the influence of boundary conditions on covariance operators derived from elliptic PDEs, Inverse Probl. Imaging, 12 (2018), pp. 1083\u20131102, https:\/\/doi.org\/10.3934\/ipi.2018045, https:\/\/arxiv.org\/abs\/1610.05280.","DOI":"10.3934\/ipi.2018045"},{"key":"ref22","doi-asserted-by":"publisher","DOI":"10.1145\/992200.992206"},{"key":"ref23","doi-asserted-by":"publisher","DOI":"10.4208\/jml.220509"},{"key":"ref24","volume-title":"Proceedings of AIAA SCITECH 2023","author":"Du X."},{"key":"ref25","doi-asserted-by":"publisher","DOI":"10.3390\/a15070241"},{"key":"ref26","unstructured":"TensorFlow: Large-Scale Machine Learning on Heterogeneous Systems, 2015, https:\/\/www.tensorflow.org\/."},{"key":"ref27","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2021.114181"},{"key":"ref28","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2023.112523"},{"key":"ref29","doi-asserted-by":"publisher","DOI":"10.1137\/23M1553170"},{"key":"ref30","unstructured":"Z. Hao, C. Ying, H. Su, J. Zhu, J. Song, and Z. Cheng, Bi-level physics-informed neural networks for PDE constrained optimization using Broyden\u2019s hypergradients, in Proceedings of the 11th International\u00a0 Conference on Learning Representations, 2023, https:\/\/arxiv.org\/abs\/2209.07075."},{"key":"ref31","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2018.02.037"},{"key":"ref32","volume-title":"Math. Model. Theory Appl.","volume":"23","author":"Hinze M.","year":"2009"},{"key":"ref33","doi-asserted-by":"publisher","DOI":"10.1016\/0045-7825(89)90111-4"},{"key":"ref34","doi-asserted-by":"crossref","unstructured":"R. Hwang, J. Y. Lee, J. Y. Shin, and H. J. Hwang, Solving PDE-constrained control problems using operator learning, in Proceedings of the AAAI Conference on Artificial Intelligence, Vol. 36, 2022, pp. 4504\u20134512.","DOI":"10.1609\/aaai.v36i4.20373"},{"key":"ref35","doi-asserted-by":"publisher","DOI":"10.1137\/22M1477751"},{"key":"ref36","doi-asserted-by":"publisher","DOI":"10.1007\/s10444-022-09981-z"},{"key":"ref37","unstructured":"D. P. Kingma and J. Ba, Adam: A method for stochastic optimization, in Proceedings of the 3rd International Conference on Learning Representations, 2015, https:\/\/arxiv.org\/abs\/1412.6980."},{"key":"ref38","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2022.115371"},{"key":"ref39","doi-asserted-by":"publisher","DOI":"10.1137\/120892362"},{"key":"ref40","doi-asserted-by":"publisher","DOI":"10.1137\/140954556"},{"key":"ref41","first-page":"1","volume":"24","author":"Kovachki N.","year":"2023","journal-title":"J. Mach. Learn. Res."},{"key":"ref42","doi-asserted-by":"publisher","DOI":"10.1137\/15M1041390"},{"key":"ref43","doi-asserted-by":"publisher","DOI":"10.1137\/16M108269X"},{"key":"ref44","doi-asserted-by":"publisher","DOI":"10.1007\/s00158-022-03477-6"},{"key":"ref45","volume-title":"Multipole graph neural operator for parametric partial differential equations","author":"Li Z.","year":"2020"},{"key":"ref46","unstructured":"Z. Li, N. Kovachki, K. Azizzadenesheli, B. Liu, K. Bhattacharya, A. Stuart, and A. Anandkumar, Fourier neural operator for parametric partial differential equations, in Proceedings of the International Conference on Learning Representations, 2021."},{"key":"ref47","first-page":"1","volume":"1","author":"Li Z.","year":"2024","journal-title":"ACM\/IMS J. Data Sci."},{"key":"ref48","doi-asserted-by":"publisher","DOI":"10.1111\/j.1467-9868.2011.00777.x"},{"key":"ref49","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-23099-8"},{"key":"ref50","doi-asserted-by":"publisher","DOI":"10.1038\/s42256-021-00302-5"},{"key":"ref51","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2022.114778"},{"key":"ref52","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevResearch.4.023210"},{"key":"ref53","doi-asserted-by":"publisher","DOI":"10.1137\/21M1397908"},{"key":"ref54","doi-asserted-by":"publisher","DOI":"10.1023\/A:1015145924517"},{"key":"ref55","doi-asserted-by":"publisher","DOI":"10.4310\/AMSA.2016.v1.n2.a4"},{"key":"ref56","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-030-77226-0"},{"key":"ref57","doi-asserted-by":"publisher","DOI":"10.1051\/m2an\/2021025"},{"key":"ref58","doi-asserted-by":"publisher","DOI":"10.1137\/18M1224076"},{"key":"ref59","doi-asserted-by":"publisher","DOI":"10.1137\/20M133957X"},{"key":"ref60","doi-asserted-by":"publisher","DOI":"10.1002\/nme.4761"},{"key":"ref61","doi-asserted-by":"publisher","DOI":"10.1002\/nla.2472"},{"key":"ref62","volume-title":"Numerical Optimization","author":"Nocedal J.","year":"2006","edition":"2"},{"key":"ref63","unstructured":"M. Ohlberger and S. Rave, Reduced basis methods: Success, limitations and future challenges, in Proceedings of the Conference ALGORITMY 2016, 2016, pp. 1\u201312, http:\/\/www.iam.fmph.uniba.sk\/amuc\/ojs\/index.php\/algoritmy\/article\/view\/389."},{"key":"ref64","doi-asserted-by":"crossref","unstructured":"T. O\u2019Leary-Roseberry, P. Chen, U. Villa, and O. Ghattas, Derivate informed neural operator: An efficient framework for high-dimensional parametric derivative learning, J. Comput. Phys., 496 (2024), 112555, http:\/\/arxiv.org\/abs\/2206.10745.","DOI":"10.1016\/j.jcp.2023.112555"},{"key":"ref65","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2022.115730"},{"key":"ref66","author":"O\u2019Leary-Roseberry T.","year":"2021","journal-title":"hIPPYflow: Dimension Reduced Surrogate Construction for Parametric PDE Maps in Python"},{"key":"ref67","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2021.114199"},{"key":"ref68","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2018.10.045"},{"key":"ref69","doi-asserted-by":"publisher","DOI":"10.1016\/j.ress.2010.01.001"},{"key":"ref70","doi-asserted-by":"publisher","DOI":"10.21314\/JOR.2000.038"},{"key":"ref71","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611976595"},{"key":"ref72","doi-asserted-by":"publisher","DOI":"10.1016\/j.engappai.2023.107615"},{"key":"ref73","doi-asserted-by":"publisher","DOI":"10.1145\/3311790.3396656"},{"key":"ref74","unstructured":"U. Villa and T. O\u2019Leary-Roseberry, A Note on the Relationship Between PDE-Based Precision Operators and Mat\u00e9rn Covariances, 2024, https:\/\/arxiv.org\/abs\/2407.00471."},{"key":"ref75","doi-asserted-by":"publisher","DOI":"10.1145\/3428447"},{"key":"ref76","unstructured":"S. Wang, M. A. Bhouri, and P. Perdikaris, Fast PDE-Constrained Optimization Via Self-Supervised Operator Learning, preprint, arXiv:2110.13297, 2021."},{"key":"ref77","doi-asserted-by":"publisher","DOI":"10.1007\/s10915-023-02145-1"},{"key":"ref78","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2022.114823"},{"key":"ref79","doi-asserted-by":"publisher","DOI":"10.1137\/18M1220996"},{"key":"ref80","unstructured":"Q. Zhao, D. B. Lindell, and G. Wetzstein, Learning to solve PDE-constrained inverse problems with graph networks,\u00a0in Proceedings of the 39th International Conference on Machine Learning, 2022, https:\/\/arxiv.org\/abs\/2206.00711."}],"container-title":["SIAM Journal on Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/23M157956X","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T16:19:45Z","timestamp":1787329185000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/23M157956X"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,7,24]]},"references-count":80,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2025,8,31]]}},"alternative-id":["10.1137\/23M157956X"],"URL":"https:\/\/doi.org\/10.1137\/23m157956x","relation":{},"ISSN":["1064-8275","1095-7197"],"issn-type":[{"value":"1064-8275","type":"print"},{"value":"1095-7197","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,7,24]]}}}