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arxiv: 2605.28684 · v1 · pith:3TZKJRNVnew · submitted 2026-05-27 · 💻 cs.LG · cs.CE· cs.NA· math.NA· physics.comp-ph

History-aware adaptive reduced-order models via incremental singular value decomposition

Amirpasha Hedayat , Ali Mohaghegh , Laura Balzano , Cheng Huang , Karthik Duraisamy This is my paper

Pith reviewed 2026-06-29 13:38 UTC · model grok-4.3

classification 💻 cs.LG cs.CEcs.NAmath.NAphysics.comp-ph
keywords reduced-order modelsincremental singular value decompositionadaptive basishistory-aware adaptationrotating detonation engineBurgers equationSod shock tubeonline learning
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The pith

Incremental singular value decomposition enables history-aware adaptive reduced-order models that update bases online from full-order snapshots.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Reduced-order models lose accuracy when dynamics move outside the training regime. This paper introduces a projection-based adaptive framework that employs incremental singular value decomposition to incorporate occasional full-order operator evaluations as correction snapshots. The iSVD update propagates automatically to reduced operators and hyper-reduction, while its evolving singular structure encodes a history of observed dynamics. On the one-dimensional Burgers equation, history-aware iSVD updates outperform instantaneous alternatives; the same pattern holds for the Sod shock tube and a stiff ten-species rotating detonation engine. For the RDE case the approach improves both predictive accuracy and computational efficiency over the direct adaptive ROM baseline, with the dominant cost arising from snapshot acquisition rather than the update step itself.

Core claim

The iSVD adaptive ROM framework updates the reduced basis online using correction snapshots from full-order evaluations, naturally propagating changes through the basis to reduced operators and hyper-reduction machinery, while retaining an encoded history of the observed dynamics through its evolving singular structure; this yields stronger performance than non-history-aware adaptation on nonlinear problems of increasing complexity including the RDE.

What carries the argument

Incremental singular value decomposition (iSVD) for online basis adaptation, which encodes prior dynamics in its singular structure and automatically updates all reduced operators.

If this is right

  • History-aware iSVD updates outperform instantaneous basis updates on the Burgers equation.
  • The performance advantage persists through the compressible Sod shock tube and RDE cases.
  • On the RDE problem the iSVD ROM exceeds the direct adaptive ROM baseline in both accuracy and efficiency.
  • The iSVD update step itself is negligible in cost; the dominant online expense is acquiring correction snapshots.
  • The method supports ROMs that remain predictive over horizons orders of magnitude longer than the initial training window.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The history-encoding property of iSVD could be paired with error estimators to trigger snapshots only when needed.
  • The same incremental mechanism might transfer to other matrix-factorization-based adaptation schemes in dynamical systems.
  • Longer predictive horizons open the possibility of using these ROMs inside real-time optimization loops without periodic full retraining.

Load-bearing premise

Occasional full-order evaluations can be obtained at a frequency sufficient to keep the adapted basis accurate while their cost remains lower than the savings from using the ROM.

What would settle it

A controlled RDE run in which the frequency of full-order snapshot corrections is successively lowered until the iSVD ROM error exceeds that of the direct adaptive ROM baseline would falsify the claimed efficiency gain.

Figures

Figures reproduced from arXiv: 2605.28684 by Ali Mohaghegh, Amirpasha Hedayat, Cheng Huang, Karthik Duraisamy, Laura Balzano.

Figure 1
Figure 1. Figure 1: Schematic of the lookahead correction signal mechanism used in the proposed adaptive ROM framework. The ROM advances on the fine time grid, while a separate coarse-time-step full-order trajectory is evolved independently using step size 𝑧Δ𝑡. At each adaptation event, the coarse FOM provides a lookahead snapshot approximating the system state 𝑧 fine steps into the future, which is then used as the correctio… view at source ↗
Figure 2
Figure 2. Figure 2: Static LSPG–QDEIM results for the Burgers problem. For all models, the training interval is 𝑡 ∈ [0, 𝑤initΔ𝑡], and the testing interval is 𝑡 ∈ [𝑤initΔ𝑡, 0.5], with Δ𝑡 = 10−3 . we keep the ROM itself small but simply provide it with more offline data, to see if a static basis becomes sufficiently predictive. Figure 2b depicts the results of this experiment. As the training window is extended, the static ROM … view at source ↗
Figure 3
Figure 3. Figure 3: Relative error of the static LSPG–QDEIM ROM for the Burgers problem. For all models, the training interval is 𝑡 ∈ [0, 𝑤initΔ𝑡], and the testing interval is 𝑡 ∈ [𝑤initΔ𝑡, 0.5], with Δ𝑡 = 10−3 [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Adaptive LSPG–QDEIM solution profiles for the Burgers problem at different adaptation windows 𝑧. For each 𝑧, each method is shown with its best-performing method-specific hyperparameter. All ROMs are trained on the interval 𝑡 ∈ [0, 0.004], and tested over 𝑡 ∈ [0.004, 0.5]. history-aware methods, where they preserve a subspace that continues to represent the physically relevant manifold of the solution, whe… view at source ↗
Figure 5
Figure 5. Figure 5: Relative error histories of the Burgers adaptive ROMs for different adaptation windows 𝑧. For each 𝑧, each method is shown with its best-performing method-specific hyperparameter. All ROMs are trained on the interval 𝑡 ∈ [0, 0.004], and tested over 𝑡 ∈ [0.004, 0.5]. preserving the most relevant information accumulated from the past dynamics. Windowed SVD and the Direct method also perform significantly bet… view at source ↗
Figure 6
Figure 6. Figure 6: Effect of the forgetting factor 𝜆 on the iSVD adaptive ROM for the Burgers problem at fixed 𝑧 = 10. All ROMs are trained on the interval 𝑡 ∈ [0, 0.004], and tested over 𝑡 ∈ [0.004, 0.5]. aggressive forgetting (smaller 𝜆) is clearly beneficial. This behavior is physically intuitive, as the Burgers dynamics move rapidly away from the minimal initial training manifold, and so the basis must adapt quickly to n… view at source ↗
Figure 7
Figure 7. Figure 7: Comparison of the correction signal error and the iSVD adaptive ROM error for the Burgers problem over different adaptation windows 𝑧. All ROMs are trained on the interval 𝑡 ∈ [0, 0.004], and tested over 𝑡 ∈ [0.004, 0.5]. This figure shows that the adaptive ROM remains more accurate than the signal that informs its basis updates [PITH_FULL_IMAGE:figures/full_fig_p021_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Sod shock tube solution profiles for adaptation window 𝑧 = 10. All ROMs are trained on the interval 𝑡 ∈ [0, 0.001], and tested over 𝑡 ∈ [0.001, 0.125]. the prediction horizon. Both the Direct adaptive ROM and the proposed iSVD adaptive ROM capture the dominant wave pattern well. Between these two, the iSVD-based ROM is consistently the more faithful approximation. This is particularly visible at later time… view at source ↗
Figure 9
Figure 9. Figure 9: Relative error histories for the Sod shock tube at adaptation window 𝑧 = 10. All ROMs are trained on the interval 𝑡 ∈ [0, 0.001], and tested over 𝑡 ∈ [0.001, 0.125] [PITH_FULL_IMAGE:figures/full_fig_p024_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Solution profiles for the one-dimensional RDE case within its transient phase. Both ROMs are trained on the interval 𝑡 ∈ [0, 5 × 10−6 ms], and tested over 𝑡 ∈ [5 × 10−6 ms, 1.5 ms]. 5.3.3. Runtime and acceleration In addition to accuracy, the RDE case provides a meaningful test of computational efficiency. Because this is the most demanding example in the paper, it is also the most compelling place to rep… view at source ↗
Figure 11
Figure 11. Figure 11: Relative error histories for the one-dimensional RDE case within its transient phase. Both ROMs are trained on the interval 𝑡 ∈ [0, 5 × 10−6 ms], and tested over 𝑡 ∈ [5 × 10−6 ms, 1.5 ms]. but also substantially faster. A major reason for this acceleration advantage is the different update frequency used by the two adaptive strategies in this RDE setting. Following the reference Direct adaptive ROM implem… view at source ↗
Figure 12
Figure 12. Figure 12: Spacetime diagram for density in the RDE case. Both ROMs are trained on the interval 𝑡 ∈ [0, 5 × 10−6 ms], and tested over 𝑡 ∈ [5 × 10−6 ms, 1.5 ms]. The time axis is restricted to the transient regime before the solution settles into approximately cyclic behavior. Bottom row shows the corresponding absolute error. 0.00 0.01 0.02 x 0.00 0.05 0.10 0.15 0.20 time [ms] FOM 0.00 0.01 0.02 x 0.00 0.05 0.10 0.1… view at source ↗
Figure 13
Figure 13. Figure 13: Spacetime diagram for velocity in the RDE case. Both ROMs are trained on the interval 𝑡 ∈ [0, 5 × 10−6 ms], and tested over 𝑡 ∈ [5 × 10−6 ms, 1.5 ms]. The time axis is restricted to the transient regime before the solution settles into approximately cyclic behavior. Bottom row shows the corresponding absolute error. For variable 𝑞𝑘 , ROM model ∈ {Direct, iSVD}, and saved time index 𝑡 𝑛 , the instantaneous… view at source ↗
Figure 14
Figure 14. Figure 14: Spacetime diagram for pressure in the RDE case. Both ROMs are trained on the interval 𝑡 ∈ [0, 5 × 10−6 ms], and tested over 𝑡 ∈ [5 × 10−6 ms, 1.5 ms]. The time axis is restricted to the transient regime before the solution settles into approximately cyclic behavior. Bottom row shows the corresponding absolute error. 0.00 0.01 0.02 x 0.00 0.05 0.10 0.15 0.20 time [ms] FOM 0.00 0.01 0.02 x 0.00 0.05 0.10 0.… view at source ↗
Figure 15
Figure 15. Figure 15: Spacetime diagram for temperature in the RDE case. Both ROMs are trained on the interval 𝑡 ∈ [0, 5× 10−6 ms], and tested over 𝑡 ∈ [5 × 10−6 ms, 1.5 ms]. The time axis is restricted to the transient regime before the solution settles into approximately cyclic behavior. Bottom row shows the corresponding absolute error. The aggregate error shown in [PITH_FULL_IMAGE:figures/full_fig_p029_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Parametric comparison of the Direct and iSVD adaptive ROMs over equivalence ratios 𝜙 = {0.8, 1.0, 1.2} and inlet pressures 𝑃in = {0.5, 1.0, 2.0, 4.0} atm. Both ROMs across all cases are trained on the interval 𝑡 ∈ [0, 5 × 10−6 ms], and tested over 𝑡 ∈ [5 × 10−6 ms, 1.5 ms]. The first two panels show the aggregate relative 𝐿2 error for the Direct and iSVD methods, respectively. The third panel shows the im… view at source ↗
Figure 17
Figure 17. Figure 17: Per-variable improvement factor for the iSVD adaptive ROM relative to the Direct adaptive ROM across the full parametric sweep. Both ROMs across all cases are trained on the interval 𝑡 ∈ [0, 5 × 10−6 ms], and tested over 𝑡 ∈ [5 × 10−6 ms, 1.5 ms]. quantity, but is observed across the primitive variables and species mass fractions. The acceleration factors remain the same as previously reported and are the… view at source ↗
read the original abstract

Reduced-order models (ROMs) can accelerate high-dimensional dynamical simulations, but their accuracy often deteriorates when online dynamics leave the regime represented by offline training data. We develop a projection-based adaptive ROM framework based on incremental singular value decomposition (iSVD), in which occasional full-order operator evaluations provide correction snapshots for online basis updates. The intrusive ROMs considered here are fully parameterized by the basis, so each update naturally propagates to reduced operators and hyper-reduction machinery. Through its evolving singular structure, iSVD retains an encoded history of the observed dynamics and is history-aware in this sense. We study the method on three nonlinear problems of increasing complexity: the one-dimensional viscous Burgers equation, the Sod shock tube, and a stiff one-dimensional ten-species rotating detonation engine (RDE). The Burgers problem is used to analyze the method and compare iSVD with alternative basis adaptation rules, showing that history-aware updates outperform instantaneous updates and that iSVD gives the strongest overall performance. The Sod and RDE cases demonstrate that these advantages persist in more challenging compressible-flow settings. For the RDE problem, the iSVD adaptive ROM improves upon the current state-of-the-art Direct adaptive ROM baseline in both predictive accuracy and computational efficiency. A cost analysis shows that the dominant online cost comes from interacting with the full-order model to obtain correction snapshots, while the iSVD update itself is negligible. These results identify iSVD as an effective mechanism for online learning of reduced subspaces and suggest a path toward ROMs that remain predictive over horizons several orders of magnitude longer than their initial training window.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The paper develops a projection-based adaptive reduced-order model (ROM) framework that uses incremental singular value decomposition (iSVD) to update the reduced basis online via occasional full-order model (FOM) correction snapshots. The approach is history-aware because iSVD encodes prior dynamics in its singular structure; updates propagate automatically to reduced operators and hyper-reduction. Numerical studies on the 1D viscous Burgers equation, Sod shock tube, and a stiff 10-species rotating detonation engine (RDE) show that iSVD outperforms instantaneous-update and direct-adaptive baselines, with the RDE case reporting gains in both predictive accuracy and online efficiency over the state-of-the-art Direct adaptive ROM.

Significance. If the efficiency claim holds, the work supplies a concrete, history-retaining mechanism for extending ROM horizons by orders of magnitude beyond the initial training window on nonlinear compressible flows. The Burgers analysis and the RDE demonstration on a stiff multi-species problem constitute useful evidence that incremental SVD can serve as an effective online subspace learner while keeping the adaptation cost negligible relative to FOM snapshot acquisition.

major comments (1)
  1. [Abstract and §5] Abstract and §5 (RDE results): the headline claim that iSVD improves both accuracy and computational efficiency over Direct adaptive ROM rests on the unquantified assumption that the frequency of required FOM correction snapshots remains low enough to produce net savings. No scaling relation, worst-case bound, or explicit count of snapshots versus simulation horizon or shock strength is provided, so the efficiency advantage cannot be verified from the reported data.
minor comments (1)
  1. [§3] Notation for the iSVD update rule and the definition of the correction snapshot selection criterion should be stated explicitly in the methods section rather than left implicit from the algorithm box.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and the focus on the efficiency claims. We address the major comment below and will revise the manuscript to improve transparency.

read point-by-point responses

  1. Referee: [Abstract and §5] Abstract and §5 (RDE results): the headline claim that iSVD improves both accuracy and computational efficiency over Direct adaptive ROM rests on the unquantified assumption that the frequency of required FOM correction snapshots remains low enough to produce net savings. No scaling relation, worst-case bound, or explicit count of snapshots versus simulation horizon or shock strength is provided, so the efficiency advantage cannot be verified from the reported data.

    Authors: We agree that the efficiency comparison would be strengthened by explicit quantification. In the RDE experiments, iSVD and the Direct adaptive ROM employ the identical adaptation schedule and therefore the same number and frequency of FOM correction snapshots; the measured wall-clock savings arise because the iSVD update itself is negligible relative to the Direct update cost, as stated in the cost analysis of §5. In the revised manuscript we will add the precise number of snapshots acquired, the total simulation horizon (in time steps and physical time), and the resulting snapshot frequency for the RDE case. We do not supply a general scaling relation or worst-case bound on required snapshot frequency, because the work is empirical and the necessary frequency is problem-dependent; we will explicitly note this scope limitation in the revised §5. revision: partial

Circularity Check

0 steps flagged

No circularity; method and claims are independently defined and empirically evaluated

full rationale

The paper introduces an iSVD-based adaptive ROM framework whose basis updates are defined directly from occasional FOM snapshots and incremental SVD mechanics. No derivation step reduces to a fitted parameter renamed as prediction, no self-citation is invoked as a uniqueness theorem or load-bearing premise, and no ansatz is smuggled via prior work. The RDE efficiency claim rests on observed snapshot costs versus ROM savings in the reported experiments rather than on any self-referential construction. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review yields no explicit free parameters, axioms, or invented entities; the approach relies on standard iSVD properties and the assumption that full-order snapshots can be obtained on demand.

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Reference graph

Works this paper leans on

71 extracted references · 66 canonical work pages · 1 internal anchor

  1. [1]

    P.Benner,S.Gugercin,K.Willcox,Asurveyofprojection-basedmodelreductionmethodsforparametricdynamicalsystems,SIAMReview 57 (4) (2015) 483–531.doi:10.1137/130932715

    work page doi:10.1137/130932715 2015

  2. [2]

    A. C. Antoulas, C. A. Beattie, S. Gügencın, Interpolatory Methods for Model Reduction, SIAM, Philadelphia, PA, 2020.doi:10.1137/1. 9781611976083

    work page doi:10.1137/1 2020

  3. [3]

    C. W. Rowley, S. T. M. Dawson, Model reduction for flow analysis and control, Annual Review of Fluid Mechanics 49 (2017) 387–417. doi:10.1146/annurev-fluid-010816-060042

    work page doi:10.1146/annurev-fluid-010816-060042 2017

  4. [4]

    K.Carlberg,M.Barone,H.Antil,Galerkinv.least-squaresPetrov–Galerkinprojectioninnonlinearmodelreduction,JournalofComputational Physics 330 (2017) 693–734.doi:10.1016/j.jcp.2016.10.033

    work page doi:10.1016/j.jcp.2016.10.033 2017

  5. [5]

    B.Peherstorfer,K.Willcox,Data-drivenoperatorinferencefornonintrusiveprojection-basedmodelreduction,ComputerMethodsinApplied Mechanics and Engineering 306 (2016) 196–215.doi:10.1016/j.cma.2016.03.025

    work page doi:10.1016/j.cma.2016.03.025 2016

  6. [6]

    Huang, K

    C. Huang, K. Duraisamy, C. L. Merkle, Component-based reduced order modeling of large-scale complex systems, Frontiers in Physics 10 (2022) 900064.doi:10.3389/fphy.2022.900064

    work page doi:10.3389/fphy.2022.900064 2022

  7. [7]

    C.W.Rowley,T.Colonius,R.M.Murray,ModelreductionforcompressibleflowsusingPODandGalerkinprojection,PhysicaD:Nonlinear Phenomena 189 (1–2) (2004) 115–129.doi:10.1016/j.physd.2003.03.001

    work page doi:10.1016/j.physd.2003.03.001 2004

  8. [8]

    M. F. Barone, I. Kalashnikova, D. J. Segalman, H. K. Thornquist, Stable Galerkin reduced order models for linearized compressible flow, Journal of Computational Physics 228 (6) (2009) 1932–1946.doi:10.1016/j.jcp.2008.11.015

    work page doi:10.1016/j.jcp.2008.11.015 2009

  9. [9]

    Arnold-Medabalimi, C

    N. Arnold-Medabalimi, C. Huang, K. Duraisamy, Large-eddy simulation and challenges for projection-based reduced-order modeling of a gas turbine model combustor, International Journal of Spray and Combustion Dynamics 14 (1–2) (2022) 153–175.doi:10.1177/ 17568277221100650

    2022

  10. [10]

    J. L. Lumley, The structure of inhomogeneous turbulent flows, in: A. M. Yaglom, V. I. Tatarski (Eds.), Atmospheric Turbulence and Radio Wave Propagation, Nauka, Moscow, 1967, pp. 166–178

    1967

  11. [11]

    Turbulence and the dynamics of coherent structures. I. Coherent structures,

    L. Sirovich, Turbulence and the dynamics of coherent structures. I–III, Quarterly of Applied Mathematics 45 (3) (1987) 561–590, parts I–III have separate DOIs: 10.1090/qam/910462, 10.1090/qam/910463, 10.1090/qam/910464

    work page doi:10.1090/qam/910462 1987

  12. [12]

    Holmes, J

    P. Holmes, J. L. Lumley, G. Berkooz, Turbulence, Coherent Structures, Dynamical Systems and Symmetry, Cambridge University Press, Cambridge, UK, 1996.doi:10.1017/CBO9780511622700

    work page doi:10.1017/cbo9780511622700 1996

  13. [13]

    Hedayat, A

    A. Hedayat, A. Padovan, K. Duraisamy, Toward adaptive non-intrusive reduced-order models: Design and challenges, arXiv preprint arXiv:2602.11378 (2026).doi:10.48550/arXiv.2602.11378

    work page doi:10.48550/arxiv.2602.11378 2026

  14. [14]

    C.Huang,K.Duraisamy,Predictivereducedordermodelingofchaoticmulti-scaleproblemsusingadaptivelysampledprojections,Journalof Computational Physics 491 (2023) 112356.doi:10.1016/j.jcp.2023.112356

    work page doi:10.1016/j.jcp.2023.112356 2023

  15. [15]

    Breaking the Kolmogorov barrier with nonlinear model reduction,

    B. Peherstorfer, Breaking the Kolmogorov barrier with nonlinear model reduction, Notices of the American Mathematical Society 69 (5) (2022) 725–733.doi:10.1090/noti2475

    work page doi:10.1090/noti2475 2022

  16. [16]

    K. Lee, K. T. Carlberg, Model reduction of dynamical systems on nonlinear manifolds using deep convolutional autoencoders, Journal of Computational Physics 404 (2020) 108973.doi:10.1016/j.jcp.2019.108973

    work page doi:10.1016/j.jcp.2019.108973 2020

  17. [17]

    Fresca, L

    S. Fresca, L. Dede’, A. Manzoni, A comprehensive deep learning-based approach to reduced order modeling of nonlinear time-dependent parametrized PDEs, Journal of Scientific Computing 87 (2) (2021) 61.doi:10.1007/s10915-021-01462-7

    work page doi:10.1007/s10915-021-01462-7 2021

  18. [18]

    Y. Kim, Y. Choi, D. Widemann, T. Zohdi, A fast and accurate physics-informed neural network reduced order model with shallow masked autoencoder, Journal of Computational Physics 451 (2022) 110841.doi:10.1016/j.jcp.2021.110841

    work page doi:10.1016/j.jcp.2021.110841 2022

  19. [19]

    Conti, G

    P. Conti, G. Gobat, S. Fresca, A. Manzoni, A. Frangi, Reduced order modeling of parametrized systems through autoencoders and SINDy approach, Computer Methods in Applied Mechanics and Engineering 411 (2023) 116072.doi:10.1016/j.cma.2023.116072

    work page doi:10.1016/j.cma.2023.116072 2023

  20. [20]

    S.E.Otto,G.R.Macchio,C.W.Rowley,Learningnonlinearprojectionsforreduced-ordermodelingofdynamicalsystemsusingconstrained autoencoders, Chaos: An Interdisciplinary Journal of Nonlinear Science 33 (11) (2023) 113130.doi:10.1063/5.0169688

    work page doi:10.1063/5.0169688 2023

  21. [21]

    Barnett and C

    J. Barnett, C. Farhat, Quadratic approximation manifold for mitigating the Kolmogorov barrier in nonlinear projection-based model order reduction, Journal of Computational Physics 464 (2022) 111348.doi:10.1016/j.jcp.2022.111348

    work page doi:10.1016/j.jcp.2022.111348 2022

  22. [22]

    Geelen, S

    R. Geelen, S. Wright, K. Willcox, Operator inference for non-intrusive model reduction with quadratic manifolds, Computer Methods in Applied Mechanics and Engineering 403 (2023) 115717.doi:10.1016/j.cma.2022.115717

    work page doi:10.1016/j.cma.2022.115717 2023

  23. [23]

    Ohlberger, S

    M. Ohlberger, S. Rave, Nonlinear reduced basis approximation of parameterized evolution equations via the method of freezing, Comptes Rendus Mathematique 351 (23–24) (2013) 901–906.doi:10.1016/j.crma.2013.10.028

    work page doi:10.1016/j.crma.2013.10.028 2013

  24. [24]

    C. W. Rowley, I. G. Kevrekidis, J. E. Marsden, K. Lust, Reduction and reconstruction for self-similar dynamical systems, Nonlinearity 16 (4) (2003) 1257–1275.doi:10.1088/0951-7715/16/4/304

    work page doi:10.1088/0951-7715/16/4/304 2003

  25. [25]

    Reiss, P

    J. Reiss, P. Schulze, J. Sesterhenn, V. Mehrmann, The shifted proper orthogonal decomposition: a mode decomposition for multiple transport phenomena, SIAM Journal on Scientific Computing 40 (3) (2018) A1322–A1344.doi:10.1137/17M1140571

    work page doi:10.1137/17m1140571 2018

  26. [26]

    N.J.Nair,M.Balajewicz,Transportedsnapshotmodelorderreductionapproachforparametric,steady-statefluidflowscontainingparameter- dependent shocks, International Journal for Numerical Methods in Engineering 117 (12) (2019) 1234–1262.doi:10.1002/nme.5998

    work page doi:10.1002/nme.5998 2019

  27. [27]

    D.Rim,B.Peherstorfer,K.T.Mandli,Manifoldapproximationsviatransportedsubspaces:modelreductionfortransport-dominatedproblems, SIAM Journal on Scientific Computing 45 (1) (2023) A170–A199.doi:10.1137/20M1316998

    work page doi:10.1137/20m1316998 2023

  28. [28]

    M. A. Mirhoseini, M. J. Zahr, Model reduction of convection-dominated partial differential equations via optimization-based implicit feature tracking, Journal of Computational Physics 473 (2023) 111739.doi:10.1016/j.jcp.2022.111739

    work page doi:10.1016/j.jcp.2022.111739 2023

  29. [29]

    Online adaptive model reduction for nonlinear systems via low-rank updates,

    B. Peherstorfer, K. Willcox, Online adaptive model reduction for nonlinear systems via low-rank updates, SIAM Journal on Scientific Computing 37 (4) (2015) A2123–A2150.doi:10.1137/140989169

    work page doi:10.1137/140989169 2015

  30. [30]

    B. Peherstorfer, Model reduction for transport-dominated problems via online adaptive bases and adaptive sampling, SIAM Journal on Scientific Computing 42 (5) (2020) A2803–A2836.doi:10.1137/19M1257275. Hedayat et al.:Preprint submitted to ElsevierPage 48 of 50 History-aware adaptive ROMs via iSVD

    work page doi:10.1137/19m1257275 2020

  31. [31]

    Mohaghegh, C

    A. Mohaghegh, C. Huang, Feature-guided sampling strategy for adaptive model order reduction of convection-dominated problems, Journal of Computational Physics 545 (2026) 114468.doi:10.1016/j.jcp.2025.114468

    work page doi:10.1016/j.jcp.2025.114468 2026

  32. [32]

    D.Amsallem,C.Farhat,Interpolationmethodforadaptingreduced-ordermodelsandapplicationtoaeroelasticity,AIAAJournal46(7)(2008) 1803–1813.doi:10.2514/1.35374

    work page doi:10.2514/1.35374 2008

  33. [33]

    Amsallem, C

    D. Amsallem, C. Farhat, An online method for interpolating linear parametric reduced-order models, SIAM Journal on Scientific Computing 33 (5) (2011) 2169–2198.doi:10.1137/100813051

    work page doi:10.1137/100813051 2011

  34. [34]

    D.Amsallem,M.J.Zahr,C.Farhat,Nonlinearmodelorderreductionbasedonlocalreduced-orderbases,InternationalJournalforNumerical Methods in Engineering 92 (10) (2012) 891–916.doi:10.1002/nme.4371

    work page doi:10.1002/nme.4371 2012

  35. [35]

    Peherstorfer, D

    B. Peherstorfer, D. Butnaru, K. Willcox, H.-J. Bungartz, Localized discrete empirical interpolation method, SIAM Journal on Scientific Computing 36 (1) (2014) A168–A192.doi:10.1137/130924408

    work page doi:10.1137/130924408 2014

  36. [36]

    Dynamic data-driven reduced-order models,

    B. Peherstorfer, K. Willcox, Dynamic data-driven reduced-order models, Computer Methods in Applied Mechanics and Engineering 291 (2015) 21–41.doi:10.1016/j.cma.2015.03.018

    work page doi:10.1016/j.cma.2015.03.018 2015

  37. [37]

    Singh, W

    R. Singh, W. I. T. Uy, B. Peherstorfer, Lookahead data-gathering strategies for online adaptive model reduction of transport-dominated problems, Chaos: An Interdisciplinary Journal of Nonlinear Science 33 (11) (2023) 113112.doi:10.1063/5.0169392

    work page doi:10.1063/5.0169392 2023

  38. [38]

    On-the-fly reduced order modeling of passive and reactive species via time-dependent manifolds,

    D. Ramezanian, A. G. Nouri, H. Babaee, On-the-fly reduced order modeling of passive and reactive species via time-dependent manifolds, Computer Methods in Applied Mechanics and Engineering 382 (2021) 113882.doi:10.1016/j.cma.2021.113882

    work page doi:10.1016/j.cma.2021.113882 2021

  39. [39]

    Real-time reduced-order modeling of stochastic partial differential equations via time-dependent subspaces,

    P. Patil, H. Babaee, Real-time reduced-order modeling of stochastic partial differential equations via time-dependent subspaces, Journal of Computational Physics 415 (2020) 109511.doi:10.1016/j.jcp.2020.109511

    work page doi:10.1016/j.jcp.2020.109511 2020

  40. [40]

    K. S. Jung, C. E. Lacey, H. Babaee, J. H. Chen, Accelerating high-fidelity simulations of chemically reacting flows using reduced-order modeling with time-dependent bases, Computer Methods in Applied Mechanics and Engineering 437 (2025) 117758.doi:10.1016/j. cma.2025.117758

    work page doi:10.1016/j 2025

  41. [41]

    Zucatti, M

    V. Zucatti, M. J. Zahr, An adaptive, training-free reduced-order model for convection-dominated problems based on hybrid snapshots, International Journal for Numerical Methods in Fluids 96 (2) (2024) 189–208.doi:10.1002/fld.5240

    work page doi:10.1002/fld.5240 2024

  42. [42]

    F. Bai, Y. Wang, A reduced order modeling method based on GNAT-embedded hybrid snapshot simulation, Mathematics and Computers in Simulation 199 (2022) 100–132.doi:10.1016/j.matcom.2022.03.006

    work page doi:10.1016/j.matcom.2022.03.006 2022

  43. [43]

    L.Feng,G.Fu,Z.Wang,AFOM/ROMhybridapproachforacceleratingnumericalsimulations,JournalofScientificComputing89(3)(2021) 61.doi:10.1007/s10915-021-01668-9

    work page doi:10.1007/s10915-021-01668-9 2021

  44. [44]

    Carlberg, Adaptiveℎ-refinement for reduced-order models, International Journal for Numerical Methods in Engineering 102 (5) (2015) 1192–1210.doi:10.1002/nme.4800

    K. Carlberg, Adaptiveℎ-refinement for reduced-order models, International Journal for Numerical Methods in Engineering 102 (5) (2015) 1192–1210.doi:10.1002/nme.4800

    work page doi:10.1002/nme.4800 2015

  45. [45]

    P. A. Etter, K. T. Carlberg, Online adaptive basis refinement and compression for reduced-order models via vector-space sieving, Computer Methods in Applied Mechanics and Engineering 364 (2020) 112931.doi:10.1016/j.cma.2020.112931

    work page doi:10.1016/j.cma.2020.112931 2020

  46. [46]

    M.Yano,T.Huang,M.J.Zahr,Agloballyconvergentmethodtoacceleratetopologyoptimizationusingon-the-flymodelreduction,Computer Methods in Applied Mechanics and Engineering 375 (2021) 113635.doi:10.1016/j.cma.2020.113635

    work page doi:10.1016/j.cma.2020.113635 2021

  47. [47]

    Comon, G

    P. Comon, G. H. Golub, Tracking a few extreme singular values and vectors in signal processing, Proceedings of the IEEE 78 (8) (1990) 1327–1343.doi:10.1109/5.58320

    work page doi:10.1109/5.58320 1990

  48. [48]

    doi:10.1109/JPROC.2018.2847041

    L.Balzano,Y.Chi,Y.M.Lu,StreamingPCAandsubspacetracking:themissingdatacase,ProceedingsoftheIEEE106(8)(2018)1293–1310. doi:10.1109/JPROC.2018.2847041

    work page doi:10.1109/jproc.2018.2847041 2018

  49. [49]

    J. R. Bunch, C. P. Nielsen, Updating the singular value decomposition, Numerische Mathematik 31 (2) (1978) 111–129.doi:10.1007/ BF01397471

    1978

  50. [50]

    Brand, Incremental singular value decomposition of uncertain data with missing values, in: European Conference on Computer Vision (ECCV), Springer, 2002, pp

    M. Brand, Incremental singular value decomposition of uncertain data with missing values, in: European Conference on Computer Vision (ECCV), Springer, 2002, pp. 707–720.doi:10.1007/3-540-47969-4_47

    work page doi:10.1007/3-540-47969-4_47 2002

  51. [51]

    Brand, Fast low-rank modifications of the thin singular value decomposition, Linear Algebra and its Applications 415 (1) (2006) 20–30

    M. Brand, Fast low-rank modifications of the thin singular value decomposition, Linear Algebra and its Applications 415 (1) (2006) 20–30. doi:10.1016/j.laa.2005.07.021

    work page doi:10.1016/j.laa.2005.07.021 2006

  52. [52]

    T. P. Krasulina, The method of stochastic approximation for the determination of the least eigenvalue of a symmetrical matrix, USSR Computational Mathematics and Mathematical Physics 9 (6) (1969) 189–195.doi:10.1016/0041-5553(69)90135-9

    work page doi:10.1016/0041-5553(69)90135-9 1969

  53. [53]

    The Noisy Power Method: A Meta Algorithm with Applications

    M. Hardt, E. Price, The noisy power method: a meta algorithm with applications, in: Advances in Neural Information Processing Systems (NeurIPS), Vol. 27, 2014.doi:10.48550/arXiv.1311.2495

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.1311.2495 2014

  54. [54]

    Allen-Zhu, Y

    Z. Allen-Zhu, Y. Li, First efficient convergence for streaming𝑘-PCA: a global, gap-free, and near-optimal rate, in: IEEE Annual Symposium on Foundations of Computer Science (FOCS), 2017, pp. 487–492.doi:10.1109/FOCS.2017.51

    work page doi:10.1109/focs.2017.51 2017

  55. [55]

    Oja, Simplified neuron model as a principal component analyzer, Journal of Mathematical Biology 15 (3) (1982) 267–273.doi: 10.1007/BF00275687

    E. Oja, Simplified neuron model as a principal component analyzer, Journal of Mathematical Biology 15 (3) (1982) 267–273.doi: 10.1007/BF00275687

    work page doi:10.1007/bf00275687 1982

  56. [56]

    704–711.doi:10.1109/ALLERTON.2010.5706976

    L.Balzano,R.Nowak,B.Recht,Onlineidentificationandtrackingofsubspacesfromhighlyincompleteinformation,in:48thAnnualAllerton Conference on Communication, Control, and Computing, 2010, pp. 704–711.doi:10.1109/ALLERTON.2010.5706976

    work page doi:10.1109/allerton.2010.5706976 2010

  57. [57]

    Balzano, S

    L. Balzano, S. J. Wright, On GROUSE and incremental SVD, in: 2013 IEEE 5th International Workshop on Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP), 2013, pp. 1–4.doi:10.1109/CAMSAP.2013.6713992

    work page doi:10.1109/camsap.2013.6713992 2013

  58. [58]

    Yang, Projection approximation subspace tracking, IEEE Transactions on Signal Processing 43 (1) (1995) 95–107.doi:10.1109/78

    B. Yang, Projection approximation subspace tracking, IEEE Transactions on Signal Processing 43 (1) (1995) 95–107.doi:10.1109/78. 365290

    work page doi:10.1109/78 1995

  59. [59]

    Y.Chi,Y.C.Eldar,R.Calderbank,PETRELS:parallelsubspaceestimationandtrackingbyrecursiveleastsquaresfrompartialobservations, IEEE Transactions on Signal Processing 61 (23) (2013) 5947–5959.doi:10.1109/TSP.2013.2282910

    work page doi:10.1109/tsp.2013.2282910 2013

  60. [60]

    Zimmermann, B

    R. Zimmermann, B. Peherstorfer, K. Willcox, Geometric subspace updates with applications to online adaptive nonlinear model reduction, SIAM Journal on Matrix Analysis and Applications 39 (1) (2018) 234–261.doi:10.1137/17M1123286

    work page doi:10.1137/17m1123286 2018

  61. [61]

    Mohaghegh, C

    A. Mohaghegh, C. Huang, Self adaptive reduced order modeling framework for rotating detonation engine simulations, in: AIAA SCITECH 2026 Forum, Orlando, FL, 2026.doi:10.2514/6.2026-1204. Hedayat et al.:Preprint submitted to ElsevierPage 49 of 50 History-aware adaptive ROMs via iSVD

    work page doi:10.2514/6.2026-1204 2026

  62. [62]

    Carlberg, C

    K. Carlberg, C. Bou-Mosleh, C. Farhat, Efficient non-linear model reduction via a least-squares petrov–galerkin projection and com- pressive tensor approximations, International Journal for Numerical Methods in Engineering 86 (2) (2011) 155–181.arXiv:https: //onlinelibrary.wiley.com/doi/pdf/10.1002/nme.3050,doi:https://doi.org/10.1002/nme.3050. URLhttps:/...

    work page doi:10.1002/nme.3050 2011

  63. [63]

    Carlberg, C

    K. Carlberg, C. Farhat, J. Cortial, D. Amsallem, The gnat method for nonlinear model reduction: Effective implementation and application to computational fluid dynamics and turbulent flows, Journal of Computational Physics 242 (2013) 623–647.doi:https://doi.org/10. 1016/j.jcp.2013.02.028. URLhttps://www.sciencedirect.com/science/article/pii/S0021999113001472

    2013

  64. [64]

    URLhttps://www.sciencedirect.com/science/article/pii/S0045782520301754

    E.J.Parish,C.R.Wentland,K.Duraisamy,Theadjointpetrov–galerkinmethodfornon-linearmodelreduction,ComputerMethodsinApplied Mechanics and Engineering 365 (2020) 112991.doi:https://doi.org/10.1016/j.cma.2020.112991. URLhttps://www.sciencedirect.com/science/article/pii/S0045782520301754

    work page doi:10.1016/j.cma.2020.112991 2020

  65. [65]

    Chaturantabut, D

    S. Chaturantabut, D. C. Sorensen, Nonlinear model reduction via discrete empirical interpolation, SIAM Journal on Scientific Computing 32 (5) (2010) 2737–2764.arXiv:https://doi.org/10.1137/090766498,doi:10.1137/090766498. URLhttps://doi.org/10.1137/090766498

    work page doi:10.1137/090766498 2010

  66. [66]

    Huang, C

    C. Huang, C. R. Wentland, K. Duraisamy, C. Merkle, Model reduction for multi-scale transport problems using model-form preserving least- squaresprojectionswithvariabletransformation,JournalofComputationalPhysics448(2022)110742.doi:https://doi.org/10.1016/ j.jcp.2021.110742. URLhttps://www.sciencedirect.com/science/article/pii/S0021999121006379

    work page arXiv 2022

  67. [67]

    Drmač, S

    Z. Drmač, S. Gugercin, A new selection operator for the discrete empirical interpolation method—improved a priori error bound and extensions, SIAM Journal on Scientific Computing 38 (2) (2016) A631–A648.arXiv:https://doi.org/10.1137/15M1019271,doi: 10.1137/15M1019271. URLhttps://doi.org/10.1137/15M1019271

    work page doi:10.1137/15m1019271 2016

  68. [68]

    Balzano, On the equivalence of oja’s algorithm and grouse, in: International Conference on Artificial Intelligence and Statistics, PMLR, 2022, pp

    L. Balzano, On the equivalence of oja’s algorithm and grouse, in: International Conference on Artificial Intelligence and Statistics, PMLR, 2022, pp. 7014–7030

    2022

  69. [69]

    Mohaghegh, C

    A. Mohaghegh, C. Huang, Compflowlab: A python code to develop and prototype new data-driven models for challenging compressible flow problems with shocks and chemical reactions (05 2026).doi:10.13140/RG.2.2.36810.12483

    work page doi:10.13140/rg.2.2.36810.12483 2026

  70. [70]

    Boris,A.Cohen, Weakand strongignition.i

    E.Oran, T.Young,J. Boris,A.Cohen, Weakand strongignition.i. numericalsimulationsof shocktubeexperiments, CombustionandFlame 48 (1982) 135–148.doi:https://doi.org/10.1016/0010-2180(82)90123-7. URLhttps://www.sciencedirect.com/science/article/pii/0010218082901237

    work page doi:10.1016/0010-2180(82)90123-7 1982

  71. [71]

    Huang, R

    C. Huang, R. G. Camacho, Investigations of parametric reduced order modeling framework for rotating detonation engines, in: AIAA SCITECH 2024 Forum, Orlando, FL, 2024.doi:10.2514/6.2024-2038. Hedayat et al.:Preprint submitted to ElsevierPage 50 of 50

    work page doi:10.2514/6.2024-2038 2024