{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,14]],"date-time":"2026-10-14T21:48:58Z","timestamp":1792014538404,"version":"4.4.0"},"reference-count":77,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2026,9,10]],"date-time":"2026-09-10T00:00:00Z","timestamp":1788998400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2026,9,10]],"date-time":"2026-09-10T00:00:00Z","timestamp":1788998400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"DOI":"10.13039\/100000181","name":"Air Force Office of Scientific Research","doi-asserted-by":"publisher","award":["FA9550-24-1-0327"],"award-info":[{"award-number":["FA9550-24-1-0327"]}],"id":[{"id":"10.13039\/100000181","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["2436499"],"award-info":[{"award-number":["2436499"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100006502","name":"Defense Sciences Office, DARPA","doi-asserted-by":"publisher","award":["HR00112430347"],"award-info":[{"award-number":["HR00112430347"]}],"id":[{"id":"10.13039\/100006502","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Sci Comput"],"published-print":{"date-parts":[[2026,11]]},"DOI":"10.1007\/s10915-026-03457-8","type":"journal-article","created":{"date-parts":[[2026,9,10]],"date-time":"2026-09-10T15:50:14Z","timestamp":1789055414000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Active Learning for Data-Driven Reduced Models of Parametric Differential Systems with Bayesian Operator Inference"],"prefix":"10.1007","volume":"109","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0231-5359","authenticated-orcid":false,"given":"Shane A.","family":"McQuarrie","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5541-437X","authenticated-orcid":false,"given":"Mengwu","family":"Guo","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2281-3067","authenticated-orcid":false,"given":"Anirban","family":"Chaudhuri","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2026,9,10]]},"reference":[{"issue":"7","key":"3457_CR1","doi-asserted-by":"publisher","first-page":"1803","DOI":"10.2514\/1.35374","volume":"46","author":"D Amsallem","year":"2008","unstructured":"Amsallem, D., Farhat, C.: Interpolation method for adapting reduced-order models and application to aeroelasticity. AIAA J. 46(7), 1803\u20131813 (2008). https:\/\/doi.org\/10.2514\/1.35374","journal-title":"AIAA J."},{"issue":"3","key":"3457_CR2","doi-asserted-by":"publisher","first-page":"2432","DOI":"10.1137\/22M1481658","volume":"22","author":"F Andreuzzi","year":"2023","unstructured":"Andreuzzi, F., Demo, N., Rozza, G.: A dynamic mode decomposition extension for the forecasting of parametric dynamical systems. SIAM J. Appl. Dyn. Syst. 22(3), 2432\u20132458 (2023). https:\/\/doi.org\/10.1137\/22M1481658","journal-title":"SIAM J. Appl. Dyn. Syst."},{"key":"3457_CR3","doi-asserted-by":"publisher","unstructured":"Aretz, N., Willcox, K.: Enforcing structure in data-driven reduced modeling through nested Operator Inference. In: 63rd IEEE Conference on Decision and Control (CDC) (2024). https:\/\/doi.org\/10.1109\/CDC56724.2024.10885857","DOI":"10.1109\/CDC56724.2024.10885857"},{"issue":"4","key":"3457_CR4","doi-asserted-by":"publisher","first-page":"49","DOI":"10.1007\/s10444-026-10322-7","volume":"52","author":"N Aretz","year":"2026","unstructured":"Aretz, N., Willcox, K.: Nested operator inference for adaptive data-driven learning of reduced-order models. Adv. Comput. Math. 52(4), 49 (2026). https:\/\/doi.org\/10.1007\/s10444-026-10322-7","journal-title":"Adv. Comput. Math."},{"key":"3457_CR5","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2022.111348","volume":"464","author":"J Barnett","year":"2022","unstructured":"Barnett, J., Farhat, C.: Quadratic approximation manifold for mitigating the Kolmogorov barrier in nonlinear projection-based model order reduction. J. Comput. Phys. 464, 111348 (2022). https:\/\/doi.org\/10.1016\/j.jcp.2022.111348","journal-title":"J. Comput. Phys."},{"issue":"4","key":"3457_CR6","doi-asserted-by":"publisher","first-page":"483","DOI":"10.1137\/130932715","volume":"57","author":"P Benner","year":"2015","unstructured":"Benner, P., G\u00fc\u011fercin, S., Willcox, K.: A survey of projection-based model reduction methods for parametric dynamical systems. SIAM Rev. 57(4), 483\u2013531 (2015). https:\/\/doi.org\/10.1137\/130932715","journal-title":"SIAM Rev."},{"key":"3457_CR7","doi-asserted-by":"publisher","DOI":"10.1016\/j.compfluid.2025.106568","volume":"290","author":"D Blais","year":"2025","unstructured":"Blais, D., Nadarajah, S., Biondic, C.: Goal-oriented adaptive sampling for projection-based reduced-order models. Comput. Fluids 290, 106568 (2025). https:\/\/doi.org\/10.1016\/j.compfluid.2025.106568","journal-title":"Comput. Fluids"},{"issue":"2","key":"3457_CR8","doi-asserted-by":"publisher","first-page":"165","DOI":"10.3934\/jcd.2015002","volume":"2","author":"SL Brunton","year":"2015","unstructured":"Brunton, S.L., Proctor, J.L., Tu, J.H., Kutz, J.N.: Compressed sensing and dynamic mode decomposition. J. Comput. Dyn. 2(2), 165 (2015). https:\/\/doi.org\/10.3934\/jcd.2015002","journal-title":"J. Comput. Dyn."},{"issue":"3","key":"3457_CR9","doi-asserted-by":"publisher","first-page":"595","DOI":"10.1051\/m2an\/2011056","volume":"46","author":"A Buffa","year":"2012","unstructured":"Buffa, A., Maday, Y., Patera, A.T., Prud\u2019homme, C., Turinici, G.: A priori convergence of the greedy algorithm for the parametrized reduced basis method. ESAIM: M2AN 46(3), 595\u2013603 (2012). https:\/\/doi.org\/10.1051\/m2an\/2011056","journal-title":"ESAIM: M2AN"},{"key":"3457_CR10","doi-asserted-by":"publisher","first-page":"147","DOI":"10.1017\/S0962492904000182","volume":"13","author":"HJ Bungartz","year":"2004","unstructured":"Bungartz, H.J., Griebel, M.: Sparse grids. Acta Numer 13, 147\u2013269 (2004). https:\/\/doi.org\/10.1017\/S0962492904000182","journal-title":"Acta Numer"},{"issue":"45","key":"3457_CR11","doi-asserted-by":"publisher","first-page":"22445","DOI":"10.1073\/pnas.1906995116","volume":"116","author":"K Champion","year":"2019","unstructured":"Champion, K., Lusch, B., Kutz, J.N., Brunton, S.L.: Data-driven discovery of coordinates and governing equations. Proc. Natl. Acad. Sci. 116(45), 22445\u201322451 (2019). https:\/\/doi.org\/10.1073\/pnas.1906995116","journal-title":"Proc. Natl. Acad. Sci."},{"key":"3457_CR12","unstructured":"Chen, P.Y., Xiang, J., Cho, D.H., Chang, Y., Pershing, G.A., Maia, H.T., Chiaramonte, M.M., Carlberg, K.T., Grinspun, E.: CROM: Continuous reduced-order modeling of PDEs using implicit neural representations. In: The Eleventh International Conference on Learning Representations (2023). https:\/\/openreview.net\/forum?id=FUORz1tG8Og"},{"key":"3457_CR13","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/j.camwa.2023.08.016","volume":"149","author":"L Cicci","year":"2023","unstructured":"Cicci, L., Fresca, S., Guo, M., Manzoni, A., Zunino, P.: Uncertainty quantification for nonlinear solid mechanics using reduced order models with Gaussian process regression. Comput. Math. Appl. 149, 1\u201323 (2023). https:\/\/doi.org\/10.1016\/j.camwa.2023.08.016","journal-title":"Comput. Math. Appl."},{"key":"3457_CR14","doi-asserted-by":"publisher","first-page":"129","DOI":"10.1613\/jair.295","volume":"4","author":"DA Cohn","year":"1996","unstructured":"Cohn, D.A., Ghahramani, Z., Jordan, M.I.: Active learning with statistical models. J. Artif. Intell. Res. 4, 129\u2013145 (1996). https:\/\/doi.org\/10.1613\/jair.295","journal-title":"J. Artif. Intell. Res."},{"key":"3457_CR15","doi-asserted-by":"publisher","unstructured":"Currin, C., Mitchell, T., Morris, M., Ylvisaker, D.: A Bayesian approach to the design and analysis of computer experiments. Tech. rep., Oak Ridge National Lab., TN (USA) (1988). https:\/\/doi.org\/10.2172\/814584","DOI":"10.2172\/814584"},{"key":"3457_CR16","doi-asserted-by":"publisher","unstructured":"DeChant, J., Geelen, R., McQuarrie, S.A., Guilleminot, J.: A dynamic subspace approach for low-rank approximation of large-scale nonlinear systems (2026). https:\/\/doi.org\/10.48550\/arXiv.2605.26025. arXiv:2605.26025","DOI":"10.48550\/arXiv.2605.26025"},{"key":"3457_CR17","doi-asserted-by":"publisher","unstructured":"den Boef, P., Manvelyan, D., Maubach, J., Schilders, W., van de Wouw, N.: Stable sparse operator inference for nonlinear structural dynamics (2024). https:\/\/doi.org\/10.48550\/arXiv.2407.21672. arXiv:2407.21672","DOI":"10.48550\/arXiv.2407.21672"},{"issue":"15","key":"3457_CR18","doi-asserted-by":"publisher","first-page":"3507","DOI":"10.1002\/nme.6977","volume":"123","author":"KRM dos Santos","year":"2022","unstructured":"dos Santos, K.R.M., Giovanis, D.G., Kontolati, K., Loukrezis, D., Shields, M.D.: Grassmannian diffusion maps based surrogate modeling via geometric harmonics. Int. J. Numer. Meth. Eng. 123(15), 3507\u20133529 (2022). https:\/\/doi.org\/10.1002\/nme.6977","journal-title":"Int. J. Numer. Meth. Eng."},{"key":"3457_CR19","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2023.112621","volume":"497","author":"J Duan","year":"2024","unstructured":"Duan, J., Hesthaven, J.S.: Non-intrusive data-driven reduced-order modeling for time-dependent parametrized problems. J. Comput. Phys. 497, 112621 (2024). https:\/\/doi.org\/10.1016\/j.jcp.2023.112621","journal-title":"J. Comput. Phys."},{"key":"3457_CR20","doi-asserted-by":"publisher","unstructured":"Errico, C.J., Padovan, A., Bodony, D.J.: GasNiTROM: Model reduction via non-intrusive optimization of oblique projection operators and guaranteed-stable latent-space dynamics (2026). https:\/\/doi.org\/10.48550\/arXiv.2603.21254. arXiv:2603.21254","DOI":"10.48550\/arXiv.2603.21254"},{"key":"3457_CR21","doi-asserted-by":"publisher","unstructured":"Farcas, I.G., Gundevia, R., Munipalli, R., Willcox, K.E.: Parametric non-intrusive reduced-order models via operator inference for large-scale rotating detonation engine simulations. In: AIAA SciTech 2023 Forum, 0172 (2023). https:\/\/doi.org\/10.2514\/6.2023-0172","DOI":"10.2514\/6.2023-0172"},{"issue":"2","key":"3457_CR22","doi-asserted-by":"publisher","first-page":"B457","DOI":"10.1137\/16M1093975","volume":"40","author":"IG Farca\u015f","year":"2018","unstructured":"Farca\u015f, I.G., Uekermann, B., Neckel, T., Bungartz, H.J.: Nonintrusive uncertainty analysis of fluid-structure interaction with spatially adaptive sparse grids and polynomial chaos expansion. SIAM J. Sci. Comput. 40(2), B457\u2013B482 (2018). https:\/\/doi.org\/10.1137\/16M1093975","journal-title":"SIAM J. Sci. Comput."},{"key":"3457_CR23","doi-asserted-by":"publisher","first-page":"483","DOI":"10.1090\/mcom\/3781","volume":"92","author":"NR Franco","year":"2023","unstructured":"Franco, N.R., Manzoni, A., Zunino, P.: A deep learning approach to reduced order modelling of parameter dependent partial differential equations. Math. Comput. 92, 483\u2013524 (2023). https:\/\/doi.org\/10.1090\/mcom\/3781","journal-title":"Math. Comput."},{"issue":"3\u20134","key":"3457_CR24","doi-asserted-by":"publisher","first-page":"106","DOI":"10.1080\/10618562.2014.918695","volume":"28","author":"T Franz","year":"2014","unstructured":"Franz, T., Zimmermann, R., G\u00f6rtz, S., Karcher, N.: Interpolation-based reduced-order modelling for steady transonic flows via manifold learning. Int. J. Comput. Fluid Dyn. 28(3\u20134), 106\u2013121 (2014). https:\/\/doi.org\/10.1080\/10618562.2014.918695","journal-title":"Int. J. Comput. Fluid Dyn."},{"key":"3457_CR25","doi-asserted-by":"publisher","first-page":"61","DOI":"10.1007\/s10915-021-01462-7","volume":"87","author":"S Fresca","year":"2021","unstructured":"Fresca, S., Dede\u2019, L., Manzoni, A.: A comprehensive deep learning-based approach to reduced order modeling of nonlinear time-dependent parametrized PDEs. J. Sci. Comput. 87, 61 (2021). https:\/\/doi.org\/10.1007\/s10915-021-01462-7","journal-title":"J. Sci. Comput."},{"key":"3457_CR26","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2022.115717","volume":"403","author":"R Geelen","year":"2023","unstructured":"Geelen, R., Wright, S., Willcox, K.: Operator inference for non-intrusive model reduction with quadratic manifolds. Comput. Methods Appl. Mech. Eng. 403, 115717 (2023). https:\/\/doi.org\/10.1016\/j.cma.2022.115717","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"3457_CR27","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2024.117033","volume":"427","author":"Y Geng","year":"2024","unstructured":"Geng, Y., Singh, J., Ju, L., Kramer, B., Wang, Z.: Gradient preserving operator inference: data-driven reduced-order models for equations with gradient structure. Comput. Methods Appl. Mech. Eng. 427, 117033 (2024). https:\/\/doi.org\/10.1016\/j.cma.2024.117033","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"3457_CR28","doi-asserted-by":"publisher","first-page":"445","DOI":"10.1017\/S0962492921000064","volume":"30","author":"O Ghattas","year":"2021","unstructured":"Ghattas, O., Willcox, K.: Learning physics-based models from data: perspectives from inverse problems and model reduction. Acta Numer 30, 445\u2013554 (2021). https:\/\/doi.org\/10.1017\/S0962492921000064","journal-title":"Acta Numer"},{"key":"3457_CR29","doi-asserted-by":"publisher","DOI":"10.1016\/j.physd.2025.134893","volume":"483","author":"P Goyal","year":"2025","unstructured":"Goyal, P., Pontes Duff, I., Benner, P.: Guaranteed stable quadratic models and their applications in SINDy and operator inference. Physica D 483, 134893 (2025). https:\/\/doi.org\/10.1016\/j.physd.2025.134893","journal-title":"Physica D"},{"issue":"3","key":"3457_CR30","doi-asserted-by":"publisher","first-page":"575","DOI":"10.1051\/m2an:2007031","volume":"41","author":"MA Grepl","year":"2007","unstructured":"Grepl, M.A., Maday, Y., Nguyen, N.C., Patera, A.T.: Efficient reduced-basis treatment of nonaffine and nonlinear partial differential equations. ESAIM: Math. Modell. Numer. Anal. 41(3), 575\u2013605 (2007). https:\/\/doi.org\/10.1051\/m2an:2007031","journal-title":"ESAIM: Math. Modell. Numer. Anal."},{"issue":"1","key":"3457_CR31","doi-asserted-by":"publisher","first-page":"157","DOI":"10.1051\/m2an:2005006","volume":"39","author":"MA Grepl","year":"2005","unstructured":"Grepl, M.A., Patera, A.T.: A posteriori error bounds for reduced-basis approximations of parametrized parabolic partial differential equations. ESAIM: Math. Modell. Numer. Anal. 39(1), 157\u2013181 (2005). https:\/\/doi.org\/10.1051\/m2an:2005006","journal-title":"ESAIM: Math. Modell. Numer. Anal."},{"issue":"1","key":"3457_CR32","doi-asserted-by":"publisher","first-page":"376","DOI":"10.1137\/24M1652490","volume":"24","author":"A Gruber","year":"2025","unstructured":"Gruber, A., Tezaur, I.: Variationally consistent Hamiltonian model reduction. SIAM J. Appl. Dyn. Syst. 24(1), 376\u2013414 (2025). https:\/\/doi.org\/10.1137\/24M1652490","journal-title":"SIAM J. Appl. Dyn. Syst."},{"issue":"2","key":"3457_CR33","doi-asserted-by":"publisher","first-page":"44","DOI":"10.1609\/aimag.v40i2.2850","volume":"40","author":"D Gunning","year":"2019","unstructured":"Gunning, D., Aha, D.W.: DARPA\u2019s explainable artificial intelligence (XAI) program. AI Mag. 40(2), 44\u201358 (2019). https:\/\/doi.org\/10.1609\/aimag.v40i2.2850","journal-title":"AI Mag."},{"key":"3457_CR34","doi-asserted-by":"publisher","first-page":"807","DOI":"10.1016\/j.cma.2018.07.017","volume":"341","author":"M Guo","year":"2018","unstructured":"Guo, M., Hesthaven, J.S.: Reduced order modeling for nonlinear structural analysis using Gaussian process regression. Comput. Methods Appl. Mech. Eng. 341, 807\u2013826 (2018). https:\/\/doi.org\/10.1016\/j.cma.2018.07.017","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"3457_CR35","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2022.115336","volume":"402","author":"M Guo","year":"2022","unstructured":"Guo, M., McQuarrie, S.A., Willcox, K.: Bayesian operator inference for data-driven reduced-order modeling. Comput. Methods Appl. Mech. Eng. 402, 115336 (2022). https:\/\/doi.org\/10.1016\/j.cma.2022.115336","journal-title":"Comput. Methods Appl. Mech. Eng."},{"issue":"7","key":"3457_CR36","doi-asserted-by":"publisher","first-page":"122","DOI":"10.1007\/s00158-024-03816-9","volume":"67","author":"Y Guo","year":"2024","unstructured":"Guo, Y., Nath, P., Mahadevan, S., Witherell, P.: Active learning for adaptive surrogate model improvement in high-dimensional problems. Struct. Multidiscip. Optim. 67(7), 122 (2024). https:\/\/doi.org\/10.1007\/s00158-024-03816-9","journal-title":"Struct. Multidiscip. Optim."},{"key":"3457_CR37","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2022.111852","volume":"475","author":"QA Huhn","year":"2023","unstructured":"Huhn, Q.A., Tano, M.E., Ragusa, J.C., Choi, Y.: Parametric dynamic mode decomposition for reduced order modeling. J. Comput. Phys. 475, 111852 (2023). https:\/\/doi.org\/10.1016\/j.jcp.2022.111852","journal-title":"J. Comput. Phys."},{"issue":"3","key":"3457_CR38","doi-asserted-by":"publisher","first-page":"A984","DOI":"10.1137\/130914619","volume":"36","author":"AC Ionita","year":"2014","unstructured":"Ionita, A.C., Antoulas, A.C.: Data-driven parametrized model reduction in the Loewner framework. SIAM J. Sci. Comput. 36(3), A984\u2013A1007 (2014). https:\/\/doi.org\/10.1137\/130914619","journal-title":"SIAM J. Sci. Comput."},{"key":"3457_CR39","doi-asserted-by":"publisher","first-page":"23","DOI":"10.1007\/s10915-023-02142-4","volume":"95","author":"A Ivagnes","year":"2023","unstructured":"Ivagnes, A., Demo, N., Rozza, G.: Towards a machine learning pipeline in reduced order modelling for inverse problems: neural networks for boundary parametrization, dimensionality reduction and solution manifold approximation. J. Sci. Comput. 95, 23 (2023). https:\/\/doi.org\/10.1007\/s10915-023-02142-4","journal-title":"J. Sci. Comput."},{"key":"3457_CR40","doi-asserted-by":"publisher","first-page":"80","DOI":"10.1016\/j.compstruc.2017.04.005","volume":"188","author":"S Jain","year":"2017","unstructured":"Jain, S., Tiso, P., Rutzmoser, J.B., Rixen, D.J.: A quadratic manifold for model order reduction of nonlinear structural dynamics. Comput. Struct. 188, 80\u201394 (2017). https:\/\/doi.org\/10.1016\/j.compstruc.2017.04.005","journal-title":"Comput. Struct."},{"key":"3457_CR41","doi-asserted-by":"crossref","unstructured":"Kaipio, J.P., Somersalo, E.: Statistical and Computational Inverse Problems, Applied Mathematical Sciences, vol. 160. Springer (2005)","DOI":"10.1007\/b138659"},{"key":"3457_CR42","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2021.114296","volume":"389","author":"P Khodabakhshi","year":"2022","unstructured":"Khodabakhshi, P., Willcox, K.E.: Non-intrusive data-driven model reduction for differential\u2013algebraic equations derived from lifting transformations. Comput. Methods Appl. Mech. Eng. 389, 114296 (2022). https:\/\/doi.org\/10.1016\/j.cma.2021.114296","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"3457_CR43","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2025.114418","volume":"543","author":"H Kim","year":"2025","unstructured":"Kim, H., Kramer, B.: Physically consistent predictive reduced-order modeling by enhancing operator inference with state constraints. J. Comput. Phys. 543, 114418 (2025). https:\/\/doi.org\/10.1016\/j.jcp.2025.114418","journal-title":"J. Comput. Phys."},{"key":"3457_CR44","doi-asserted-by":"publisher","first-page":"981","DOI":"10.1137\/20M1364849","volume":"20","author":"B Kramer","year":"2021","unstructured":"Kramer, B.: Stability domains for quadratic-bilinear reduced-order models. SIAM J. Appl. Dyn. Syst. 20, 981\u2013996 (2021). https:\/\/doi.org\/10.1137\/20M1364849","journal-title":"SIAM J. Appl. Dyn. Syst."},{"key":"3457_CR45","doi-asserted-by":"publisher","first-page":"521","DOI":"10.1146\/annurev-fluid-121021-025220","volume":"56","author":"B Kramer","year":"2024","unstructured":"Kramer, B., Peherstorfer, B., Willcox, K.: Learning nonlinear reduced models from data with operator inference. Annu. Rev. Fluid Mech. 56, 521\u2013548 (2024). https:\/\/doi.org\/10.1146\/annurev-fluid-121021-025220","journal-title":"Annu. Rev. Fluid Mech."},{"key":"3457_CR46","doi-asserted-by":"publisher","DOI":"10.1007\/s00158-026-04310-0","author":"B Kramer","year":"2026","unstructured":"Kramer, B., Qian, E.: Reduced-order modeling for engineering systems: survey and opportunities for digital twins. Struct. Multidiscip. Optim. (2026). https:\/\/doi.org\/10.1007\/s00158-026-04310-0","journal-title":"Struct. Multidiscip. Optim."},{"key":"3457_CR47","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2019.108973","volume":"404","author":"K Lee","year":"2020","unstructured":"Lee, K., Carlberg, K.T.: Model reduction of dynamical systems on nonlinear manifolds using deep convolutional autoencoders. J. Comput. Phys. 404, 108973 (2020). https:\/\/doi.org\/10.1016\/j.jcp.2019.108973","journal-title":"J. Comput. Phys."},{"issue":"5","key":"3457_CR48","doi-asserted-by":"publisher","first-page":"2523","DOI":"10.1137\/090775622","volume":"32","author":"C Lieberman","year":"2010","unstructured":"Lieberman, C., Willcox, K., Ghattas, O.: Parameter and state model reduction for large-scale statistical inverse problems. SIAM J. Sci. Comput. 32(5), 2523\u20132542 (2010). https:\/\/doi.org\/10.1137\/090775622","journal-title":"SIAM J. Sci. Comput."},{"key":"3457_CR49","doi-asserted-by":"publisher","first-page":"70","DOI":"10.1007\/s10915-023-02200-x","volume":"95","author":"Y Lin","year":"2023","unstructured":"Lin, Y., Gao, Z., Chen, Y., Sun, X.: A dynamic mode decomposition based reduced-order model for parameterized time-dependent partial differential equations. J. Sci. Comput. 95, 70 (2023). https:\/\/doi.org\/10.1007\/s10915-023-02200-x","journal-title":"J. Sci. Comput."},{"key":"3457_CR50","unstructured":"MacKay, D.J.: Bayesian methods for adaptive models. Ph.D. thesis, California Institute of Technology (1992)"},{"key":"3457_CR51","doi-asserted-by":"publisher","unstructured":"McQuarrie, S.A.: Data-driven parametric reduced-order models: Operator inference for reactive flow applications. Ph.D. thesis, The University of Texas at Austin (2023). https:\/\/doi.org\/10.26153\/tsw\/50172","DOI":"10.26153\/tsw\/50172"},{"key":"3457_CR52","doi-asserted-by":"publisher","DOI":"10.1016\/j.physd.2025.134572","volume":"475","author":"SA McQuarrie","year":"2025","unstructured":"McQuarrie, S.A., Chaudhuri, A., Willcox, K.E., Guo, M.: Bayesian learning with Gaussian processes for low-dimensional representations of time-dependent nonlinear systems. Physica D 475, 134572 (2025). https:\/\/doi.org\/10.1016\/j.physd.2025.134572","journal-title":"Physica D"},{"key":"3457_CR53","doi-asserted-by":"publisher","first-page":"194","DOI":"10.1080\/03036758.2020.1863237","volume":"51","author":"SA McQuarrie","year":"2021","unstructured":"McQuarrie, S.A., Huang, C., Willcox, K.: Data-driven reduced-order models via regularised operator inference for a single-injector combustion process. J. R. Soc. N. Z. 51, 194\u2013211 (2021). https:\/\/doi.org\/10.1080\/03036758.2020.1863237","journal-title":"J. R. Soc. N. Z."},{"key":"3457_CR54","doi-asserted-by":"publisher","first-page":"A1917","DOI":"10.1137\/21M1452810","volume":"45","author":"SA McQuarrie","year":"2023","unstructured":"McQuarrie, S.A., Khodabakhshi, P., Willcox, K.: Nonintrusive reduced-order models for parametric partial differential equations via data-driven operator inference. SIAM J. Sci. Comput. 45, A1917\u2013A1946 (2023). https:\/\/doi.org\/10.1137\/21M1452810","journal-title":"SIAM J. Sci. Comput."},{"key":"3457_CR55","doi-asserted-by":"publisher","unstructured":"National Academy of Engineering and National Academies of Sciences, Engineering, and Medicine: Foundational Research Gaps and Future Directions for Digital Twins. The National Academies Press, Washington, DC (2024). https:\/\/doi.org\/10.17226\/26894","DOI":"10.17226\/26894"},{"issue":"4","key":"3457_CR56","doi-asserted-by":"publisher","first-page":"3052","DOI":"10.1137\/24M1628414","volume":"23","author":"A Padovan","year":"2024","unstructured":"Padovan, A., Vollmer, B., Bodony, D.J.: Data-driven model reduction via non-intrusive optimization of projection operators and reduced-order dynamics. SIAM J. Appl. Dyn. Syst. 23(4), 3052\u20133076 (2024). https:\/\/doi.org\/10.1137\/24M1628414","journal-title":"SIAM J. Appl. Dyn. Syst."},{"key":"3457_CR57","doi-asserted-by":"publisher","first-page":"196","DOI":"10.1016\/j.cma.2016.03.025","volume":"306","author":"B Peherstorfer","year":"2016","unstructured":"Peherstorfer, B., Willcox, K.: Data-driven operator inference for nonintrusive projection-based model reduction. Comput. Methods Appl. Mech. Eng. 306, 196\u2013215 (2016). https:\/\/doi.org\/10.1016\/j.cma.2016.03.025","journal-title":"Comput. Methods Appl. Mech. Eng."},{"issue":"3","key":"3457_CR58","doi-asserted-by":"publisher","first-page":"550","DOI":"10.1137\/16M1082469","volume":"60","author":"B Peherstorfer","year":"2018","unstructured":"Peherstorfer, B., Willcox, K., Gunzburger, M.: Survey of multifidelity methods in uncertainty propagation, inference, and optimization. SIAM Rev. 60(3), 550\u2013591 (2018). https:\/\/doi.org\/10.1137\/16M1082469","journal-title":"SIAM Rev."},{"key":"3457_CR59","doi-asserted-by":"publisher","first-page":"A1934","DOI":"10.1137\/21M1393972","volume":"44","author":"E Qian","year":"2022","unstructured":"Qian, E., Farcas, I.G., Willcox, K.: Reduced operator inference for nonlinear partial differential equations. SIAM J. Sci. Comput. 44, A1934-a1959 (2022). https:\/\/doi.org\/10.1137\/21M1393972","journal-title":"SIAM J. Sci. Comput."},{"issue":"1","key":"3457_CR60","doi-asserted-by":"publisher","first-page":"3","DOI":"10.1186\/2190-5983-1-3","volume":"1","author":"A Quarteroni","year":"2011","unstructured":"Quarteroni, A., Rozza, G., Manzoni, A.: Certified reduced basis approximation for parametrized partial differential equations and applications. J. Math. Ind. 1(1), 3 (2011). https:\/\/doi.org\/10.1186\/2190-5983-1-3","journal-title":"J. Math. Ind."},{"key":"3457_CR61","volume-title":"Gaussian Processes for Machine Learning","author":"CE Rasmussen","year":"2006","unstructured":"Rasmussen, C.E., Williams, C.K.: Gaussian Processes for Machine Learning. MIT Press, Cambridge, MA (2006)"},{"issue":"4","key":"3457_CR62","doi-asserted-by":"publisher","first-page":"409","DOI":"10.1214\/ss\/1177012413","volume":"4","author":"J Sacks","year":"1989","unstructured":"Sacks, J., Welch, W.J., Mitchell, T.J., Wynn, H.P.: Design and analysis of computer experiments. Stat. Sci. 4(4), 409\u2013423 (1989). https:\/\/doi.org\/10.1214\/ss\/1177012413","journal-title":"Stat. Sci."},{"key":"3457_CR63","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2022.115836","volume":"404","author":"N Sawant","year":"2023","unstructured":"Sawant, N., Kramer, B., Peherstorfer, B.: Physics-informed regularization and structure preservation for learning stable reduced models from data with operator inference. Comput. Methods Appl. Mech. Eng. 404, 115836 (2023). https:\/\/doi.org\/10.1016\/j.cma.2022.115836","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"3457_CR64","doi-asserted-by":"publisher","DOI":"10.1063\/1.4913868","author":"T Sayadi","year":"2015","unstructured":"Sayadi, T., Schmid, P.J., Richecoeur, F., Durox, D.: Parametrized data-driven decomposition for bifurcation analysis, with application to thermo-acoustically unstable systems. Phys. Fluids (2015). https:\/\/doi.org\/10.1063\/1.4913868","journal-title":"Phys. Fluids"},{"key":"3457_CR65","doi-asserted-by":"publisher","first-page":"225","DOI":"10.1146\/annurev-fluid-030121-015835","volume":"54","author":"PJ Schmid","year":"2022","unstructured":"Schmid, P.J.: Dynamic mode decomposition and its variants. Annu. Rev. Fluid Mech. 54, 225\u2013254 (2022). https:\/\/doi.org\/10.1146\/annurev-fluid-030121-015835","journal-title":"Annu. Rev. Fluid Mech."},{"key":"3457_CR66","doi-asserted-by":"publisher","unstructured":"Schwerdtner, P., Mohan, P., Bessac, J., de\u00a0Frahan, M.T.H., Peherstorfer, B.: Operator inference aware quadratic manifolds with isotropic reduced coordinates for nonintrusive model reduction (2025). https:\/\/doi.org\/10.48550\/arXiv.2507.20463","DOI":"10.48550\/arXiv.2507.20463"},{"key":"3457_CR67","doi-asserted-by":"publisher","unstructured":"Schwerdtner, P., Peherstorfer, B.: Greedy construction of quadratic manifolds for nonlinear dimensionality reduction and nonlinear model reduction (2024). https:\/\/doi.org\/10.48550\/arXiv.2403.06732. arXiv:2403.06732","DOI":"10.48550\/arXiv.2403.06732"},{"key":"3457_CR68","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2024.116865","volume":"423","author":"H Sharma","year":"2024","unstructured":"Sharma, H., Najera-Flores, D.A., Todd, M.D., Kramer, B.: Lagrangian operator inference enhanced with structure-preserving machine learning for nonintrusive model reduction of mechanical systems. Comput. Methods Appl. Mech. Eng. 423, 116865 (2024). (j.cma.2024.116865)","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"3457_CR69","doi-asserted-by":"publisher","DOI":"10.1016\/j.physd.2021.133122","volume":"431","author":"H Sharma","year":"2022","unstructured":"Sharma, H., Wang, Z., Kramer, B.: Hamiltonian operator inference: physics-preserving learning of reduced-order models for canonical Hamiltonian systems. Phys. D 431, 133122 (2022). https:\/\/doi.org\/10.1016\/j.physd.2021.133122","journal-title":"Phys. D"},{"key":"3457_CR70","doi-asserted-by":"publisher","first-page":"451","DOI":"10.1017\/S0962492910000061","volume":"19","author":"AM Stuart","year":"2010","unstructured":"Stuart, A.M.: Inverse problems: a Bayesian perspective. Acta Numer 19, 451\u2013559 (2010). https:\/\/doi.org\/10.1017\/S0962492910000061","journal-title":"Acta Numer"},{"issue":"1\u20132","key":"3457_CR71","doi-asserted-by":"publisher","first-page":"85","DOI":"10.1016\/S0377-0427(00)00393-9","volume":"123","author":"CF Van Loan","year":"2000","unstructured":"Van Loan, C.F.: The ubiquitous Kronecker product. J. Comput. Appl. Math. 123(1\u20132), 85\u2013100 (2000). https:\/\/doi.org\/10.1016\/S0377-0427(00)00393-9","journal-title":"J. Comput. Appl. Math."},{"key":"3457_CR72","doi-asserted-by":"publisher","unstructured":"Vijaywargiya, A., McQuarrie, S.A., Gruber, A.: Tensor parametric Hamiltonian operator inference. (2025). https:\/\/doi.org\/10.48550\/arXiv.2502.10888. arXiv:2502.10888","DOI":"10.48550\/arXiv.2502.10888"},{"key":"3457_CR73","doi-asserted-by":"publisher","first-page":"2803","DOI":"10.1002\/fld.4998","volume":"93","author":"S Y\u0131ld\u0131z","year":"2021","unstructured":"Y\u0131ld\u0131z, S., Goyal, P., Benner, P., Karas\u00f6zen, B.: Learning reduced-order dynamics for parametrized shallow water equations from data. Int. J. Numer. Meth. Fluids 93, 2803\u20132821 (2021). https:\/\/doi.org\/10.1002\/fld.4998","journal-title":"Int. J. Numer. Meth. Fluids"},{"issue":"5","key":"3457_CR74","doi-asserted-by":"publisher","first-page":"1111","DOI":"10.1002\/nme.4770","volume":"102","author":"MJ Zahr","year":"2015","unstructured":"Zahr, M.J., Farhat, C.: Progressive construction of a parametric reduced-order model for PDE-constrained optimization. Int. J. Numer. Meth. Eng. 102(5), 1111\u20131135 (2015). https:\/\/doi.org\/10.1002\/nme.4770","journal-title":"Int. J. Numer. Meth. Eng."},{"key":"3457_CR75","doi-asserted-by":"publisher","DOI":"10.2514\/1.J065798","author":"BG Zastrow","year":"2025","unstructured":"Zastrow, B.G., Chaudhuri, A., Willcox, K.E., Ashley, A., Henson, M.C.: Block-structured operator inference for coupled multiphysics model reduction. AIAA J. (2025). https:\/\/doi.org\/10.2514\/1.J065798","journal-title":"AIAA J."},{"key":"3457_CR76","doi-asserted-by":"publisher","DOI":"10.1017\/dce.2022.39","volume":"4","author":"Q Zhuang","year":"2023","unstructured":"Zhuang, Q., Hartmann, D., Bungartz, H.J., Lorenzi, J.M.: Active-learning-based nonintrusive model order reduction. Data-Centric Eng. 4, e2 (2023). https:\/\/doi.org\/10.1017\/dce.2022.39","journal-title":"Data-Centric Eng."},{"key":"3457_CR77","doi-asserted-by":"publisher","unstructured":"Zimmermann, R.: Manifold interpolation. In: P.\u00a0Benner, S.\u00a0Grivet-Talocia, A.\u00a0Quarteroni, G.\u00a0Rozza, W.\u00a0Schilders, L.M. Silveira (eds.) Model Order Reduction Volume 1: System- and Data-Driven Methods and Algorithms, pp. 229\u2013274. De Gruyter, Berlin, Boston (2021). https:\/\/doi.org\/10.1515\/9783110498967-007","DOI":"10.1515\/9783110498967-007"}],"container-title":["Journal of Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-026-03457-8.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10915-026-03457-8","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-026-03457-8.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,10,14]],"date-time":"2026-10-14T21:09:00Z","timestamp":1792012140000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10915-026-03457-8"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,9,10]]},"references-count":77,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2026,11]]}},"alternative-id":["3457"],"URL":"https:\/\/doi.org\/10.1007\/s10915-026-03457-8","relation":{},"ISSN":["0885-7474","1573-7691"],"issn-type":[{"value":"0885-7474","type":"print"},{"value":"1573-7691","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,9,10]]},"assertion":[{"value":"23 February 2026","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"16 July 2026","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"24 August 2026","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"10 September 2026","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"The authors declare that they have no conflict of interest.","order":1,"name":"Ethics","label":"Conflict of interest","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"Not applicable.","order":2,"name":"Ethics","label":"Ethics approval and consent to participate","group":{"name":"EthicsHeading","label":"Declarations"}}],"article-number":"42"}}