Hostname: page-component-76d6cb85b7-dqfph Total loading time: 0 Render date: 2026-07-19T08:19:02.192Z Has data issue: false hasContentIssue false
  • English
  • Français

Nonlinear model reduction for transport-dominated problems

Part of: Partial differential equations, initial value and time-dependent initial-boundary value problems Approximations and expansions

Published online by Cambridge University Press:  19 June 2026

Jan S. Hesthaven Open the ORCID record for Jan S. Hesthaven [Opens in a new window] ,
Benjamin Peherstorfer Open the ORCID record for Benjamin Peherstorfer [Opens in a new window]  and
Benjamin Unger Open the ORCID record for Benjamin Unger [Opens in a new window]
Jan S. Hesthaven
Affiliation:
Karlsruhe Institute of Technology, Kaiserstr. 12, 76131 Karlsruhe, Germany E-mail: jan.hesthaven@kit.edu
Benjamin Peherstorfer
Affiliation:
Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY 10012, USA E-mail: pehersto@cims.nyu.edu
Benjamin Unger
Affiliation:
Karlsruhe Institute of Technology, Englerstr. 2, 76131 Karlsruhe, Germany E-mail: benjamin.unger@kit.edu
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

This article surveys nonlinear model reduction methods that remain effective in regimes where linear reduced-space approximations are intrinsically inefficient, such as transport-dominated problems with wave-like phenomena and moving coherent structures, which are commonly associated with the Kolmogorov barrier. The article organizes nonlinear model reduction techniques around three key elements – nonlinear parametrizations, reduced dynamics and online solvers – and categorizes existing approaches into transformation-based methods, online adaptive techniques, and formulations that combine generic nonlinear parametrizations with instantaneous residual minimization.

MSC classification

Primary: 65M99: None of the above, but in this section 41A46: Approximation by arbitrary nonlinear expressions; widths and entropy

Information

Type
Research Article
Information
Acta Numerica , Volume 35 , July 2026 , pp. 173 - 272
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0), which permits unrestricted re-use, distribution and reproduction, provided the original article is properly cited.
Copyright
© The Author(s), 2026. Published by Cambridge University Press

References

Abgrall, R., Amsallem, D. and Crisovan, R. (2016), Robust model reduction by l 1-norm minimization and approximation via dictionaries: Application to nonlinear hyperbolic problems, Adv. Model. Simul. Engrg Sci. 3, art. 1.Google Scholar
Adamyan, V. M., Arov, D. Z. and Krein, M. G. (1971), Analytic properties of Schmidt pairs for a Hankel operator and the generalized Schur–Takagi problem, Math. USSR Sbornik 15, 31–73.10.1070/SM1971v015n01ABEH001531CrossRefGoogle Scholar
Aghili, J., Atokple, J. Z., Billaud-Friess, M., Garnier, G., Mula, O. and Tognon, N. (2025), A dynamical neural Galerkin scheme for filtering problems, ESAIM Proc. Surveys 81, 2–15.10.1051/proc/202581002CrossRefGoogle Scholar
Aitzhan, A., Nouri, A. G., Givi, P. and Babaee, H. (2025), Reduced order modeling of turbulent reacting flows on low-rank matrix manifolds, J. Comput. Phys. 521, art. 113549.10.1016/j.jcp.2024.113549CrossRefGoogle Scholar
Alireza Mirhoseini, M. and Zahr, M. J. (2023), Model reduction of convection-dominated partial differential equations via optimization-based implicit feature tracking, J. Comput. Phys. 473, art. 111739.10.1016/j.jcp.2022.111739CrossRefGoogle Scholar
Amsallem, D. and Farhat, C. (2008), Interpolation method for adapting reduced-order models and application to aeroelasticity, AIAA J. 46, 1803–1813.10.2514/1.35374CrossRefGoogle Scholar
Amsallem, D. and Haasdonk, B. (2016), PEBL-ROM: Projection-error based local reduced-order models, Adv. Model. Simul. Engrg Sci. 3, 1–25.Google Scholar
Amsallem, D., Zahr, M. J. and Farhat, C. (2012), Nonlinear model order reduction based on local reduced-order bases, Internat. J. Numer. Methods Engrg 92, 891–916.10.1002/nme.4371CrossRefGoogle Scholar
Amsallem, D., Zahr, M. J. and Washabaugh, K. (2015), Fast local reduced basis updates for the efficient reduction of nonlinear systems with hyper-reduction, Adv. Comput. Math. 41, 1187–1230.10.1007/s10444-015-9409-0CrossRefGoogle Scholar
Anderson, W. and Farazmand, M. (2022), Evolution of nonlinear reduced-order solutions for PDEs with conserved quantities, SIAM J. Sci. Comput. 44, A176–A197.10.1137/21M1415972CrossRefGoogle Scholar
Anderson, W. and Farazmand, M. (2024), Fast and scalable computation of shape-morphing nonlinear solutions with application to evolutional neural networks, J. Comput. Phys. 498, art. 112649.10.1016/j.jcp.2023.112649CrossRefGoogle Scholar
Antoulas, A. C. (2005), Approximation of Large-Scale Dynamical Systems , Advances in Design and Control, SIAM.10.1137/1.9780898718713CrossRefGoogle Scholar
Antoulas, A. C., Beattie, C. and Gugercin, S. (2020), Interpolatory Methods for Model Reduction , Computational Science & Engineering, SIAM.10.1137/1.9781611976083CrossRefGoogle Scholar
Antoulas, A. C., Sorensen, D. C. and Gugercin, S. (2001), A survey of model reduction methods for large-scale systems, Contemp. Math. 280, 193–220.10.1090/conm/280/04630CrossRefGoogle Scholar
Arbes, F., Greif, C. and Urban, K. (2025), The Kolmogorov N-width for linear transport: Exact representation and the influence of the data, Adv. Comput. Math. 51, art. 13.10.1007/s10444-025-10224-0CrossRefGoogle Scholar
Argaud, J.-P., Bouriquet, B., de Caso, F., Gong, H., Maday, Y. and Mula, O. (2018), Sensor placement in nuclear reactors based on the generalized empirical interpolation method, J. Comput. Phys. 363, 354–370.10.1016/j.jcp.2018.02.050CrossRefGoogle Scholar
Arian, E., Fahl, M. and Sachs, E. W. (2000), Trust-region proper orthogonal decomposition for flow control. Technical report 2000-25, ICASE.Google Scholar
Arnold, A. and Jahnke, T. (2014), On the approximation of high-dimensional differential equations in the hierarchical Tucker format, BIT Numer. Math. 54, 305–341.10.1007/s10543-013-0444-2CrossRefGoogle Scholar
Astrid, P., Weiland, S., Willcox, K. and Backx, T. (2004), Missing point estimation in models described by proper orthogonal decomposition, in 43rd IEEE Conference on Decision and Control (CDC) , Vol. 2, pp. 1767–1772. Available at doi:10.1109/CDC.2004.1430301.Google Scholar
Astrid, P., Weiland, S., Willcox, K. and Backx, T. (2008), Missing point estimation in models described by proper orthogonal decomposition, IEEE Trans. Automat. Control 53, 2237–2251.10.1109/TAC.2008.2006102CrossRefGoogle Scholar
Babaee, H. and Sapsis, T. P. (2016), A minimization principle for the description of modes associated with finite-time instabilities, Proc. R. Soc. A 472(2186), art. 20150779.10.1098/rspa.2015.0779CrossRefGoogle ScholarPubMed
Babuvška, I. and Rheinboldt, W. C. (1978), Error estimates for adaptive finite element computations, SIAM J. Numer. Anal. 15, 736–754.10.1137/0715049CrossRefGoogle Scholar
Bachmayr, M. and Cohen, A. (2017), Kolmogorov widths and low-rank approximations of parametric elliptic PDEs, Math. Comp. 86, 701–724.10.1090/mcom/3132CrossRefGoogle Scholar
Bachmayr, M., Dahmen, W. and Oster, M. (2025a), Variationally correct neural residual regression for parametric PDEs: On the viability of controlled accuracy, IMA J. Numer. Anal. Available at doi:10.1093/imanum/draf073.CrossRefGoogle Scholar
Bachmayr, M., Eisenmann, H. and Uschmajew, A. (2025b), Dynamical low-rank tensor approximations to high-dimensional parabolic problems: Existence and convergence of spatial discretizations, Numer. Math. 157, 781–822.10.1007/s00211-025-01465-8CrossRefGoogle Scholar
Balabanov, O. and Nouy, A. (2021), Randomized linear algebra for model reduction II: Minimal residual methods and dictionary-based approximation, Adv. Comput. Math. 47, art. 26.10.1007/s10444-020-09836-5CrossRefGoogle Scholar
Balajewicz, M., Tezaur, I. and Dowell, E. (2016), Minimal subspace rotation on the Stiefel manifold for stabilization and enhancement of projection-based reduced order models for the compressible Navier–Stokes equations, J. Comput. Phys. 321, 224–241.10.1016/j.jcp.2016.05.037CrossRefGoogle Scholar
Ballout, H., Maday, Y. and Prud’homme, C. (2024), Nonlinear compressive reduced basis approximation for multi-parameter elliptic problem, in Multiscale, Nonlinear and Adaptive Approximation II (DeVore, R. and Kunoth, A., eds), Springer, pp. 55–73.10.1007/978-3-031-75802-7_4CrossRefGoogle Scholar
Balzano, L., Nowak, R. and Recht, B. (2010), Online identification and tracking of subspaces from highly incomplete information, in Proceedings of the 48th Annual Allerton Conference on Communication, Control, and Computing (Allerton) , IEEE, pp. 704–711.Google Scholar
Barnett, J. and Farhat, C. (2022), Quadratic approximation manifold for mitigating the Kolmogorov barrier in nonlinear projection-based model order reduction, J. Comput. Phys. 464, art. 111348.10.1016/j.jcp.2022.111348CrossRefGoogle Scholar
Barnett, J., Farhat, C. and Maday, Y. (2023a), Neural-network-augmented projection-based model order reduction for mitigating the Kolmogorov barrier to reducibility, J. Comput. Phys. 492, art. 112420.10.1016/j.jcp.2023.112420CrossRefGoogle Scholar
Barnett, J. L., Farhat, C. and Maday, Y. (2023b), Mitigating the Kolmogorov barrier for the reduction of aerodynamic models using neural-network-augmented reduced-order models, in AIAA SCITECH 2023 Forum . Available at doi:10.2514/6.2023-0535.CrossRefGoogle Scholar
Barrault, M., Maday, Y., Nguyen, N. C. and Patera, A. T. (2004), An ‘empirical interpolation’ method: Application to efficient reduced-basis discretization of partial differential equations, C.R. Math. Acad. Sci. Paris 339, 667–672.10.1016/j.crma.2004.08.006CrossRefGoogle Scholar
Battisti, B., Blickhan, T., Enchery, G., Ehrlacher, V., Lombardi, D. and Mula, O. (2023), Wasserstein model reduction approach for parametrized flow problems in porous media, ESAIM Proc. Surveys 73, 28–47.10.1051/proc/202373028CrossRefGoogle Scholar
Baur, U., Beattie, C., Benner, P. and Gugercin, S. (2011), Interpolatory projection methods for parameterized model reduction, SIAM J. Sci. Comput. 33, 2489–2518.10.1137/090776925CrossRefGoogle Scholar
Beck, M. H., Jäckle, A., Worth, G. A. and Meyer, H.-D. (2000), The multiconfiguration time-dependent hartree (MCTDH) method: A highly efficient algorithm for propagating wavepackets, Phys. Rep. 324, 1–105.10.1016/S0370-1573(99)00047-2CrossRefGoogle Scholar
Bengio, Y., Courville, A. and Vincent, P. (2013), Representation learning: A review and new perspectives, IEEE Trans. Pattern Anal. Mach. Intell. 35, 1798–1828.10.1109/TPAMI.2013.50CrossRefGoogle ScholarPubMed
Benner, P. and Breiten, T. (2015), Two-sided projection methods for nonlinear model order reduction, SIAM J. Sci. Comput. 37, B239–B260.10.1137/14097255XCrossRefGoogle Scholar
Benner, P. and Werner, S. W. R. (2020), Hankel-norm approximation of large-scale descriptor systems, Adv. Comput. Math. 46, art. 40.10.1007/s10444-020-09750-wCrossRefGoogle Scholar
Benner, P., Cohen, A., Ohlberger, M. and Willcox, K. (2017), Model Reduction and Approximation , Advances in Design and Control, SIAM.10.1137/1.9781611974829CrossRefGoogle Scholar
Benner, P., Goyal, P. and Gugercin, S. (2018), ${\mathrm{\mathscr{H}}}_2$ -quasi-optimal model order reduction for quadratic-bilinear control systems, SIAM J. Matrix Anal. Appl. 39, 983–1032.10.1137/16M1098280CrossRefGoogle Scholar
Benner, P., Goyal, P., Heiland, J. and Pontes Duff, I. (2023), A quadratic decoder approach to nonintrusive reduced-order modeling of nonlinear dynamical systems, Proc. Appl. Math. Mech. 23, art. e202200049.10.1002/pamm.202200049CrossRefGoogle Scholar
Benner, P., Grivet-Talocia, S., Quarteroni, A., Rozza, G., Schilders, W. and Silveira, L. M. (2021a), Model Order Reduction , Vol. 1 , System- and Data-Driven Methods and Algorithms, De Gruyter.Google Scholar
Benner, P., Grivet-Talocia, S., Quarteroni, A., Rozza, G., Schilders, W. and Silveira, L. M. (2021b), Model Order Reduction , Vol. 2, Snapshot-Based Methods and Algorithms, De Gruyter.Google Scholar
Benner, P., Gugercin, S. and Willcox, K. (2015), A survey of projection-based model reduction methods for parametric dynamical systems, SIAM Rev. 57, 483–531.10.1137/130932715CrossRefGoogle Scholar
Bensalah, A., Nouy, A. and Soffo, J. (2025), Nonlinear manifold approximation using compositional polynomial networks. Available at arXiv:2502.05088. To appear in SIAM J. Sci. Comput. 10.1137/25M1734002CrossRefGoogle Scholar
Berger, M. J. and Colella, P. (1989), Local adaptive mesh refinement for shock hydrodynamics, J. Comput. Phys. 82, 64–84.10.1016/0021-9991(89)90035-1CrossRefGoogle Scholar
Berger, M. J. and Oliger, J. (1984), Adaptive mesh refinement for hyperbolic partial differential equations, J. Comput. Phys. 53, 484–512.10.1016/0021-9991(84)90073-1CrossRefGoogle Scholar
Berman, J. and Peherstorfer, B. (2023), Randomized sparse Neural Galerkin schemes for solving evolution equations with deep networks, in Advances in Neural Information Processing Systems 36 (Oh, A. et al., eds), Curran Associates, pp. 4097–4114.10.52202/075280-0179CrossRefGoogle Scholar
Berman, J. and Peherstorfer, B. (2024), CoLoRA: Continuous low-rank adaptation for reduced implicit neural modeling of parameterized partial differential equations, in Proceedings of the 41st International Conference on Machine Learning (Salakhutdinov, R. et al., eds), Vol. 235 of Proceedings of Machine Learning Research, PMLR, pp. 3565–3583.Google Scholar
Berman, J., Schwerdtner, P. and Peherstorfer, B. (2024), Neural Galerkin schemes for sequential-in-time solving of partial differential equations with deep networks, in Numerical Analysis Meets Machine Learning (Mishra, S. and Townsend, A., eds), Vol. 25 of Handbook of Numerical Analysis, Elsevier, pp. 389–418.Google Scholar
Beyn, W. J. and Thümmler, V. (2004), Freezing solutions of equivariant evolution equations, SIAM J. Appl. Dyn. Syst. 3, 85–116.10.1137/030600515CrossRefGoogle Scholar
Billaud-Friess, M. and Nouy, A. (2017), Dynamical model reduction method for solving parameter-dependent dynamical systems, SIAM J. Sci. Comput. 39, A1766–A1792.10.1137/16M1071493CrossRefGoogle Scholar
Binev, P., Cohen, A., Dahmen, W., DeVore, R., Petrova, G. and Wojtaszczyk, P. (2011), Convergence rates for greedy algorithms in reduced basis methods, SIAM J. Math. Anal. 43, 1457–1472.10.1137/100795772CrossRefGoogle Scholar
Black, F., Schulze, P. and Unger, B. (2020), Projection-based model reduction with dynamically transformed modes, ESAIM Math. Model. Numer. Anal. 54, 2011–2043.10.1051/m2an/2020046CrossRefGoogle Scholar
Black, F., Schulze, P. and Unger, B. (2021a), Efficient wildland fire simulation via nonlinear model order reduction, Fluids 6, art. 280.10.3390/fluids6080280CrossRefGoogle Scholar
Black, F., Schulze, P. and Unger, B. (2021b), Model order reduction with dynamically transformed modes for the wave equation, Proc. Appl. Math. Mech. 20, art. e202000321.10.1002/pamm.202000321CrossRefGoogle Scholar
Black, F., Schulze, P. and Unger, B. (2022), Modal decomposition of flow data via gradient-based transport optimization, in Active Flow and Combustion Control 2021 (King, R. and Peitsch, D., eds), Springer, pp. 203–224.10.1007/978-3-030-90727-3_13CrossRefGoogle Scholar
Blickhan, T. (2024), A registration method for reduced basis problems using linear optimal transport, SIAM J. Sci. Comput. 46, A3177–A3204.10.1137/23M1570715CrossRefGoogle Scholar
Bon, D., Caris, B. and Mula, O. (2025), Stable nonlinear dynamical approximation with dynamical sampling. Available at arXiv:2505.11938.Google Scholar
Bonito, A., Cohen, A., DeVore, R., Guignard, D., Jantsch, P. and Petrova, G. (2021), Nonlinear methods for model reduction, ESAIM Math. Model. Numer. Anal. 55, 507–531.10.1051/m2an/2020057CrossRefGoogle Scholar
Braess, D. (1986), Nonlinear Approximation Theory , Springer.10.1007/978-3-642-61609-9CrossRefGoogle Scholar
Broeckhove, J., Lathouwers, L., Kesteloot, E. and Van Leuven, P. (1988), On the equivalence of time-dependent variational principles, Chem. Phys. Lett. 149, 547–550.10.1016/0009-2614(88)80380-4CrossRefGoogle Scholar
Bruna, J., Peherstorfer, B. and Vanden-Eijnden, E. (2024), Neural Galerkin schemes with active learning for high-dimensional evolution equations, J. Comput. Phys. 496, art. 112588.10.1016/j.jcp.2023.112588CrossRefGoogle Scholar
Brunken, J., Smetana, K. and Urban, K. (2019), (Parametrized) first order transport equations: Realization of optimally stable Petrov–Galerkin methods, SIAM J. Sci. Comput. 41, A592–A621.10.1137/18M1176269CrossRefGoogle Scholar
Buchfink, P., Glas, S. and Haasdonk, B. (2023), Symplectic model reduction of Hamiltonian systems on nonlinear manifolds and approximation with weakly symplectic autoencoder, SIAM J. Sci. Comput. 45, A289–A311.10.1137/21M1466657CrossRefGoogle Scholar
Buchfink, P., Glas, S. and Haasdonk, B. (2024a), Approximation bounds for model reduction on polynomially mapped manifolds, C.R. Math. Acad. Sci. Paris 362, 1211–1226.10.5802/crmath.632CrossRefGoogle Scholar
Buchfink, P., Glas, S., Haasdonk, B. and Unger, B. (2024b), Model reduction on manifolds: A differential geometric framework, Phys. D 468, art. 134299.10.1016/j.physd.2024.134299CrossRefGoogle Scholar
Buffa, A., Maday, Y., Patera, A. T., Prud’homme, C. and Turinici, G. (2012), A priori convergence of the greedy algorithm for the parametrized reduced basis method, ESAIM Math. Model. Numer. Anal. 46, 595–603.10.1051/m2an/2011056CrossRefGoogle Scholar
Buhr, A. and Smetana, K. (2018), Randomized local model order reduction, SIAM J. Sci. Comput. 40, A2120–A2151.10.1137/17M1138480CrossRefGoogle Scholar
Buhr, A., Engwer, C., Ohlberger, M. and Rave, S. (2017), ArbiLoMod, a simulation technique designed for arbitrary local modifications, SIAM J. Sci. Comput. 39, A1435–A1465.10.1137/15M1054213CrossRefGoogle Scholar
Bui-Thanh, T., Willcox, K., Ghattas, O. and van Bloemen Waanders, B. (2007), Goal-oriented, model-constrained optimization for reduction of large-scale systems, J. Comput. Phys. 224, 880–896.10.1016/j.jcp.2006.10.026CrossRefGoogle Scholar
Burela, S., Krah, P. and Reiss, J. (2025), Parametric model order reduction for a wildland fire model via the shifted POD-based deep learning method, Adv. Comput. Math. 51, art. 9.10.1007/s10444-025-10220-4CrossRefGoogle Scholar
Cagniart, N., Maday, Y. and Stamm, B. (2019), Model order reduction for problems with large convection effects, in Contributions to Partial Differential Equations and Applications (Chetverushkin, B. N. et al., eds), Computational Methods in Applied Sciences, Springer, pp. 131–150.10.1007/978-3-319-78325-3_10CrossRefGoogle Scholar
Cao, X., Saltik, M. B. and Weiland, S. (2015), Hankel model reduction for descriptor systems, in 54th IEEE Conference on Decision and Control (CDC) , pp. 4668–4673. Available at doi:10.1109/CDC.2015.7402947.Google Scholar
Carlberg, K. (2015), Adaptive h-refinement for reduced-order models, Internat. J. Numer. Methods Engrg 102, 1192–1210.10.1002/nme.4800CrossRefGoogle Scholar
Carlberg, K., Barone, M. and Antil, H. (2017), Galerkin v. least-squares Petrov–Galerkin projection in nonlinear model reduction, J. Comput. Phys. 330, 693–734.10.1016/j.jcp.2016.10.033CrossRefGoogle Scholar
Carlberg, K., Bou-Mosleh, C. and Farhat, C. (2011), Efficient non-linear model reduction via a least-squares Petrov–Galerkin projection and compressive tensor approximations, Internat. J. Numer. Methods Engrg 86, 155–181.10.1002/nme.3050CrossRefGoogle Scholar
Carrel, B. (2026), Randomized methods for dynamical low-rank approximation, J. Comput. Phys. 544, art. 114421.10.1016/j.jcp.2025.114421CrossRefGoogle Scholar
Cenedese, M., Axås, J., Bäuerlein, B., Avila, K. and Haller, G. (2022), Data-driven modeling and prediction of non-linearizable dynamics via spectral submanifolds, Nat. Commun. 13, art. 872.10.1038/s41467-022-28518-yCrossRefGoogle ScholarPubMed
Ceruti, G. and Lubich, C. (2022), An unconventional robust integrator for dynamical low-rank approximation, BIT Numer. Math. 62, 23–44.10.1007/s10543-021-00873-0CrossRefGoogle Scholar
Ceruti, G., Einkemmer, L., Kusch, J. and Lubich, C. (2024a), A robust second-order low-rank BUG integrator based on the midpoint rule, BIT Numer. Math. 64, art. 30.10.1007/s10543-024-01032-xCrossRefGoogle Scholar
Ceruti, G., Kusch, J. and Lubich, C. (2022), A rank-adaptive robust integrator for dynamical low-rank approximation, BIT Numer. Math. 62, 1149–1174.10.1007/s10543-021-00907-7CrossRefGoogle Scholar
Ceruti, G., Kusch, J. and Lubich, C. (2024b), A parallel rank-adaptive integrator for dynamical low-rank approximation, SIAM J. Sci. Comput. 46, B205–B228.10.1137/23M1565103CrossRefGoogle Scholar
Charous, A. and Lermusiaux, P. F. J. (2023), Dynamically orthogonal Runge–Kutta schemes with perturbative retractions for the dynamical low-rank approximation, SIAM J. Sci. Comput. 45, A872–A897.10.1137/21M1431229CrossRefGoogle Scholar
Charous, A. and Lermusiaux, P. F. J. (2024), Stable rank-adaptive dynamically orthogonal Runge–Kutta schemes, SIAM J. Sci. Comput. 46, A529–A560.10.1137/22M1534948CrossRefGoogle Scholar
Chaturantabut, S. and Sorensen, D. C. (2010), Nonlinear model reduction via discrete empirical interpolation, SIAM J. Sci. Comput. 32, 2737–2764.10.1137/090766498CrossRefGoogle Scholar
Chen, H., Wu, R., Grinspun, E., Zheng, C. and Chen, P. Y. (2023a), Implicit neural spatial representations for time-dependent PDEs, in Proceedings of the 40th International Conference on Machine Learning (Krause, A. et al., eds), Vol. 202 of Proceedings of Machine Learning Research, PMLR, pp. 5162–5177.Google Scholar
Chen, P. Y., Xiang, J., Cho, D. H., Chang, Y., Pershing, G. A., Maia, H. T., Chiaramonte, M. M., Carlberg, K. T. and Grinspun, E. (2023b), CROM: Continuous reduced-order modeling of PDEs using implicit neural representations, in Eleventh International Conference on Learning Representations (ICLR 2023) . Available at https://openreview.net/forum?id=FUORz1tG8Og.Google Scholar
Chen, Y., Gottlieb, S., Ji, L. and Maday, Y. (2021), An EIM-degradation free reduced basis method via over collocation and residual hyper reduction-based error estimation, J. Comput. Phys. 444, art. 110545.10.1016/j.jcp.2021.110545CrossRefGoogle Scholar
Chen, Z., Mccarran, J., Vizcaino, E., Soljacic, M. and Luo, D. (2024), TENG: Time-evolving natural gradient for solving PDEs with deep neural nets toward machine precision, in Proceedings of the 41st International Conference on Machine Learning (Salakhutdinov, R. et al., eds), Vol. 235 of Proceedings of Machine Learning Research, PMLR, pp. 7143–7162.Google Scholar
Cho, W., Lee, K., Rim, D. and Park, N. (2023), Hypernetwork-based meta-learning for low-rank physics-informed neural networks, in Thirty-Seventh Conference on Neural Information Processing Systems . Available at https://openreview.net/forum?id=dzqKAM2sKa.Google Scholar
Cohen, A. and DeVore, R. (2015), Approximation of high-dimensional parametric PDEs, Acta Numer. 24, 1–159.10.1017/S0962492915000033CrossRefGoogle Scholar
Cohen, A. and DeVore, R. (2016), Kolmogorov widths under holomorphic mappings, IMA J. Numer. Anal. 36, 1–12.Google Scholar
Cohen, A., Dahmen, W., DeVore, R. and Nichols, J. (2020), Reduced basis greedy selection using random training sets, ESAIM Math. Model. Numer. Anal. 54, 1509–1524.10.1051/m2an/2020004CrossRefGoogle Scholar
Cohen, A., Dahmen, W., Mula, O. and Nichols, J. (2022a), Nonlinear reduced models for state and parameter estimation, SIAM/ASA J. Uncertain. Quantif. 10, 227–267.10.1137/20M1380818CrossRefGoogle Scholar
Cohen, A., DeVore, R. and Schwab, C. (2011), Analytic regularity and polynomial approximation of parametric and stochastic elliptic PDEs, Anal. Appl. 9, 11–47.10.1142/S0219530511001728CrossRefGoogle Scholar
Cohen, A., DeVore, R., Petrova, G. and Wojtaszczyk, P. (2022b), Optimal stable nonlinear approximation, Found. Comput. Math 22, 607–648.10.1007/s10208-021-09494-zCrossRefGoogle Scholar
Cohen, A., Farhat, C., Maday, Y. and Somacal, A. (2024), Nonlinear compressive reduced basis approximation for PDE’s, C.R. Méc. 352, 267–279.Google Scholar
Constantine, P. G. (2015), Active Subspaces: Emerging Ideas for Dimension Reduction in Parameter Studies , SIAM Spotlights, SIAM.10.1137/1.9781611973860CrossRefGoogle Scholar
Copeland, D. M., Cheung, S. W., Huynh, K. and Choi, Y. (2022), Reduced order models for Lagrangian hydrodynamics, Comput. Methods Appl. Mech. Engrg 388, art. 114259.10.1016/j.cma.2021.114259CrossRefGoogle Scholar
Cortinovis, A., Kressner, D., Massei, S. and Peherstorfer, B. (2020), Quasi-optimal sampling to learn basis updates for online adaptive model reduction with adaptive empirical interpolation, in American Control Conference (ACC) 2020 , IEEE, pp. 2472–2477.10.23919/ACC45564.2020.9147832CrossRefGoogle Scholar
Coughlin, J., Hu, J. and Shumlak, U. (2024), Robust and conservative dynamical low-rank methods for the Vlasov equation via a novel macro–micro decomposition, J. Comput. Phys. 509, art. 113055.10.1016/j.jcp.2024.113055CrossRefGoogle Scholar
Dahmen, W., Plesken, C. and Welper, G. (2014), Double greedy algorithms: Reduced basis methods for transport dominated problems, ESAIM Math. Model. Numer. Anal. 48, 623–663.10.1051/m2an/2013103CrossRefGoogle Scholar
Daniel, T., Casenave, F., Akkari, N. and Ryckelynck, D. (2020), Model order reduction assisted by deep neural networks (ROM-net), Adv. Model. Simul. Engrg Sci. 7, art. 16.Google Scholar
de Avila Belbute-Peres, F., Chen, Y. and Sha, F. (2021), HyperPINN: Learning parameterized differential equations with physics-informed hypernetworks, in The Symbiosis of Deep Learning and Differential Equations . Available at https://openreview.net/ forum?id=LxUuRDUhRjM.Google Scholar
Deuflhard, P. and Weiser, M. (2012), Adaptive Numerical Solution of PDEs , De Gruyter.10.1515/9783110283112CrossRefGoogle Scholar
DeVore, R. A. (1998), Nonlinear approximation, Acta Numer. 7, 51–150.10.1017/S0962492900002816CrossRefGoogle Scholar
DeVore, R. A., Kyriazis, G., Leviatan, D. and Tikhomirov, V. M. (1993), Wavelet compression and nonlinear n-widths, Adv. Comput. Math. 1, 197–214.10.1007/BF02071385CrossRefGoogle Scholar
DeVore, R., Hanin, B. and Petrova, G. (2021), Neural network approximation, Acta Numer. 30, 327–444.10.1017/S0962492921000052CrossRefGoogle Scholar
Dihlmann, M., Drohmann, M. and Haasdonk, B. (2011), Model reduction of parametrized evolution problems using the reduced basis method with adaptive time partitioning, in Adaptive Modeling and Simulation 2011 (ADMOS 2011) (Aubry, D. et al., eds), International Center for Numerical Methods in Engineering (CIMNE). Google Scholar
Dirac, P. A. (1930), Note on exchange phenomena in the Thomas atom, Math. Proc. Camb. Phil. Soc. 26, 376–385.10.1017/S0305004100016108CrossRefGoogle Scholar
Do, M.-H., Feydy, J. and Mula, O. (2025), Sparse Wasserstein barycenters and application to reduced order modeling, J. Sci. Comput. 102, art. 64.10.1007/s10915-024-02766-0CrossRefGoogle Scholar
Dong, Y., Schwerdtner, P. and Peherstorfer, B. (2025), Randomized time stepping of nonlinearly parametrized solutions of evolution problems. Available at arXiv:2512.19009.Google Scholar
Drmač, Z. and Gugercin, S. (2016), A new selection operator for the Discrete Empirical Interpolation Method: Improved a priori error bound and extensions, SIAM J. Sci. Comput. 38, A631–A648.10.1137/15M1019271CrossRefGoogle Scholar
Drohmann, M., Haasdonk, B. and Ohlberger, M. (2011), Adaptive reduced basis methods for nonlinear convection–diffusion equations, in Finite Volumes for Complex Applications VI Problems & Perspectives (Fořt, J. et al., eds), Springer, pp. 369–377.10.1007/978-3-642-20671-9_39CrossRefGoogle Scholar
Drohmann, M., Haasdonk, B. and Ohlberger, M. (2012), Reduced basis approximation for nonlinear parametrized evolution equations based on empirical operator interpolation, SIAM J. Sci. Comput. 34, A937–A969.10.1137/10081157XCrossRefGoogle Scholar
Du, Y. and Zaki, T. A. (2021), Evolutional deep neural network, Phys. Rev. E 104, art. 045303.10.1103/PhysRevE.104.045303CrossRefGoogle ScholarPubMed
Eftang, J. L. and Stamm, B. (2012), Parameter multi-domain ‘hp’ empirical interpolation, Internat. J. Numer. Methods Engrg 90, 412–428.10.1002/nme.3327CrossRefGoogle Scholar
Eftang, J. L., Knezevic, D. J. and Patera, A. T. (2011), An hp certified reduced basis method for parametrized parabolic partial differential equations, Math. Comput. Model. Dyn. Sys. 17, 395–422.10.1080/13873954.2011.547670CrossRefGoogle Scholar
Eftang, J. L., Patera, A. T. and Rønquist, E. M. (2010), An hp certified reduced basis method for parametrized elliptic partial differential equations, SIAM J. Sci. Comput. 32, 3170–3200.10.1137/090780122CrossRefGoogle Scholar
Ehrlacher, V., Lombardi, D., Mula, O. and Vialard, F.-X. (2020), Nonlinear model reduction on metric spaces: Application to one-dimensional conservative PDEs in Wasserstein spaces, ESAIM Math. Model. Numer. Anal. 54, 2159–2197.10.1051/m2an/2020013CrossRefGoogle Scholar
Einkemmer, L. and Joseph, I. (2021), A mass, momentum, and energy conservative dynamical low-rank scheme for the Vlasov equation, J. Comput. Phys. 443, art. 110495.10.1016/j.jcp.2021.110495CrossRefGoogle Scholar
Einkemmer, L. and Lubich, C. (2019), A quasi-conservative dynamical low-rank algorithm for the Vlasov equation, SIAM J. Sci. Comput. 41, B1061–B1081.10.1137/18M1218686CrossRefGoogle Scholar
Einkemmer, L., Hu, J. and Wang, Y. (2021), An asymptotic-preserving dynamical low-rank method for the multi-scale multi-dimensional linear transport equation, J. Comput. Phys. 439, art. 110353.10.1016/j.jcp.2021.110353CrossRefGoogle Scholar
Einkemmer, L., Kormann, K., Kusch, J., McClarren, R. G. and Qiu, J.-M. (2025), A review of low-rank methods for time-dependent kinetic simulations, J. Comput. Phys. 538, art. 114191.10.1016/j.jcp.2025.114191CrossRefGoogle Scholar
Einkemmer, L., Ostermann, A. and Scalone, C. (2023), A robust and conservative dynamical low-rank algorithm, J. Comput. Phys. 484, art. 112060.10.1016/j.jcp.2023.112060CrossRefGoogle Scholar
Engwer, C., Ohlberger, M. and Renelt, L. (2024), Model order reduction of an ultraweak and optimally stable variational formulation for parametrized reactive transport problems, SIAM J. Sci. Comput. 46, A3205–A3229.10.1137/23M1613402CrossRefGoogle Scholar
Ern, A. and Guermond, J.-L. (2004), Theory and Practice of Finite Elements , Springer.10.1007/978-1-4757-4355-5CrossRefGoogle Scholar
Etter, P. A. and Carlberg, K. T. (2020), Online adaptive basis refinement and compression for reduced-order models via vector-space sieving, Comput. Methods Appl. Mech. Engrg 364, art. 112931.10.1016/j.cma.2020.112931CrossRefGoogle Scholar
Everson, R. and Sirovich, L. (1995), The Karhunen–Loève procedure for gappy data, J. Opt. Soc. Amer. A 12, 1657–1664.10.1364/JOSAA.12.001657CrossRefGoogle Scholar
Farhat, C., Avery, P., Chapman, T. and Cortial, J. (2014), Dimensional reduction of nonlinear finite element dynamic models with finite rotations and energy-based mesh sampling and weighting for computational efficiency, Internat. J. Numer. Methods Engrg 98, 625–662.10.1002/nme.4668CrossRefGoogle Scholar
Farhat, C., Chapman, T. and Avery, P. (2015), Structure-preserving, stability, and accuracy properties of the energy-conserving sampling and weighting method for the hyper reduction of nonlinear finite element dynamic models, Internat. J. Numer. Methods Engrg 102, 1077–1110.10.1002/nme.4820CrossRefGoogle Scholar
Feischl, M., Lasser, C., Lubich, C. and Nick, J. (2024), Regularized dynamical parametric approximation. Available at arXiv:2403.19234.Google Scholar
Feppon, F. and Lermusiaux, P. F. J. (2018), Dynamically orthogonal numerical schemes for efficient stochastic advection and Lagrangian transport, SIAM Rev. 60, 595–625.10.1137/16M1109394CrossRefGoogle Scholar
Ferrero, A., Taddei, T. and Zhang, L. (2022), Registration-based model reduction of parameterized two-dimensional conservation laws, J. Comput. Phys. 457, art. 111068.10.1016/j.jcp.2022.111068CrossRefGoogle Scholar
Finzi, M. A., Potapczynski, A., Choptuik, M. and Wilson, A. G. (2023), A stable and scalable method for solving initial value PDEs with neural networks, in Eleventh International Conference on Learning Representations (ICLR 2023) . Available at https://openreview.net/ forum?id=vsMyHUq_C1c.Google Scholar
Francis, B. A. (1987), A Course in H ∞ Control Theory, Springer.Google Scholar
Franco, N. R., Manzoni, A. and Zunino, P. (2023), A deep learning approach to Reduced Order Modelling of parameter dependent partial differential equations, Math. Comp. 92, 483–524.10.1090/mcom/3781CrossRefGoogle Scholar
Frenkel, J. (1934), Wave Mechanics: Advanced General Theory , Oxford.Google Scholar
Fresca, S. and Manzoni, A. (2022), POD-DL-ROM: Enhancing deep learning-based reduced order models for nonlinear parametrized PDEs by proper orthogonal decomposition, Comput. Methods Appl. Mech. Engrg 388, art. 114181.10.1016/j.cma.2021.114181CrossRefGoogle Scholar
Fresca, S., Dede’, L. and Manzoni, A. (2021), A comprehensive deep learning-based approach to reduced order modeling of nonlinear time-dependent parametrized PDEs, J. Sci. Comput. 87, art. 61.10.1007/s10915-021-01462-7CrossRefGoogle Scholar
Geelen, R. and Willcox, K. (2022), Localized non-intrusive reduced-order modelling in the operator inference framework, Philos. Trans. R. Soc. A 380(2229), art. 20210206.10.1098/rsta.2021.0206CrossRefGoogle ScholarPubMed
Geelen, R., Balzano, L. and Willcox, K. (2023a), Learning latent representations in high-dimensional state spaces using polynomial manifold constructions, in 62nd IEEE Conference on Decision and Control (CDC) , pp. 4960–4965. Available at doi: 10.1109/CDC49753.2023.10384209.Google Scholar
Geelen, R., Balzano, L., Wright, S. and Willcox, K. (2024), Learning physics-based reduced-order models from data using nonlinear manifolds, Chaos 34, art. 033122.10.1063/5.0170105CrossRefGoogle ScholarPubMed
Geelen, R., Wright, S. and Willcox, K. (2023b), Operator inference for non-intrusive model reduction with quadratic manifolds, Comput. Methods Appl. Mech. Engrg 403, art. 115717.10.1016/j.cma.2022.115717CrossRefGoogle Scholar
Gerbeau, J.-F. and Lombardi, D. (2014), Approximated Lax pairs for the reduced order integration of nonlinear evolution equations, J. Comput. Phys. 265, 246–269.10.1016/j.jcp.2014.01.047CrossRefGoogle Scholar
Ghattas, O. and Willcox, K. (2021), Learning physics-based models from data: Perspectives from inverse problems and model reduction, Acta Numer. 30, 445–554.10.1017/S0962492921000064CrossRefGoogle Scholar
Glas, S. and Mu, H. (2025), Piece-wise symplectic model reduction on quadratically embedded manifolds, in Numerical Mathematics and Advanced Applications (ENUMATH 2023) (Sequeira, A. et al., eds), Vol. 1, Springer, pp. 355–364.Google Scholar
Glas, S. and Unger, B. (2025), Leveraging time and parameters for nonlinear model reduction methods, IFAC-PapersOnLine 59, 550–555.10.1016/j.ifacol.2025.11.092CrossRefGoogle Scholar
Glas, S., Mayerhofer, A. and Urban, K. (2017), Two ways to treat time in reduced basis methods, in Model Reduction of Parametrized Systems (Benner, P. et al., eds), Springer, pp. 1–16.Google Scholar
Glover, K. (1984), All optimal Hankel-norm approximations of linear multivariable systems and their l ∞-error bounds, Internat. J. Control 39, 1115–1193.10.1080/00207178408933239CrossRefGoogle Scholar
Goodfellow, I., Bengio, Y. and Courville, A. (2016), Deep Learning , MIT Press. Available at http://www.deeplearningbook.org.Google Scholar
Gosea, I. V., Gugercin, S. and Unger, B. (2021), Parametric model reduction via rational interpolation along parameters, in 60th IEEE Conference on Decision and Control (CDC) , pp. 6895–6900. Available at doi:10.1109/CDC45484.2021.9682841.Google Scholar
Gowrachari, H., Demo, N., Stabile, G. and Rozza, G. (2025), Non-intrusive model reduction of advection-dominated hyperbolic problems using neural network shift augmented manifold transformation, Comput. Fluids 300, art. 106758.10.1016/j.compfluid.2025.106758CrossRefGoogle Scholar
Goyal, P. and Benner, P. (2024), Generalized quadratic embeddings for nonlinear dynamics using deep learning, Phys. D 463, art. 134158.10.1016/j.physd.2024.134158CrossRefGoogle Scholar
Greif, C. and Urban, K. (2019), Decay of the Kolmogorov n-width for wave problems, Appl. Math. Lett. 96, 216–222.10.1016/j.aml.2019.05.013CrossRefGoogle Scholar
Greif, C., Junk, P. and Urban, K. (2022), Linear/ridge expansions: Enhancing linear approximations by ridge functions, Adv. Comput. Math. 48, art. 15.10.1007/s10444-022-09936-4CrossRefGoogle Scholar
Grepl, M. A. and Patera, A. T. (2005), A posteriori error bounds for reduced-basis approximations of parametrized parabolic partial differential equations, ESAIM Math. Model. Numer. Anal. 39, 157–181.10.1051/m2an:2005006CrossRefGoogle Scholar
Grimberg, S., Farhat, C., Tezaur, R. and Bou-Mosleh, C. (2021), Mesh sampling and weighting for the hyperreduction of nonlinear Petrov–Galerkin reduced-order models with local reduced-order bases, Internat. J. Numer. Methods Engrg 122, 1846–1874.10.1002/nme.6603CrossRefGoogle Scholar
Gruber, A., Gunzburger, M., Ju, L. and Wang, Z. (2022), A comparison of neural network architectures for data-driven reduced-order modeling, Comput. Methods Appl. Mech. Engrg 393, art. 114764.10.1016/j.cma.2022.114764CrossRefGoogle Scholar
Gruhlke, R., Nouy, A. and Trunschke, P. (2024), Optimal sampling for stochastic and natural gradient descent. Available at arXiv:2402.03113.Google Scholar
Grundel, S. and Sarna, N. (2024), Hyper-reduction for parametrized transport dominated problems via adaptive reduced meshes, Partial Differ. Equ. Appl. 5, art. 3.10.1007/s42985-023-00270-yCrossRefGoogle Scholar
Gu, C. (2011), QLMOR: A projection-based nonlinear model order reduction approach using quadratic–linear representation of nonlinear systems, IEEE Trans. Comput.-Aided Des. Integr. Circuits Syst. 30, 1307–1320.10.1109/TCAD.2011.2142184CrossRefGoogle Scholar
Gu, C. (2012), Model order reduction of nonlinear dynamical systems. PhD thesis, UCB/EECS-2012-217, EECS Department, University of California, Berkeley.Google Scholar
Gubisch, M. and Volkwein, S. (2017), Proper orthogonal decomposition for linear-quadratic optimal control, in Model Reduction and Approximation (Benner, P. et al., eds), SIAM, pp. 3–63.Google Scholar
Guglielmi, N. and Manucci, M. (2023), Model order reduction in contour integral methods for parametric PDEs, SIAM J. Sci. Comput. 45, A1711–A1740.10.1137/22M1520189CrossRefGoogle Scholar
Guglielmi, N., López-Fernández, M. and Manucci, M. (2021), Pseudospectral roaming contour integral methods for convection–diffusion equations, J. Sci. Comput. 89, art. 22.10.1007/s10915-021-01601-0CrossRefGoogle Scholar
Haasdonk, B. (2017), Reduced basis methods for parametrized PDEs: A tutorial introduction for stationary and instationary problems, in Model Reduction and Approximation (Benner, P. et al., eds), SIAM, pp. 65–136.10.1137/1.9781611974829.ch2CrossRefGoogle Scholar
Haasdonk, B. and Ohlberger, M. (2008), Reduced basis method for finite volume approximations of parametrized linear evolution equations, ESAIM Math. Model. Numer. Anal. 42, 277–302.10.1051/m2an:2008001CrossRefGoogle Scholar
Haasdonk, B. and Ohlberger, M. (2011), Efficient reduced models and a posteriori error estimation for parametrized dynamical systems by offline/online decomposition, Math. Comput. Model. Dyn. Sys. 17, 145–161.10.1080/13873954.2010.514703CrossRefGoogle Scholar
Haasdonk, B., Dihlmann, M. and Ohlberger, M. (2011), A training set and multiple bases generation approach for parameterized model reduction based on adaptive grids in parameter space, Math. Comput. Model. Dyn. Sys. 17, 423–442.10.1080/13873954.2011.547674CrossRefGoogle Scholar
Haasdonk, B., Ohlberger, M. and Rozza, G. (2008), A reduced basis method for evolution schemes with parameter-dependent explicit operators, Electron. Trans. Numer. Anal. 32, 145–161.Google Scholar
Hartman, D. and Mestha, L. K. (2017), A deep learning framework for model reduction of dynamical systems, in 2017 IEEE Conference on Control Technology and Applications (CCTA) , IEEE, pp. 1917–1922.10.1109/CCTA.2017.8062736CrossRefGoogle Scholar
Hauck, C. D. and Schnake, S. (2023), A predictor–corrector strategy for adaptivity in dynamical low-rank approximations, SIAM J. Matrix Anal. Appl. 44, 971–1005.10.1137/22M1519493CrossRefGoogle Scholar
Heller, E. J. (1976), Time dependent variational approach to semiclassical dynamics, J. Chem. Phys. 64, 63–73.10.1063/1.431911CrossRefGoogle Scholar
Herkert, R., Buchfink, P. and Haasdonk, B. (2024), Dictionary-based online-adaptive structure-preserving model order reduction for parametric Hamiltonian systems, Adv. Comput. Math. 50, art. 12.10.1007/s10444-023-10102-7CrossRefGoogle Scholar
Hess, M., Alla, A., Quaini, A., Rozza, G. and Gunzburger, M. (2019), A localized reduced-order modeling approach for PDEs with bifurcating solutions, Comput. Methods Appl. Mech. Engrg 351, 379–403.10.1016/j.cma.2019.03.050CrossRefGoogle Scholar
Hesthaven, J. S. and Pagliantini, C. (2021), Structure-preserving reduced basis methods for Poisson systems, Math. Comp. 90, 1701–1740.10.1090/mcom/3618CrossRefGoogle Scholar
Hesthaven, J. S., Pagliantini, C. and Ripamonti, N. (2022a), Rank-adaptive structure-preserving model order reduction of Hamiltonian systems, ESAIM Math. Model. Numer. Anal. 56, 617–650.10.1051/m2an/2022013CrossRefGoogle Scholar
Hesthaven, J. S., Pagliantini, C. and Ripamonti, N. (2024), Adaptive symplectic model order reduction of parametric particle-based Vlasov–Poisson equation, Math. Comp. 93, 1153–1202.10.1090/mcom/3885CrossRefGoogle Scholar
Hesthaven, J. S., Pagliantini, C. and Rozza, G. (2022b), Reduced basis methods for time-dependent problems, Acta Numer. 31, 265–345.10.1017/S0962492922000058CrossRefGoogle Scholar
Hesthaven, J. S., Rozza, G. and Stamm, B. (2016), Certified Reduced Basis Methods for Parametrized Partial Differential Equations , SpringerBriefs in Mathematics, Springer.10.1007/978-3-319-22470-1CrossRefGoogle Scholar
Hesthaven, J. S., Stamm, B. and Zhang, S. (2014), Efficient greedy algorithms for high-dimensional parameter spaces with applications to empirical interpolation and reduced basis methods, ESAIM Math. Model. Numer. Anal. 48, 259–283.10.1051/m2an/2013100CrossRefGoogle Scholar
Hillebrecht, B. and Unger, B. (2025a), Prediction error certification for PINNs: Theory, computation, and application to Stokes flow. Available at arXiv:2508.07994.Google Scholar
Hillebrecht, B. and Unger, B. (2025b), Rigorous a posteriori error bounds for PDE-defined PINNs, IEEE Trans. Neural Netw. Learn. Syst. 36, 1583–1593.10.1109/TNNLS.2023.3335837CrossRefGoogle Scholar
Hochbruck, M., Neher, M. and Schrammer, S. (2023), Rank-adaptive dynamical low-rank integrators for first-order and second-order matrix differential equations, BIT Numer. Math. 63, art. 9.Google Scholar
Holmes, P., Lumley, J. L. and Berkooz, G. (1996), Turbulence, Coherent Structures, Dynamical Systems and Symmetry , Cambridge University Press.10.1017/CBO9780511622700CrossRefGoogle Scholar
Hu, E. J., Shen, Y., Wallis, P., Allen-Zhu, Z., Li, Y., Wang, S., Wang, L. and Chen, W. (2022), LoRA: Low-rank adaptation of large language models, in International Conference on Learning Representations (ICLR 2022) . Available at https://openreview.net/ forum?id=nZeVKeeFYf9.Google Scholar
Huang, C. and Duraisamy, K. (2023), Predictive reduced order modeling of chaotic multi-scale problems using adaptively sampled projections, J. Comput. Phys. 491, art. 112356.10.1016/j.jcp.2023.112356CrossRefGoogle Scholar
Huang, C., Xu, J., Duraisamy, K. and Merkle, C. (2018), Exploration of reduced-order models for rocket combustion applications, in 2018 AIAA Aerospace Sciences Meeting . Available at doi:10.2514/6.2018-1183.Google Scholar
Hund, M., Mitchell, T., Mlinarić, P. and Saak, J. (2022), Optimization-based parametric model order reduction via ${\mathrm{\mathscr{H}}}_2\otimes {\mathrm{\mathcal{L}}}_2$ first-order necessary conditions, SIAM J. Sci. Comput. 44, A1554–A1578.10.1137/21M140290XCrossRefGoogle Scholar
Iollo, A. and Lombardi, D. (2014), Advection modes by optimal mass transfer, Phys. Rev. E 89, art. 022923.10.1103/PhysRevE.89.022923CrossRefGoogle ScholarPubMed
Iollo, A., Labatut, J., Mounoud, P. and Taddei, T. (2026), Mathematical aspects of registration methods in bounded domains. Available at arXiv:2601.03010.Google Scholar
Issan, O. and Kramer, B. (2023), Predicting solar wind streams from the inner-heliosphere to earth via shifted operator inference, J. Comput. Phys. 473, art. 111689.10.1016/j.jcp.2022.111689CrossRefGoogle Scholar
Jain, S., Tiso, P., Rutzmoser, J. B. and Rixen, D. J. (2017), A quadratic manifold for model order reduction of nonlinear structural dynamics, Comput. Struct. 188, 80–94.10.1016/j.compstruc.2017.04.005CrossRefGoogle Scholar
Kashima, K. (2016), Nonlinear model reduction by deep autoencoder of noise response data, in 55th Conference on Decision and Control (CDC) , pp. 5750–5755. Available at doi:10.1109/CDC.2016.7799153.Google Scholar
Kast, M. and Hesthaven, J. S. (2024), Positional embeddings for solving PDEs with evolutional deep neural networks, J. Comput. Phys. 508, art. 112986.10.1016/j.jcp.2024.112986CrossRefGoogle Scholar
Kaulmann, S. and Haasdonk, B. (2013), Online greedy reduced basis construction using dictionaries, in Adaptive Modeling and Simulation 2013 (ADMOS 2013) (Moitinho de Almeida, J. P. et al., eds), International Center for Numerical Methods in Engineering (CIMNE), pp. 365–376.Google Scholar
Kay, K. G. (1989), The matrix singularity problem in the time-dependent variational method, Chem. Phys. 137, 165–175.10.1016/0301-0104(89)87102-2CrossRefGoogle Scholar
Kazashi, Y., Nobile, F. and Zoccolan, F. (2025), Dynamical low-rank approximation for stochastic differential equations, Math. Comp. 94, 1335–1375.Google Scholar
Khamlich, M., Pichi, F. and Rozza, G. (2025), Optimal transport–inspired deep learning framework for slow-decaying Kolmogorov n-width problems: Exploiting Sinkhorn loss and Wasserstein kernel, SIAM J. Sci. Comput. 47, C235–C264.10.1137/23M1604680CrossRefGoogle Scholar
Kieri, E., Lubich, C. and Walach, H. (2016), Discretized dynamical low-rank approximation in the presence of small singular values, SIAM J. Numer. Anal. 54, 1020–1038.10.1137/15M1026791CrossRefGoogle Scholar
Kim, Y., Choi, Y., Widemann, D. and Zohdi, T. (2022), A fast and accurate physics-informed neural network reduced order model with shallow masked autoencoder, J. Comput. Phys. 451, art. 110841.10.1016/j.jcp.2021.110841CrossRefGoogle Scholar
Kirby, M. and Armbruster, D. (1992), Reconstructing phase space from PDE simulations, Z. Angew. Math. Phys. 43, 999–1022.10.1007/BF00916425CrossRefGoogle Scholar
Kleikamp, H., Ohlberger, M. and Rave, S. (2022), Nonlinear model order reduction using diffeomorphic transformations of a space–time domain, in Recent Advances in Model Reduction and Surrogate Modeling (MATHMOD 2022) , ARGESIM.Google Scholar
Klein, B. and Ohlberger, M. (2026), Multi-fidelity learning of reduced order models for parabolic PDE constrained optimization, Adv. Comput. Math. 52, art. 19.10.1007/s10444-026-10296-6CrossRefGoogle Scholar
Klein, R., Sanderse, B., Costa, P., Pecnik, R. and Henkes, R. (2025), Entropy-stable model reduction of one-dimensional hyperbolic systems using rational quadratic manifolds, J. Comput. Phys. 528, art. 113817.10.1016/j.jcp.2025.113817CrossRefGoogle Scholar
Koch, O. and Lubich, C. (2007), Dynamical low-rank approximation, SIAM J. Matrix Anal. Appl. 29, 434–454.10.1137/050639703CrossRefGoogle Scholar
Koch, O. and Lubich, C. (2010), Dynamical tensor approximation, SIAM J. Matrix Anal. Appl. 31, 2360–2375.10.1137/09076578XCrossRefGoogle Scholar
Kolmogoroff, A. (1936), Über die beste Annäherung von Funktionen einer gegebenen Funktionenklasse, Ann. of Math. (2) 37, 107–110.10.2307/1968691CrossRefGoogle Scholar
Kolouri, S., Park, S. R. and Rohde, G. K. (2016), The Radon cumulative distribution transform and it application to image classification, IEEE Trans. Image Process. 25, 920–934.10.1109/TIP.2015.2509419CrossRefGoogle ScholarPubMed
Krah, P., Büchholz, S., Häringer, M. and Reiss, J. (2023), Front transport reduction for complex moving fronts, J. Sci. Comput. 96, art. 28.10.1007/s10915-023-02210-9CrossRefGoogle Scholar
Krah, P., Marmin, A., Zorawski, B., Reiss, J. and Schneider, K. (2025), A robust shifted proper orthogonal decomposition: Proximal methods for decomposing flows with multiple transports, SIAM J. Sci. Comput. 47, A633–A656.10.1137/24M164392XCrossRefGoogle Scholar
Kramer, B. and Willcox, K. E. (2019), Nonlinear model order reduction via lifting transformations and proper orthogonal decomposition, AIAA J. 57, 2297–2307.10.2514/1.J057791CrossRefGoogle Scholar
Kramer, B., Peherstorfer, B. and Willcox, K. E. (2024), Learning nonlinear reduced models from data with operator inference, Annu. Rev. Fluid Mech. 56, 521–548.10.1146/annurev-fluid-121021-025220CrossRefGoogle Scholar
Kramer, P. and Saraceno, M. (1981), Geometry of the Time-Dependent Variational Principle in Quantum Mechanics , Vol. 140 of Lecture Notes in Physics, Springer.10.1007/3-540-10579-4CrossRefGoogle Scholar
Kunisch, K. and Volkwein, S. (2008), Proper orthogonal decomposition for optimality systems, ESAIM Math. Model. Numer. Anal. 42, 1–23.10.1051/m2an:2007054CrossRefGoogle Scholar
Kvaal, S., Lasser, C., Pedersen, T. B. and Adamowicz, L. (2023), No need for a grid: Adaptive fully-flexible gaussians for the time-dependent Schrödinger equation. Available at arXiv:2207.00271.Google Scholar
Lam, H. Y., Ceruti, G. and Kressner, D. (2025), Randomized low-rank Runge–Kutta methods, SIAM J. Matrix Anal. Appl. 46, 1587–1615.10.1137/24M1692691CrossRefGoogle Scholar
Lasser, C. and Lubich, C. (2020), Computing quantum dynamics in the semiclassical regime, Acta Numer. 29, 229–401.10.1017/S0962492920000033CrossRefGoogle Scholar
LeCun, Y., Bengio, Y. and Hinton, G. (2015), Deep learning, Nature 521(7553), 436–444.10.1038/nature14539CrossRefGoogle ScholarPubMed
Lee, K. and Carlberg, K. T. (2020), Model reduction of dynamical systems on nonlinear manifolds using deep convolutional autoencoders, J. Comput. Phys. 404, art. 108973.10.1016/j.jcp.2019.108973CrossRefGoogle Scholar
Lee, K. and Carlberg, K. T. (2021), Deep conservation: A latent-dynamics model for exact satisfaction of physical conservation laws, Proc. AAAI Conf. Artif. Intell. 35, 277–285.Google Scholar
Lindsey, M. (2025), MNE: Overparametrized neural evolution with applications to diffusion processes and sampling. Available at arXiv:2502.03645.Google Scholar
Long, T., Barnett, R., Jefferson-Loveday, R., Stabile, G. and Icardi, M. (2025), A reduced-order model for advection-dominated problems based on the Radon Cumulative Distribution Transform, Adv. Comput. Math. 51, art. 5.10.1007/s10444-024-10209-5CrossRefGoogle Scholar
Lu, H. and Tartakovsky, D. M. (2020), Lagrangian dynamic mode decomposition for construction of reduced-order models of advection-dominated phenomena, J. Comput. Phys. 407, art. 109229.10.1016/j.jcp.2020.109229CrossRefGoogle Scholar
Lu, H. and Tartakovsky, D. M. (2021), Dynamic mode decomposition for construction of reduced-order models of hyperbolic problems with shocks, J. Mach. Learn. Model. Comput. 2, 1–29.10.1615/JMachLearnModelComput.2021036132CrossRefGoogle Scholar
Lubich, C. (2005), On variational approximations in quantum molecular dynamics, Math. Comp. 74, 765–779.10.1090/S0025-5718-04-01685-0CrossRefGoogle Scholar
Lubich, C. (2008), From Quantum to Classical Molecular Dynamics: Reduced Models and Numerical Analysis , Vol. 12 of Zurich Lectures in Advanced Mathematics, European Mathematical Society.10.4171/067CrossRefGoogle Scholar
Lubich, C. and Nick, J. (2025), Regularized dynamical parametric approximation of stiff evolution problems. Available at arXiv:2501.12118.Google Scholar
Lubich, C. and Oseledets, I. V. (2014), A projector-splitting integrator for dynamical low-rank approximation, BIT Numer. Math. 54, 171–188.10.1007/s10543-013-0454-0CrossRefGoogle Scholar
Lubich, C., Oseledets, I. V. and Vandereycken, B. (2015), Time integration of tensor trains, SIAM J. Numer. Anal. 53, 917–941.10.1137/140976546CrossRefGoogle Scholar
Lubich, C., Rohwedder, T., Schneider, R. and Vandereycken, B. (2013), Dynamical approximation by hierarchical Tucker and tensor-train tensors, SIAM J. Matrix Anal. Appl. 34, 470–494.10.1137/120885723CrossRefGoogle Scholar
Maday, Y. and Rønquist, E. M. (2002), A reduced-basis element method, J. Sci. Comput. 17, 447–459.10.1023/A:1015197908587CrossRefGoogle Scholar
Maday, Y., Manzoni, A. and Quarteroni, A. (2016), An online intrinsic stabilization strategy for the reduced basis approximation of parametrized advection-dominated problems, C.R. Math. Acad. Sci. Paris 354, 1188–1194.10.1016/j.crma.2016.10.008CrossRefGoogle Scholar
Maday, Y., Patera, A. T. and Turinici, G. (2002), Global a priori convergence theory for reduced-basis approximations of single-parameter symmetric coercive elliptic partial differential equations, C.R. Math. Acad. Sci. Paris 335, 289–294.10.1016/S1631-073X(02)02466-4CrossRefGoogle Scholar
Maulik, R., Lusch, B. and Balaprakash, P. (2021), Reduced-order modeling of advection-dominated systems with recurrent neural networks and convolutional autoencoders, Phys. Fluids 33, art. 037106.10.1063/5.0039986CrossRefGoogle Scholar
McLachlan, A. D. (1964), A variational solution of the time-dependent Schrodinger equation, Mol. Phys. 8, 39–44.10.1080/00268976400100041CrossRefGoogle Scholar
Mehrmann, V. and Unger, B. (2023), Control of port-Hamiltonian differential-algebraic systems and applications, Acta Numer. 32, 395–515.10.1017/S0962492922000083CrossRefGoogle Scholar
Mendible, A., Brunton, S. L., Aravkin, A. Y., Lowrie, W. and Kutz, J. N. (2020), Dimensionality reduction and reduced-order modeling for traveling wave physics, Theor. Comput. Fluid Dyn. 34, 385–400.10.1007/s00162-020-00529-9CrossRefGoogle Scholar
Meyer, H.-D., Manthe, U. and Cederbaum, L. (1990), The multi-configurational time-dependent Hartree approach, Chem. Phys. Lett. 165, 73–78.10.1016/0009-2614(90)87014-ICrossRefGoogle Scholar
Miller, K. and Miller, R. N. (1981), Moving finite elements I, SIAM J. Numer. Anal. 18, 1019–1032.10.1137/0718070CrossRefGoogle Scholar
Mohaghegh, A. and Huang, C. (2026), Feature-guided sampling strategy for adaptive model order reduction of convection-dominated problems, J. Comput. Phys. 545, art. 114468.10.1016/j.jcp.2025.114468CrossRefGoogle Scholar
Mojgani, R. and Balajewicz, M. (2017), Lagrangian basis method for dimensionality reduction of convection dominated nonlinear flows. Available at arXiv:1701.04343.Google Scholar
Moore, B. (1981), Principal component analysis in linear systems: Controllability, observability, and model reduction, IEEE Trans. Automat. Control 26, 17–32.10.1109/TAC.1981.1102568CrossRefGoogle Scholar
Mowlavi, S. and Sapsis, T. P. (2018), Model order reduction for stochastic dynamical systems with continuous symmetries, SIAM J. Sci. Comput. 40, A1669–A1695.10.1137/17M1126576CrossRefGoogle Scholar
Mullis, C. and Roberts, R. (1976), Synthesis of minimum roundoff noise fixed point digital filters, IEEE Trans. Circuits Syst. 23, 551–562.10.1109/TCS.1976.1084254CrossRefGoogle Scholar
Musharbash, E. and Nobile, F. (2018), Dual dynamically orthogonal approximation of incompressible Navier Stokes equations with random boundary conditions, J. Comput. Phys. 354, 135–162.10.1016/j.jcp.2017.09.061CrossRefGoogle Scholar
Musharbash, E., Nobile, F. and Vidličková, E. (2020), Symplectic dynamical low rank approximation of wave equations with random parameters, BIT Numer. Math. 60, 1153–1201.10.1007/s10543-020-00811-6CrossRefGoogle Scholar
Musharbash, E., Nobile, F. and Zhou, T. (2015), Error analysis of the dynamically orthogonal approximation of time dependent random PDEs, SIAM J. Sci. Comput. 37, A776–A810.10.1137/140967787CrossRefGoogle Scholar
Nabian, M. A., Gladstone, R. J. and Meidani, H. (2021), Efficient training of physics-informed neural networks via importance sampling, Comput.-Aided Civ. Engrg 36, 962–977.10.1111/mice.12685CrossRefGoogle Scholar
Naderi, M. H. and Babaee, H. (2023), Adaptive sparse interpolation for accelerating nonlinear stochastic reduced-order modeling with time-dependent bases, Comput. Methods Appl. Mech. Engrg 405, art. 115813.10.1016/j.cma.2022.115813CrossRefGoogle Scholar
Nicolini, J. L. and Teixeira, F. L. (2021), Reduced-order modeling of advection-dominated kinetic plasma problems by shifted proper orthogonal decomposition, IEEE Trans. Plasma Sci. 49, 3689–3699.10.1109/TPS.2021.3115657CrossRefGoogle Scholar
Nobile, F. and Trindade, T. T. (2025a), Error estimates for SUPG-stabilised dynamical low rank approximations, in Numerical Mathematics and Advanced Applications (ENUMATH 2023) (Sequeira, A. et al., eds), Vol. 2, Springer, pp. 438–447.Google Scholar
Nobile, F. and Trindade, T. T. (2025b), Petrov–Galerkin dynamical low rank approximation: SUPG stabilisation of advection-dominated problems, Comput. Methods Appl. Mech. Engrg 433, art. 117495.10.1016/j.cma.2024.117495CrossRefGoogle Scholar
Nonino, M., Ballarin, F., Rozza, G. and Maday, Y. (2023), A reduced basis method by means of transport maps for a fluid–structure interaction problem with slowly decaying Kolmogorov n-width, Adv. Comput. Sci. Engrg 1, 36–58.10.3934/acse.2023002CrossRefGoogle Scholar
Nouy, A. (2010), A priori model reduction through proper generalized decomposition for solving time-dependent partial differential equations, Comput. Methods Appl. Mech. Engrg 199, 1603–1626.10.1016/j.cma.2010.01.009CrossRefGoogle Scholar
Nouy, A. and Pasco, A. (2024), Dictionary-based model reduction for state estimation, Adv. Comput. Math. 50, art. 32.10.1007/s10444-024-10129-4CrossRefGoogle Scholar
Ohlberger, M. and Rave, S. (2013), Nonlinear reduced basis approximation of parameterized evolution equations via the method of freezing, C.R. Math. Acad. Sci. Paris 351, 901–906.10.1016/j.crma.2013.10.028CrossRefGoogle Scholar
Ohlberger, M. and Rave, S. (2016), Reduced basis methods: Success, limitations and future challenges, in Proceedings of the Conference Algoritmy . Available at http://www.iam.fmph.uniba.sk/amuc/ojs/index.php/algoritmy/article/view/389.Google Scholar
Ohlberger, M. and Schindler, F. (2015), Error control for the localized reduced basis multiscale method with adaptive on-line enrichment, SIAM J. Sci. Comput. 37, A2865–A2895.10.1137/151003660CrossRefGoogle Scholar
Ohlberger, M. and Schindler, F. (2017), Non-conforming localized model reduction with online enrichment: Towards optimal complexity in PDE constrained optimization, in Finite Volumes for Complex Applications VIII: Hyperbolic, Elliptic and Parabolic Problems (Cancès, C. and Omnes, P., eds), Springer, pp. 357–365.10.1007/978-3-319-57394-6_38CrossRefGoogle Scholar
Otto, S. E., Macchio, G. R. and Rowley, C. W. (2023), Learning nonlinear projections for reduced-order modeling of dynamical systems using constrained autoencoders, Chaos 33, art. 113130.10.1063/5.0169688CrossRefGoogle ScholarPubMed
Otto, S. E., Padovan, A. and Rowley, C. W. (2022), Optimizing oblique projections for nonlinear systems using trajectories, SIAM J. Sci. Comput. 44, A1681–A1702.10.1137/21M1425815CrossRefGoogle Scholar
Pacciarini, P. and Rozza, G. (2014), Stabilized reduced basis method for parametrized advection–diffusion PDEs, Comput. Methods Appl. Mech. Engrg 274, 1–18.10.1016/j.cma.2014.02.005CrossRefGoogle Scholar
Pagliantini, C. (2021), Dynamical reduced basis methods for Hamiltonian systems, Numer. Math. 148, 409–448.10.1007/s00211-021-01211-wCrossRefGoogle Scholar
Pagliantini, C. and Vismara, F. (2025), Fully adaptive structure-preserving hyper-reduction of parametric Hamiltonian systems, SIAM J. Sci. Comput. 47, A124–A152.10.1137/23M1601225CrossRefGoogle Scholar
Pagliantini, C. and Vismara, F. (2026), Adaptive hyper-reduction of non-sparse operators: Application to parametric particle-based kinetic plasma models, J. Comput. Phys. 549, art. 114609.10.1016/j.jcp.2025.114609CrossRefGoogle Scholar
Pan, S., Brunton, S. L. and Kutz, J. N. (2023), Neural implicit flow: A mesh-agnostic dimensionality reduction paradigm of spatio-temporal data, J. Mach. Learn. Res. 24, 1–60.Google Scholar
Papapicco, D., Demo, N., Girfoglio, M., Stabile, G. and Rozza, G. (2022), The neural network shifted-proper orthogonal decomposition: A machine learning approach for non-linear reduction of hyperbolic equations, Comput. Methods Appl. Mech. Engrg 392, art. 114687.10.1016/j.cma.2022.114687CrossRefGoogle Scholar
Parish, E. J. and Carlberg, K. T. (2021), Windowed least-squares model reduction for dynamical systems, J. Comput. Phys. 426, art. 109939.10.1016/j.jcp.2020.109939CrossRefGoogle Scholar
Pazy, A. (1983), Semigroups of Linear Operators and Applications to Partial Differential Equations , Applied Mathematical Sciences, Springer.10.1007/978-1-4612-5561-1CrossRefGoogle Scholar
Peherstorfer, B. (2020), Model reduction for transport-dominated problems via online adaptive bases and adaptive sampling, SIAM J. Sci. Comput. 42, A2803–A2836.10.1137/19M1257275CrossRefGoogle Scholar
Peherstorfer, B. (2022), Breaking the Kolmogorov barrier with nonlinear model reduction, Notices Amer. Math. Soc. 69, 725–733.10.1090/noti2475CrossRefGoogle Scholar
Peherstorfer, B. and Willcox, K. (2015a), Dynamic data-driven reduced-order models, Comput. Methods Appl. Mech. Engrg 291, 21–41.10.1016/j.cma.2015.03.018CrossRefGoogle Scholar
Peherstorfer, B. and Willcox, K. (2015b), Online adaptive model reduction for nonlinear systems via low-rank updates, SIAM J. Sci. Comput. 37, A2123–A2150.10.1137/140989169CrossRefGoogle Scholar
Peherstorfer, B., Butnaru, D., Willcox, K. and Bungartz, H.-J. (2014), Localized discrete empirical interpolation method, SIAM J. Sci. Comput. 36, A168–A192.10.1137/130924408CrossRefGoogle Scholar
Peherstorfer, B., Drmač, Z. and Gugercin, S. (2020), Stability of discrete empirical interpolation and gappy proper orthogonal decomposition with randomized and deterministic sampling points, SIAM J. Sci. Comput. 42, A2837–A2864.10.1137/19M1307391CrossRefGoogle Scholar
Peherstorfer, B., Willcox, K. and Gunzburger, M. (2018), Survey of multifidelity methods in uncertainty propagation, inference, and optimization, SIAM Rev. 60, 550–591.10.1137/16M1082469CrossRefGoogle Scholar
Perez, E., Strub, F., de Vries, H., Dumoulin, V. and Courville, A. (2018), FiLM: Visual reasoning with a general conditioning layer, in Proceedings of the Thirty-Second AAAI Conference on Artificial Intelligence and Thirtieth Innovative Applications of Artificial Intelligence Conference and Eighth AAAI Symposium on Educational Advances in Artificial Intelligence , AAAI’18/IAAI’18/EAAI’18, AAAI Press.Google Scholar
Phuong Huynh, D. B., Knezevic, D. J. and Patera, A. T. (2013), A static condensation reduced basis element method: Approximation and a posteriori error estimation, ESAIM Math. Model. Numer. Anal. 47, 213–251.10.1051/m2an/2012022CrossRefGoogle Scholar
Pinkus, A. (1985), n-Widths in Approximation Theory , Ergebnisse der Mathematik und ihrer Grenzgebiete, Springer.10.1007/978-3-642-69894-1CrossRefGoogle Scholar
Pinkus, A. (1999), Approximation theory of the MLP model in neural networks, Acta Numer. 8, 143–195.10.1017/S0962492900002919CrossRefGoogle Scholar
Prud’homme, C., Rovas, D. V., Veroy, K., Machiels, L., Maday, Y., Patera, A. T. and Turinici, G. (2001), Reliable real-time solution of parametrized partial differential equations: Reduced-basis output bound methods, J. Fluids Engrg 124, 70–80.10.1115/1.1448332CrossRefGoogle Scholar
Qian, E., Grepl, M., Veroy, K. and Willcox, K. (2017), A certified trust region reduced basis approach to PDE-constrained optimization, SIAM J. Sci. Comput. 39, S434–S460.10.1137/16M1081981CrossRefGoogle Scholar
Qian, E., Kramer, B., Peherstorfer, B. and Willcox, K. (2020), Lift & learn: Physics-informed machine learning for large-scale nonlinear dynamical systems, Phys. D 406, art. 132401.10.1016/j.physd.2020.132401CrossRefGoogle Scholar
Quarteroni, A. and Rozza, G., eds (2014), Reduced Order Methods for Modeling and Computational Reduction , Modeling, Simulation and Applications, Springer.10.1007/978-3-319-02090-7CrossRefGoogle Scholar
Quarteroni, A., Manzoni, A. and Negri, F. (2016), Reduced Basis Methods for Partial Differential Equations: An Introduction , UNITEXT, Springer.10.1007/978-3-319-15431-2CrossRefGoogle Scholar
Raissi, M., Perdikaris, P. and Karniadakis, G. (2019), Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations, J. Comput. Phys. 378, 686–707.10.1016/j.jcp.2018.10.045CrossRefGoogle Scholar
Ramezanian, D., Nouri, A. G. and Babaee, H. (2021), On-the-fly reduced order modeling of passive and reactive species via time-dependent manifolds, Comput. Methods Appl. Mech. Engrg 382, art. 113882.10.1016/j.cma.2021.113882CrossRefGoogle Scholar
Rapún, M.-L. and Vega, J. M. (2010), Reduced order models based on local POD plus Galerkin projection, J. Comput. Phys. 229, 3046–3063.10.1016/j.jcp.2009.12.029CrossRefGoogle Scholar
Razavi, A. H. and Yano, M. (2025), Registration-based nonlinear model reduction of parametrized aerodynamics problems with applications to transonic Euler and RANS flows, J. Comput. Phys. 521, art. 113576.10.1016/j.jcp.2024.113576CrossRefGoogle Scholar
Regazzoni, F., Pagani, S., Salvador, M., Dede’, L. and Quarteroni, A. (2024), Learning the intrinsic dynamics of spatio-temporal processes through latent dynamics networks, Nat. Commun. 15, art. 1834.10.1038/s41467-024-45323-xCrossRefGoogle ScholarPubMed
Reiss, J. (2021), Optimization-based modal decomposition for systems with multiple transports, SIAM J. Sci. Comput. 43, A2079–A2101.10.1137/20M1322005CrossRefGoogle Scholar
Reiss, J., Schulze, P., Sesterhenn, J. and Mehrmann, V. (2018), The shifted proper orthogonal decomposition: A mode decomposition for multiple transport phenomena, SIAM J. Sci. Comput. 40, A1322–A1344.10.1137/17M1140571CrossRefGoogle Scholar
Ren, J., Wolf, W. R. and Mao, X. (2021), Model reduction of traveling-wave problems via Radon cumulative distribution transform, Phys. Rev. Fluids 6, art. L082501.10.1103/PhysRevFluids.6.L082501CrossRefGoogle Scholar
Rim, D. (2018), Dimensional splitting of hyperbolic partial differential equations using the Radon transform, SIAM J. Sci. Comput. 40, A4184–A4207.10.1137/17M1135633CrossRefGoogle Scholar
Rim, D. and Welper, G. (2026), A low-rank neural representation of entropy solutions, Adv. Comput. Math. 52, art. 2.10.1007/s10444-025-10274-4CrossRefGoogle Scholar
Rim, D., Moe, S. and LeVeque, R. J. (2018), Transport reversal for model reduction of hyperbolic partial differential equations, SIAM/ASA J. Uncertain. Quantif. 6, 118–150.10.1137/17M1113679CrossRefGoogle Scholar
Rim, D., Peherstorfer, B. and Mandli, K. T. (2023), Manifold approximations via transported subspaces: Model reduction for transport-dominated problems, SIAM J. Sci. Comput. 45, A170–A199.10.1137/20M1316998CrossRefGoogle Scholar
Romor, F., Stabile, G. and Rozza, G. (2023), Non-linear manifold reduced-order models with convolutional autoencoders and reduced over-collocation method, J. Sci. Comput. 94, 74.10.1007/s10915-023-02128-2CrossRefGoogle Scholar
Rothe, E. (1930), Zweidimensionale parabolische Randwertaufgaben als Grenzfall eindimensionaler Randwertaufgaben, Math. Ann. 102, 650–670.10.1007/BF01782368CrossRefGoogle Scholar
Rotskoff, G. M., Mitchell, A. R. and Vanden-Eijnden, E. (2022), Active importance sampling for variational objectives dominated by rare events: Consequences for optimization and generalization, in Proceedings of the 2nd Mathematical and Scientific Machine Learning Conference (Bruna, J. et al., eds), Vol. 145 of Proceedings of Machine Learning Research, PMLR, pp. 757–780.Google Scholar
Rowan, K., Schatzki, L., Zaklama, T., Suzuki, Y., Watanabe, K. and Varga, K. (2020), Simulation of a hydrogen atom in a laser field using the time-dependent variational principle, Phys. Rev. E 101, art. 023313.10.1103/PhysRevE.101.023313CrossRefGoogle Scholar
Rowley, C. W. and Marsden, J. E. (2000), Reconstruction equations and the Karhunen–Loève expansion for systems with symmetry, Phys. D 142, 1–19.10.1016/S0167-2789(00)00042-7CrossRefGoogle Scholar
Rowley, C. W., Kevrekidis, I. G., Marsden, J. E. and Lust, K. (2003), Reduction and reconstruction for self-similar dynamical systems, Nonlinearity 16, art. 1257.10.1088/0951-7715/16/4/304CrossRefGoogle Scholar
Rozza, G., Huynh, D. B. P. and Patera, A. T. (2008), Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations, Arch. Comput. Methods Engrg 15, 229–275.10.1007/s11831-008-9019-9CrossRefGoogle Scholar
Rumelhart, D. E., Hinton, G. E. and Williams, R. J. (1986), Learning representations by back-propagating errors, Nature 323(6088), 533–536.10.1038/323533a0CrossRefGoogle Scholar
Rutzmoser, J. B., Rixen, D. J., Tiso, P. and Jain, S. (2017), Generalization of quadratic manifolds for reduced order modeling of nonlinear structural dynamics, Comput. Struct. 192, 196–209.10.1016/j.compstruc.2017.06.003CrossRefGoogle Scholar
Sapsis, T. P. and Lermusiaux, P. F. J. (2009), Dynamically orthogonal field equations for continuous stochastic dynamical systems, Phys. D 238, 2347–2360.10.1016/j.physd.2009.09.017CrossRefGoogle Scholar
Sarna, N. and Benner, P. (2021), Data-driven model order reduction for problems with parameter-dependent jump-discontinuities, Comput. Methods Appl. Mech. Engrg 387, art. 114168.10.1016/j.cma.2021.114168CrossRefGoogle Scholar
Sarna, N., Giesselmann, J. and Benner, P. (2024), Data-driven snapshot calibration via monotonic feature matching, Finite Elem. Anal. Des 230, art. 104065.10.1016/j.finel.2023.104065CrossRefGoogle Scholar
Sawada, S., Heather, R., Jackson, B. and Metiu, H. (1985), A strategy for time dependent quantum mechanical calculations using a Gaussian wave packet representation of the wave function, J. Chem. Phys. 83, 3009–3027.10.1063/1.449204CrossRefGoogle Scholar
Schlegel, M. and Noack, B. R. (2015), On long-term boundedness of Galerkin models, J. Fluid Mech. 765, 325–352.10.1017/jfm.2014.736CrossRefGoogle Scholar
Schölkopf, B. and Smola, A. J. (2001), Learning with Kernels: Support Vector Machines, Regularization, Optimization, and Beyond , MIT Press.10.7551/mitpress/4175.001.0001CrossRefGoogle Scholar
Schulze, P. (2023), Energy-based model reduction of transport-dominated phenomena. Dissertation, Technische Universität Berlin.Google Scholar
Schulze, P., Reiss, J. and Mehrmann, V. (2019), Model reduction for a pulsed detonation combuster via shifted proper orthogonal decomposition, in Active Flow and Combustion Control 2018 (King, R., ed.), Springer, pp. 271–286.10.1007/978-3-319-98177-2_17CrossRefGoogle Scholar
Schulze, P., Unger, B., Beattie, C. and Gugercin, S. (2018), Data-driven structured realization, Linear Algebra Appl. 537, 250–286.10.1016/j.laa.2017.09.030CrossRefGoogle Scholar
Schwerdtner, P. and Peherstorfer, B. (2024), Greedy construction of quadratic manifolds for nonlinear dimensionality reduction and nonlinear model reduction. Available at arXiv:2403.06732.Google Scholar
Schwerdtner, P., Gugercin, S. and Peherstorfer, B. (2025a), Empirical sparse regression on quadratic manifolds, SIAM J. Sci. Comput. 47, A3085–A3107.10.1137/24M1717270CrossRefGoogle Scholar
Schwerdtner, P., Mohan, P., Pachalieva, A., Bessac, J., O’Malley, D. and Peherstorfer, B. (2025b), Online learning of quadratic manifolds from streaming data for nonlinear dimensionality reduction and nonlinear model reduction, Proc. R. Soc. A 481(2314), art. 20240670.10.1098/rspa.2024.0670CrossRefGoogle Scholar
Schwerdtner, P., Schulze, P., Berman, J. and Peherstorfer, B. (2024), Nonlinear embeddings for conserving Hamiltonians and other quantities with Neural Galerkin schemes, SIAM J. Sci. Comput. 46, C583–C607.10.1137/23M1607799CrossRefGoogle Scholar
Sesterhenn, J. and Shahirpour, A. (2019), A characteristic dynamic mode decomposition, Theor. Comput. Fluid Dyn. 33, 281–305.10.1007/s00162-019-00494-yCrossRefGoogle Scholar
Sharma, H., Mu, H., Buchfink, P., Geelen, R., Glas, S. and Kramer, B. (2023), Symplectic model reduction of Hamiltonian systems using data-driven quadratic manifolds, Comput. Methods Appl. Mech. Engrg 417, art. 116402.10.1016/j.cma.2023.116402CrossRefGoogle Scholar
Shimizu, Y. S. and Parish, E. J. (2021), Windowed space–time least-squares Petrov–Galerkin model order reduction for nonlinear dynamical systems, Comput. Methods Appl. Mech. Engrg 386, art. 114050.10.1016/j.cma.2021.114050CrossRefGoogle Scholar
Simoncini, V. (2016), Computational methods for linear matrix equations, SIAM Rev. 58, 377–441.10.1137/130912839CrossRefGoogle Scholar
Singh, R., Uy, W. I. T. and Peherstorfer, B. (2023), Lookahead data-gathering strategies for online adaptive model reduction of transport-dominated problems, Chaos 33, art. 113112.10.1063/5.0169392CrossRefGoogle ScholarPubMed
Sirovich, L. (1987), Turbulence and the dynamics of coherent structures 1: Coherent structures, Q. Appl. Math. 45, 561–571.10.1090/qam/910462CrossRefGoogle Scholar
Sitzmann, V., Martel, J., Bergman, A., Lindell, D. and Wetzstein, G. (2020), Implicit neural representations with periodic activation functions, in Advances in Neural Information Processing Systems 33 (Larochelle, H. et al., eds), Curran Associates, pp. 7462–7473.Google Scholar
Smetana, K. and Patera, A. T. (2016), Optimal local approximation spaces for component-based static condensation procedures, SIAM J. Sci. Comput. 38, A3318–A3356.10.1137/15M1009603CrossRefGoogle Scholar
Smetana, K. and Taddei, T. (2023), Localized model reduction for nonlinear elliptic partial differential equations: Localized training, partition of unity, and adaptive enrichment, SIAM J. Sci. Comput. 45, A1300–A1331.10.1137/22M148402XCrossRefGoogle Scholar
Taddei, T. (2020), A registration method for model order reduction: Data compression and geometry reduction, SIAM J. Sci. Comput. 42, A997–A1027.10.1137/19M1271270CrossRefGoogle Scholar
Taddei, T. and Zhang, L. (2021), Space–time registration-based model reduction of parameterized one-dimensional hyperbolic PDEs, ESAIM Math. Model. Numer. Anal. 55, 99–130.10.1051/m2an/2020073CrossRefGoogle Scholar
Taddei, T., Perotto, S. and Quarteroni, A. (2015), Reduced basis techniques for nonlinear conservation laws, ESAIM Math. Model. Numer. Anal. 49, 787–814.10.1051/m2an/2014054CrossRefGoogle Scholar
Taddei, T., Tifouti, I. and Barral, N. (2025), Local registration-based model reduction of parameterized PDEs with spatio-parameter adaptivity, in AIAA SCITECH 2025 Forum . Available at doi:10.2514/6.2025-1571.Google Scholar
Telgarsky, M. (2016), benefits of depth in neural networks, in 29th Annual Conference on Learning Theory (Feldman, V. et al., eds), Vol. 49 of Proceedings of Machine Learning Research, PMLR, pp. 1517–1539.Google Scholar
Tomasetto, M., Williams, J. P., Braghin, F., Manzoni, A. and Kutz, J. N. (2025), Reduced order modeling with shallow recurrent decoder networks, Nat. Commun. 16, art. 10260.10.1038/s41467-025-65126-yCrossRefGoogle ScholarPubMed
Torlo, D. (2025), Model reduction for advection dominated hyperbolic problems in an ALE framework: Offline, online phases and error estimator, in Numerical Mathematics and Advanced Applications (ENUMATH 2023) (Sequeira, A. et al., eds), Vol. 2, pp. 409–419.Google Scholar
Trefethen, L. N. (2019), Approximation Theory and Approximation Practice , SIAM.10.1137/1.9781611975949CrossRefGoogle Scholar
Unger, B. and Gugercin, S. (2019), Kolmogorov n-widths for linear dynamical systems, Adv. Comput. Math. 45, 2273–2286.10.1007/s10444-019-09701-0CrossRefGoogle Scholar
Urban, K. and Patera, A. T. (2012), A new error bound for reduced basis approximation of parabolic partial differential equations, C.R. Math. Acad. Sci. Paris 350, 203–207.10.1016/j.crma.2012.01.026CrossRefGoogle Scholar
Uy, W. I. T., Wentland, C. R., Huang, C. and Peherstorfer, B. (2024), Reduced models with nonlinear approximations of latent dynamics for model premixed flame problems, in Reduction, Approximation, Machine Learning, Surrogates, Emulators and Simulators: RAMSES (Rozza, G. et al., eds), Springer, pp. 241–259.10.1007/978-3-031-55060-7_10CrossRefGoogle Scholar
Van Gastelen, T., Edeling, W. and Sanderse, B. (2026), Modeling advection-dominated flows with space-local reduced-order models, Comput. Fluids 305, art. 106911.10.1016/j.compfluid.2025.106911CrossRefGoogle Scholar
Vapnik, V. (1991), Principles of risk minimization for learning theory, in Advances in Neural Information Processing Systems 4 (Moody, J. et al., eds), Morgan Kaufmann.Google Scholar
Veroy, K. and Patera, A. (2005), Certified real-time solution of the parametrized steady incompressible Navier–Stokes equations: Rigorous reduced-basis a posteriori error bounds, Internat. J. Numer. Methods Fluids 47, 773–788.10.1002/fld.867CrossRefGoogle Scholar
Washabaugh, K., Amsallem, D., Zahr, M. and Farhat, C. (2012), Nonlinear model reduction for CFD problems using local reduced-order bases, in 42nd AIAA Fluid Dynamics Conference and Exhibit . Available at doi:10.2514/6.2012-2686.CrossRefGoogle Scholar
Weder, P., Schwerdtner, P. and Peherstorfer, B. (2025), Nonlinear model reduction with Neural Galerkin schemes on quadratic manifolds, J. Comput. Phys. 539, art. 114249.10.1016/j.jcp.2025.114249CrossRefGoogle Scholar
Welper, G. (2017), Interpolation of functions with parameter dependent jumps by transformed snapshots, SIAM J. Sci. Comput. 39, A1225–A1250.10.1137/16M1059904CrossRefGoogle Scholar
Welper, G. (2020), Transformed snapshot interpolation with high resolution transforms, SIAM J. Sci. Comput. 42, A2037–A2061.10.1137/19M126356XCrossRefGoogle Scholar
Wen, Y., Vanden-Eijnden, E. and Peherstorfer, B. (2024), Coupling parameter and particle dynamics for adaptive sampling in neural Galerkin schemes, Phys. D 462, art. 134129.10.1016/j.physd.2024.134129CrossRefGoogle Scholar
Wentland, C. R., Duraisamy, K. and Huang, C. (2023), Scalable projection-based reduced-order models for large multiscale fluid systems, AIAA J. 61, 4499–4523.10.2514/1.J062869CrossRefGoogle Scholar
Wieland, B. (2015), Implicit partitioning methods for unknown parameter sets, Adv. Comput. Math. 41, 1159–1186.10.1007/s10444-015-9404-5CrossRefGoogle Scholar
Wittmuess, P., Tarin, C., Keck, A., Arnold, E. and Sawodny, O. (2016), Parametric model order reduction via balanced truncation with Taylor series representation, IEEE Trans. Automat. Control 61, 3438–3451.10.1109/TAC.2016.2521361CrossRefGoogle Scholar
Wu, C., Zhu, M., Tan, Q., Kartha, Y. and Lu, L. (2023), A comprehensive study of non-adaptive and residual-based adaptive sampling for physics-informed neural networks, Comput. Methods Appl. Mech. Engrg 403, art. 115671.10.1016/j.cma.2022.115671CrossRefGoogle Scholar
Xu, J. and Duraisamy, K. (2020), Multi-level convolutional autoencoder networks for parametric prediction of spatio-temporal dynamics, Comput. Methods Appl. Mech. Engrg 372, art. 113379.10.1016/j.cma.2020.113379CrossRefGoogle Scholar
Yıldız, S., Goyal, P., Bendokat, T. and Benner, P. (2024), Data-driven identification of quadratic representations for nonlinear Hamiltonian systems using weakly symplectic liftings, J. Mach. Learn. Model. Comput. 5, 45–71.10.1615/JMachLearnModelComput.2024052810CrossRefGoogle Scholar
Zahr, M. J. and Farhat, C. (2015), Progressive construction of a parametric reduced-order model for PDE-constrained optimization, Internat. J. Numer. Methods Engrg 102, 1111–1135.10.1002/nme.4770CrossRefGoogle Scholar
Zhang, H., Chen, Y., Vanden-Eijnden, E. and Peherstorfer, B. (2024), Sequential-in-time training of nonlinear parametrizations for solving time-dependent partial differential equations. Available at arXiv:2404.01145. To appear in SIAM Rev.Google Scholar
Zhao, T., Sun, C., Cohen, A., Stokes, J. and Veerapaneni, S. (2024), Quantum-inspired variational algorithms for partial differential equations: Application to financial derivative pricing, Quant. Finance 24, 1–11.10.1080/14697688.2023.2259954CrossRefGoogle Scholar
Zimmermann, R. and Willcox, K. (2016), An accelerated greedy missing point estimation procedure, SIAM J. Sci. Comput. 38, A2827–A2850.10.1137/15M1042899CrossRefGoogle Scholar
Zimmermann, R., Peherstorfer, B. and Willcox, K. (2018), Geometric subspace updates with applications to online adaptive nonlinear model reduction, SIAM J. Matrix Anal. Appl. 39, 234–261.10.1137/17M1123286CrossRefGoogle Scholar
Zucatti, V. and Zahr, M. J. (2024), An adaptive, training-free reduced-order model for convection-dominated problems based on hybrid snapshots, Internat. J. Numer. Methods Fluids 96, 189–208.10.1002/fld.5240CrossRefGoogle Scholar
Zuyev, A., Feng, L. and Benner, P. (2024), Estimates of the Kolmogorov n-width for nonlinear transformations with application to distributed-parameter control systems, IEEE Control Systems Lett. 8, 1877–1882.10.1109/LCSYS.2024.3418314CrossRefGoogle Scholar