{ "id": "ep38ehd3e6", "slug": "reduced-order-modeling", "title": "Reduced-order modeling", "updated": "2026-10-15", "topic_path": [ { "id": "technology", "label": "Technology and the built world", "api_url": "https://www.edgepedia.org/api/v1/topics/technology" }, { "id": "technology.computing", "label": "Computing and digital systems", "api_url": "https://www.edgepedia.org/api/v1/topics/technology.computing" }, { "id": "technology.computing.ai", "label": "Artificial intelligence and data", "api_url": "https://www.edgepedia.org/api/v1/topics/technology.computing.ai" }, { "id": "technology.computing.ai.algorithms", "label": "Algorithms and computational methods", "api_url": "https://www.edgepedia.org/api/v1/topics/technology.computing.ai.algorithms" }, { "id": "technology.computing.ai.numerical_string_geometry", "label": "Numerical, string, and geometric algorithms", "api_url": "https://www.edgepedia.org/api/v1/topics/technology.computing.ai.numerical_string_geometry" }, { "id": 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"geo.nongeo.t1946.technology.computing.ai.numerical_string_geometry.numerical_analysis_methods", "label": "Numerical methods and approximation", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.technology.computing.ai.numerical_string_geometry.numerical_analysis_methods" } ] }, { "id": "geo.nongeo.t2001.technology.computing.ai.numerical_string_geometry", "label": "Non-geographic · 2001 to 2020: Numerical, string, and geometric algorithms", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t2001.technology.computing.ai.numerical_string_geometry", "path": [ { "id": "geo.nongeo", "label": "Non-geographic", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo" }, { "id": "geo.nongeo.t2001", "label": "Non-geographic · 2001 to 2020", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t2001" }, { "id": "geo.nongeo.t2001.technology", "label": "Technology and the built world", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t2001.technology" }, { "id": "geo.nongeo.t2001.technology.computing", "label": "Computing and digital systems", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t2001.technology.computing" }, { "id": "geo.nongeo.t2001.technology.computing.ai", "label": "Artificial intelligence and data", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t2001.technology.computing.ai" }, { "id": "geo.nongeo.t2001.technology.computing.ai.algorithms", "label": "Algorithms and computational methods", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t2001.technology.computing.ai.algorithms" }, { "id": "geo.nongeo.t2001.technology.computing.ai.numerical_string_geometry", "label": "Numerical, string, and geometric algorithms", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t2001.technology.computing.ai.numerical_string_geometry" } ] } ], "excerpt": "Reduced-order modeling (ROM) builds low-dimensional surrogates of high-dimensional simulations of complex systems, running orders of magnitude faster than the full-order model while retaining most of its accuracy, most commonly by POD-Galerkin projection.", "snippet": "Reduced-order modeling (ROM) builds low-dimensional surrogates of high-dimensional simulations of complex systems, running orders of magnitude faster than the full-order model while retaining most of its accuracy, most commonly by POD-Galerkin projection.", "node": "technology.computing.ai.numerical_string_geometry.numerical_analysis_methods", "markdown": "# Reduced-order modeling\n\nA reduced-order model (ROM) is a low-dimensional surrogate of a high-dimensional simulation of a complex system, built so that predictions, control calculations, and design studies run orders of magnitude faster than the full-order model (FOM) while retaining most of its accuracy. The motivation is cost: the number of degrees of freedom of a turbulent flow is estimated as \\( N \\sim Re^{9/4} \\), so direct simulation at high [Reynolds number](https://www.edgepedia.org/reynolds-number) is prohibitive for tasks requiring many repeated evaluations.[1](https://www.sandia.gov/app/uploads/sites/127/2023/03/SAND_Report_ROM_2014.pdf) • [2](https://sites.me.ucsb.edu/~moehlis/moehlis_papers/tutorial_final.pdf) Parametric ROMs characterize system response across parameter values for design, control, optimization, and uncertainty quantification.[3](https://dspace.mit.edu/bitstream/handle/1721.1/100939/Benner-2015-Survey%20of%20projection-based.pdf;jsessionid=5E0B593E87F281D7E715DE863C34C5D6?sequence=1)\n\n| Key fact | Value |\n|---|---|\n| Purpose | Fast surrogate for design, control, optimization, uncertainty quantification[3](https://dspace.mit.edu/bitstream/handle/1721.1/100939/Benner-2015-Survey%20of%20projection-based.pdf;jsessionid=5E0B593E87F281D7E715DE863C34C5D6?sequence=1) |\n| Typical online speedup | Up to \\( 10^{5} \\) for some fluid problems[4](https://arxiv.org/html/2406.00559v2); 5 h FOM vs 1 s ROM in a medical-device flow[5](https://link.springer.com/article/10.1007/s40314-025-03344-2) |\n| Most established method | POD-Galerkin projection[6](https://deca.upc.edu/en/people/ramon.codina/publications-1/monographs-and-book-chapter/chapweb11.pdf) |\n| Two construction families | Intrusive (manipulates governing equations) vs non-intrusive (data only)[4](https://arxiv.org/html/2406.00559v2) |\n| Nonlinear-term cost | Removed by hyper-reduction (EIM, DEIM, gappy POD, ECSW)[7](https://web.stanford.edu/group/frg/course_work/CME345/CA-AA216-CME345-Ch9.pdf) |\n| Main failure modes | Convection-dominated flows, stability loss, extrapolation outside training data[5](https://link.springer.com/article/10.1007/s40314-025-03344-2) • [8](https://www.annualreviews.org/content/journals/10.1146/annurev-fluid-010816-060042) |\n\n## How it works\n\nThe core idea is projection. A set of snapshots, solutions of the FOM for chosen parameters or times, is collected into a matrix, and the leading left singular vectors of that snapshot matrix (equivalently, the eigenvectors of its [correlation matrix](https://www.edgepedia.org/correlation-matrix)) are computed to form an orthonormal POD basis; a rank-\\( r \\) truncation of the singular value decomposition gives the best rank-\\( r \\) approximation of the snapshot matrix in the least-squares sense; this is the proper orthogonal decomposition (POD), also known as principal component analysis or the Karhunen-Loève expansion.[9](https://ar5iv.labs.arxiv.org/html/1911.08954) The state is approximated with a reduced variable \\( y \\) of small dimension, and the governing equations are projected onto the span of \\( V \\). If the residual is tested against the same functions used in the ansatz, the method is Galerkin; if the test space differs, it is Petrov-Galerkin.[9](https://ar5iv.labs.arxiv.org/html/1911.08954) • [8](https://www.annualreviews.org/content/journals/10.1146/annurev-fluid-010816-060042) Among all linear decompositions, POD captures on average the most kinetic energy possible for a projection on a given number of modes.[10](http://www.sfu.ca/~rwwitten/papers/HolmesLBMW_physrep1997.pdf)\n\nFor nonlinear problems, evaluating the nonlinear term still costs \\( O(N) \\) per step, so hyper-reduction is required. DEIM approximates the nonlinear function as an oblique projection, \\( \\tilde{f}(\\cdot,t) = V_{f}(P^{T}V_{f})^{-1}P^{T}f(\\cdot,t) \\), so the online cost no longer scales with the size \\( N \\) of the high-dimensional model; the trade-off is that the online phase is software-intrusive.[7](https://web.stanford.edu/group/frg/course_work/CME345/CA-AA216-CME345-Ch9.pdf)\n\n## How it is done\n\nROM construction is split into an offline stage, where high-fidelity simulations for selected parameter values are computed once and the low-dimensional representation is extracted, and an online stage where solutions for new parameters are predicted efficiently; intrusive ROMs require access to the governing equations and code, while non-intrusive ROMs are purely data-driven.[4](https://arxiv.org/html/2406.00559v2) • [6](https://deca.upc.edu/en/people/ramon.codina/publications-1/monographs-and-book-chapter/chapweb11.pdf) The basis can be built by POD, whose computation scales as \\( O(N \\cdot N_{\\delta}^{2}) \\), or by the greedy algorithm, which adds one basis function per iteration where an error estimator attains its maximum, requiring one FOM solution per iteration; convergence is exponential when the solution set has an exponentially small Kolmogorov \\( N \\)-width.[11](https://iris.sissa.it/retrieve/5d2da36a-d26a-435c-9123-b48972881153/2407.03325v2.pdf) • [9](https://ar5iv.labs.arxiv.org/html/1911.08954) Affine parameter dependency lets all computations that depend on the model size be moved into the offline phase, with the online solve costing independently of the discretization size \\( N_{h} \\), fast enough for mobile and embedded devices; non-affine problems are handled by the empirical interpolation method.[9](https://ar5iv.labs.arxiv.org/html/1911.08954) A data-driven alternative, Operator Inference, follows three steps: snapshot generation, POD basis construction, and regression that learns reduced-order matrices from projected snapshot data.[12](https://www.osti.gov/pages/biblio/2580095)\n\n## Origin\n\n[Dynamic mode decomposition](https://www.edgepedia.org/dynamic-mode-decomposition) (DMD) was reported by Peter J. Schmid in 2010 in the Journal of Fluid Mechanics as a method to extract dynamic information from flow fields from simulation or experiment.[13](https://doi.org/10.1017/s0022112010001217) A data-driven approximation of the Koopman operator extending DMD was reported by Matthew O. Williams, Ioannis G. Kevrekidis, and Clarence W. Rowley in 2015 in the Journal of Nonlinear Science.[14](https://doi.org/10.1007/s00332-015-9258-5) Optimal mode decomposition for unsteady flows was reported by A. Wynn and colleagues in 2013 in the Journal of Fluid Mechanics.[15](https://doi.org/10.1017/jfm.2013.426) A characteristic dynamic mode decomposition was reported by Jörn Sesterhenn and Amir Shahirpour in 2019 in Theoretical and Computational Fluid Dynamics.[16](https://doi.org/10.1007/s00162-019-00494-y)\n\nBalanced proper orthogonal decomposition (BPOD) was reported by C. W. Rowley in 2005 in the International Journal of Bifurcation and Chaos.[17](https://doi.org/10.1142/s0218127405012429) An energy-based inner product for stable Galerkin projection of compressible flows was reported by Clarence W. Rowley, Tim Colonius, and Richard M. Murray in 2003 in Physica D.[18](https://doi.org/10.1016/j.physd.2003.03.001) Gappy POD was reported by R. Everson and L. Sirovich in 1995 in the Journal of the Optical Society of America A as a Karhunen-Loève procedure for gappy data.[19](https://doi.org/10.1364/josaa.12.001657) The empirical interpolation method was reported by Maxime Barrault and colleagues in 2004 in Comptes Rendus Mathématique.[20](https://doi.org/10.1016/j.crma.2004.08.006) DEIM was reported by Saifon Chaturantabut and Danny C. Sorensen in 2010 in the SIAM Journal on Scientific Computing.[21](https://doi.org/10.1137/090766498) Operator Inference was reported by Benjamin Peherstorfer and [Karen Willcox](https://www.edgepedia.org/karen-willcox) in 2016 in Computer Methods in Applied Mechanics and Engineering.[22](https://doi.org/10.1016/j.cma.2016.03.025) Linearly recurrent autoencoder networks for learning dynamics were reported by Samuel E. Otto and Clarence W. Rowley in 2019 in the SIAM Journal on Applied Dynamical Systems.[23](https://doi.org/10.1137/18m1177846) RONOM (Reduced-Order Neural Operator Modeling) was reported by Sven Dummer, Dongwei Ye, and Christoph Brune in 2026.[24](https://doi.org/10.1137/25m1777700) POD itself and balanced truncation long predate these publications; reviews describe POD as the Karhunen-Loève decomposition or principal component analysis, with roots in nineteenth-century matrix diagonalization related to the SVD, and balanced truncation as a control-theoretic procedure.[8](https://www.annualreviews.org/content/journals/10.1146/annurev-fluid-010816-060042) • [25](https://people.cs.vt.edu/~asandu/Public/Qual2011/Approx/Antoulas_2007_POD.pdf)\n\n## Variants\n\n**POD-Galerkin** is considered the most well established and commonly used ROM method.[6](https://deca.upc.edu/en/people/ramon.codina/publications-1/monographs-and-book-chapter/chapweb11.pdf) **Balanced truncation** trades off controllability and observability and carries a priori error bounds close to the minimum achievable, but requires dense Gramians. BPOD approximates it with empirical Gramians from simulation data at cost similar to POD; the [eigensystem realization algorithm](https://www.edgepedia.org/eigensystem-realization-algorithm) (ERA) produces models equivalent to BPOD without adjoint responses, enabling use on experimental data.[8](https://www.annualreviews.org/content/journals/10.1146/annurev-fluid-010816-060042) • [17](https://doi.org/10.1142/s0218127405012429) • [26](https://ar5iv.labs.arxiv.org/html/1702.01453)\n\n**DMD** is a factorization and dimensionality reduction technique for data sequences that extracts coherent structures and reduces complex evolution to dominant features; it is purely data-driven, whereas POD and BPOD projection models require the governing equations.[27](https://www.annualreviews.org/content/journals/10.1146/annurev-fluid-030121-015835) • [8](https://www.annualreviews.org/content/journals/10.1146/annurev-fluid-010816-060042) The Koopman operator, an infinite-dimensional linear operator that completely characterizes nonlinear dynamics, extends DMD to nonlinear systems.[8](https://www.annualreviews.org/content/journals/10.1146/annurev-fluid-010816-060042) **Hyper-reduction** methods include EIM, DEIM (a greedy algorithm traceable to gappy POD, originally designed for image reconstruction), the missing point estimator, cubature-based approximation, energy-conserving sampling and weighting (ECSW), and AMR-based hyperreduction.[6](https://deca.upc.edu/en/people/ramon.codina/publications-1/monographs-and-book-chapter/chapweb11.pdf) **Neural approaches** include autoencoders and operator learning: DeepONet pairs a branch net encoding input functions at fixed sensor points with a trunk net encoding output coordinates, and the [Fourier neural operator](https://www.edgepedia.org/fourier-neural-operator) (FNO) parameterizes the integral kernel in Fourier space with FFT-based layers.[28](https://www.nature.com/articles/s41467-024-49411-w)\n\n## Applications\n\nParametric reduced models for aircraft aeroelasticity enable rapid characterization of the flight envelope, calculations that would otherwise require many weeks of computation time.[3](https://dspace.mit.edu/bitstream/handle/1721.1/100939/Benner-2015-Survey%20of%20projection-based.pdf;jsessionid=5E0B593E87F281D7E715DE863C34C5D6?sequence=1) In structural dynamics, ECSW with a POD-based model of dimension \\( k = 100 \\) (from \\( N = 1{,}399{,}056 \\)) delivered a speedup factor of 28,935 at 96% relative accuracy for an air-blast V-hull case.[7](https://web.stanford.edu/group/frg/course_work/CME345/CA-AA216-CME345-Ch9.pdf) In biomedical flows, a patient-specific coronary arteries case reduced the FOM simulation of about \\( 1.8 \\times 10^{5} \\) s to about 9 s online with local POD-RBF, a speedup of about \\( 10^{4} \\), while cutting mean error from 17-18% to 12% versus global POD-RBF.[11](https://iris.sissa.it/retrieve/5d2da36a-d26a-435c-9123-b48972881153/2407.03325v2.pdf) DMD has been applied beyond fluids to video surveillance, epidemiology, neurobiology, and financial engineering.[27](https://www.annualreviews.org/content/journals/10.1146/annurev-fluid-030121-015835)\n\n## Limitations and alternatives\n\nSpeedups are large but condition-dependent. ROMs are very efficient when \\( O(10) \\) basis functions suffice, typical of diffusion-dominated flows; convection-dominated flows need many more modes, degrading efficiency.[5](https://link.springer.com/article/10.1007/s40314-025-03344-2) In a linearized channel flow, the first three POD modes contain 99.65% of the data energy, yet POD-truncation models are quite inaccurate, while balanced truncation, BPOD, and DMD/ERA give very accurate order-three models; POD is usually not optimal for Galerkin projection because low-energy states can strongly influence dynamics, and it performs particularly poorly for non-normal systems with large transient growth, as in shear flows.[8](https://www.annualreviews.org/content/journals/10.1146/annurev-fluid-010816-060042)\n\n**Stability** is a recurring failure mode. POD and BPOD ROMs lack an a priori stability guarantee in general, a real problem in compressible and high-Reynolds-number flows; a compressible POD/Galerkin ROM can be stable for one number of modes but unstable for another. For asymptotically stable linear systems, balanced truncation preserves asymptotic stability in the reduced-order model, but classical dense-Gramian implementations scale poorly with system size; computational feasibility depends on the system and the algorithm, and low-rank techniques extend balanced truncation to large-scale systems beyond the reach of dense methods. It is generally limited to linear problems.[1](https://www.sandia.gov/app/uploads/sites/127/2023/03/SAND_Report_ROM_2014.pdf) ROMs inconsistent with the FOM's discretization of the nonlinearity carry extra error terms and can lock, failing to converge to the FOM as modes increase.[29](https://export.arxiv.org/pdf/2111.06749v2.pdf) The relative speedups of intrusive and non-intrusive ROMs depend on the methods and problem, and error-estimation methods exist only for some non-intrusive ROMs, with availability and guarantees depending on the method and assumptions.[4](https://arxiv.org/html/2406.00559v2) • [30](https://link.springer.com/article/10.1007/s00158-022-03282-1) • [31](https://www.numdam.org/articles/10.1051/m2an/2021010/) L-DeepONet can interpolate in time but not in space, and physics-informed learning is difficult because governing equations are unknown in the latent space.[28](https://www.nature.com/articles/s41467-024-49411-w)\n\n## References\n\n1. [Reduced Order Modeling for Prediction and Control of Large-Scale Systems (Sandia report SAND2014)](https://www.sandia.gov/app/uploads/sites/127/2023/03/SAND_Report_ROM_2014.pdf)\n2. [Low-dimensional Modelling of Turbulence Using the Proper Orthogonal Decomposition: A Tutorial (Smith, Moehlis, Holmes)](https://sites.me.ucsb.edu/~moehlis/moehlis_papers/tutorial_final.pdf)\n3. [A Survey of Projection-Based Model Reduction Methods for Parametric Dynamical Systems (Benner et al.)](https://dspace.mit.edu/bitstream/handle/1721.1/100939/Benner-2015-Survey%20of%20projection-based.pdf;jsessionid=5E0B593E87F281D7E715DE863C34C5D6?sequence=1)\n4. [A brief review of Reduced Order Models using intrusive and non-intrusive techniques (2024)](https://arxiv.org/html/2406.00559v2)\n5. [Stabilized POD reduced order models for convection-dominated incompressible flows (Computational and Applied Mathematics, 2025)](https://link.springer.com/article/10.1007/s40314-025-03344-2)\n6. [Reduced Order Modeling (book chapter, Codina et al.)](https://deca.upc.edu/en/people/ramon.codina/publications-1/monographs-and-book-chapter/chapweb11.pdf)\n7. [AA216/CME345: Hyperreduction of Projection-Based Reduced-Order Models (Stanford course notes)](https://web.stanford.edu/group/frg/course_work/CME345/CA-AA216-CME345-Ch9.pdf)\n8. [Model Reduction for Flow Analysis and Control (Annual Review of Fluid Mechanics)](https://www.annualreviews.org/content/journals/10.1146/annurev-fluid-010816-060042)\n9. [Basic Ideas and Tools for Projection-Based Model Reduction of Parametric Partial Differential Equations](https://ar5iv.labs.arxiv.org/html/1911.08954)\n10. [Turbulence, Coherent Structures, Dynamical Systems and Symmetry (Holmes et al., Physics Reports 1997)](http://www.sfu.ca/~rwwitten/papers/HolmesLBMW_physrep1997.pdf)\n11. [On the accuracy and efficiency of reduced order models: Towards real-world applications (Siena et al., Advances in Applied Mechanics, 2024)](https://iris.sissa.it/retrieve/5d2da36a-d26a-435c-9123-b48972881153/2407.03325v2.pdf)\n12. [Learning Nonlinear Reduced Models from Data with Operator Inference (Annual Review of Fluid Mechanics, 2024)](https://www.osti.gov/pages/biblio/2580095)\n13. [PETER J. SCHMID (2010). Dynamic mode decomposition of numerical and experimental data. Journal of Fluid Mechanics.](https://doi.org/10.1017/s0022112010001217)\n14. [Matthew O. Williams, Ioannis G. Kevrekidis, Clarence W. Rowley (2015). A Data–Driven Approximation of the Koopman Operator: Extending Dynamic Mode Decomposition. Journal of Nonlinear Science.](https://doi.org/10.1007/s00332-015-9258-5)\n15. [A. Wynn and colleagues (2013). Optimal mode decomposition for unsteady flows. Journal of Fluid Mechanics.](https://doi.org/10.1017/jfm.2013.426)\n16. [Jörn Sesterhenn, Amir Shahirpour (2019). A characteristic dynamic mode decomposition. Theoretical and Computational Fluid Dynamics.](https://doi.org/10.1007/s00162-019-00494-y)\n17. [C. W. ROWLEY (2005). MODEL REDUCTION FOR FLUIDS, USING BALANCED PROPER ORTHOGONAL DECOMPOSITION. International Journal of Bifurcation and Chaos.](https://doi.org/10.1142/s0218127405012429)\n18. [Clarence W. Rowley, Tim Colonius, Richard M. Murray (2003). Model reduction for compressible flows using POD and Galerkin projection. Physica D Nonlinear Phenomena.](https://doi.org/10.1016/j.physd.2003.03.001)\n19. [R. Everson, L. Sirovich (1995). Karhunen–Loève procedure for gappy data. Journal of the Optical Society of America A.](https://doi.org/10.1364/josaa.12.001657)\n20. [Maxime Barrault and colleagues (2004). An ‘empirical interpolation’ method: application to efficient reduced-basis discretization of partial differential equations. Comptes Rendus Mathématique.](https://doi.org/10.1016/j.crma.2004.08.006)\n21. [Saifon Chaturantabut, Danny C. Sorensen (2010). Nonlinear Model Reduction via Discrete Empirical Interpolation. SIAM Journal on Scientific Computing.](https://doi.org/10.1137/090766498)\n22. [Benjamin Peherstorfer, Karen Willcox (2016). Data-driven operator inference for nonintrusive projection-based model reduction. Computer Methods in Applied Mechanics and Engineering.](https://doi.org/10.1016/j.cma.2016.03.025)\n23. [Samuel E. Otto, Clarence W. Rowley (2019). Linearly Recurrent Autoencoder Networks for Learning Dynamics. SIAM Journal on Applied Dynamical Systems.](https://doi.org/10.1137/18m1177846)\n24. [Dummer, Sven; id_orcid 0000-0002-5276-4155, Ye, Dongwei; id_orcid 0000-0002-4903-1487, Brune, Christoph; id_orcid 0000-0003-0145-5069 (2026). RONOM:Reduced-Order Neural Operator Modeling. University of Twente Research Information.](https://doi.org/10.1137/25m1777700)\n25. [A survey of model reduction methods for large-scale systems (Antoulas)](https://people.cs.vt.edu/~asandu/Public/Qual2011/Approx/Antoulas_2007_POD.pdf)\n26. [Modal Analysis of Fluid Flows: An Overview (Taira et al.)](https://ar5iv.labs.arxiv.org/html/1702.01453)\n27. [Dynamic Mode Decomposition and Its Variants (Annual Review of Fluid Mechanics)](https://www.annualreviews.org/content/journals/10.1146/annurev-fluid-030121-015835)\n28. [Learning nonlinear operators in latent spaces for real-time predictions of complex dynamics in physical systems (Nature Communications, 2024)](https://www.nature.com/articles/s41467-024-49411-w)\n29. [FOM-ROM consistency study (arXiv 2111.06749v2)](https://export.arxiv.org/pdf/2111.06749v2.pdf)\n30. [Client Challenge](https://link.springer.com/article/10.1007/s00158-022-03282-1)\n31. [numdam.org](https://www.numdam.org/articles/10.1051/m2an/2021010/)\n\n---\n*Topic: Encyclopedia › Technology and the built world › Computing and digital systems › Artificial intelligence and data › Algorithms and computational methods › Numerical, string, and geometric algorithms › Numerical methods and approximation*\n\n*Initially written Sep 29, 2026 · Edited: Oct 15, 2026*\n\n*Copyright 2026 EdgeChat AI, a subsidiary of Biostate AI.*\n\nLicense: Edgepedia Community License 1.0, https://www.edgepedia.org/edgepedia/license\n", "same_as": [ "https://sites.me.ucsb.edu/~moehlis/moehlis_papers/tutorial_final.pdf", "https://deca.upc.edu/en/people/ramon.codina/publications-1/monographs-and-book-chapter/chapweb11.pdf", "https://people.cs.vt.edu/~asandu/Public/Qual2011/Approx/Antoulas_2007_POD.pdf" ], "url": "https://www.edgepedia.org/reduced-order-modeling", "markdown_url": "https://www.edgepedia.org/reduced-order-modeling.md", "license": { "name": "Edgepedia Community License 1.0", "url": "https://www.edgepedia.org/edgepedia/license", "summary": "Free with credit, commercial use included. AI training is open to everyone. For other uses, organizations over USD 100M in revenue or 100M monthly users license separately.", "spdx": "LicenseRef-Edgepedia-Community-1.0" }, "credit": "\"Reduced-order modeling\", Edgepedia (EdgeChat), https://www.edgepedia.org/reduced-order-modeling. Edgepedia Community License 1.0.", "credit_md": "\"[Reduced-order modeling](https://www.edgepedia.org/reduced-order-modeling)\", Edgepedia (EdgeChat), [https://www.edgepedia.org/reduced-order-modeling](https://www.edgepedia.org/reduced-order-modeling). [Edgepedia Community License 1.0](https://www.edgepedia.org/edgepedia/license).", "credit_html": "\"Reduced-order modeling\", Edgepedia (EdgeChat), https://www.edgepedia.org/reduced-order-modeling. Edgepedia Community License 1.0.", "speakable": "Reduced-order modeling builds low-dimensional surrogates of high-dimensional simulations of complex systems, running orders of magnitude faster than the full-order model while retaining most of its accuracy, most commonly by POD-Galerkin projection." }