Pith. sign in

REVIEW 2 major objections 4 minor 43 references

Hyper-reduction-free reduced-order Newton solvers for projection-based model-order reduction of nonlinear dynamical systems

T0 review · 2 major / 4 minor · reviewed 2026-08-02 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read For polynomial dynamical systems, reduced Newton solvers can be computed exactly without hyper-reduction, because all projected residual and Jacobian operators are precomputable offline and online cost scales only with the reduced dimension

desk verdict Useful but narrowly scoped: new explicit HRF reduced Newton operators for Galerkin/LSPG, with an abstract that overreaches on N-dependence and HRF-LSPG speed. read the letter →

arxiv 2603.03420 v2 pith:I474ZPDT submitted 2026-03-03 cs.CE

Liam K. Magargal , Parisa Khodabakhshi , Steven N. Rodriguez This is my paper · ORCID
classification cs.CE MSC 65M22
keywords projection-basedmodel-orderreductionhyper-reduction-freepolynomialnonlinearsystemsGalerkinprojectionleast-squaresPetrov-GalerkinreducedNewtonsolverliftingtransformationoffline-onlinedecomposition
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper tries to establish that a large class of nonlinear reduced-order models, those whose full-order equations are polynomial with affine parameter dependence, can be solved with a Newton method in the reduced space without the usual extra approximation layer called hyper-reduction. The authors show that every operator appearing in the projected residual and its Jacobian can be computed once offline, so the online cost depends only on the reduced dimension, not on the size of the full-order model. This means Galerkin and least-squares Petrov-Galerkin projections are solved exactly, with no sparse-sampling error and no hyper-reduction tolerance to tune. In two test problems, the Galerkin variant matches full-projection accuracy while outperforming sparse-sampling approaches in speed; the lifted cubic heat equation demonstrates how non-polynomial nonlinearities can be brought into the required form.

What carries the argument

The central device is a pair of Kronecker-splitting identities: writing x̂⊗Iₙ as Σₗ Gₗ x̂ₗ and Iₙ⊗x̂ as Σₗ Hₗ x̂ₗ, where Gₗ and Hₗ are fixed Boolean block matrices of size n²×n. Substituting these into the projected Jacobian turns every term into a product of an offline-precomputed reduced operator and a scalar component of the reduced state; the same expansion handles the residual. This reduces the Newton solve to sums over reduced-coordinate indices only, and for LSPG it introduces one additional nested sum relative to Galerkin.

What would settle it

For a fixed polynomial full-order model and reduced basis, compute the direct projected residual Φᵀr(Φx̂) and projected Jacobian Φᵀ(∂r/∂x)(Φx̂)Φ at several random reduced states and parameter values, and compare them numerically with the HRF formulas; any difference beyond machine precision would falsify the claimed exactness. Separately, fix the reduced dimension n and double N while keeping the operators fixed in structure; if per-Newton wall-clock time grows measurably, the claimed independence from N is false.

Watch

Extended reading notes

Core claim

For semidiscrete systems of the form dx/dt = C + A x + F(x⊗x) + B u + N(u⊗x), with operators that depend affinely on parameters, the paper derives explicit expressions for the projected residual and Newton Jacobian of a reduced-order model. Using Kronecker-product identities, each quadratic and bilinear term is rewritten as a sum over the reduced state components. Substituting these identities into the Galerkin and LSPG Newton systems yields operators, such as ΦᵀF(Φ⊗Φ)Gₗ and ΦᵀN(I⊗Φ), that can be precomputed offline and depend only on the reduced dimension n and the number of inputs, not on the full-order dimension N. The online Newton iteration therefore evaluates the exact projected residu

Load-bearing premise

The full-order model must be known and expressible, exactly or through an exact lifting transformation, as a polynomial system with affine parameter dependence; if the operators are inaccessible or the nonlinearity cannot be lifted, the offline precomputation breaks down.

Editorial extensions

If this is right

  • The projected Galerkin and LSPG Newton systems are solved exactly: no sparse-sampling approximation enters the reduced residual or Jacobian, so the ROM evaluation error relative to the projection is effectively zero up to floating-point arithmetic.
  • For the targeted polynomial class, online cost per Newton iteration is independent of the full-order dimension N, making very large full-order models practical in many-query settings.
  • Cubic and higher-degree polynomial terms can be handled by summing over Kronecker products of the reduced state, and lifting transformations extend the method to non-polynomial nonlinearities such as cubic reaction terms.
  • Hyper-reduction tolerance tuning disappears for these systems, and in the paper's experiments the Galerkin variant avoids the convergence failures that loose sparse-sampling tolerances produce.
  • The LSPG variant is exact but carries extra nested sums over reduced coordinates, so the paper's data favor Galerkin for computational speed while retaining LSPG as a stability-oriented alternative.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If full-order operators are only partially accessible, the same precomputation idea could in principle be coupled with data-inferred full-space operators; the paper flags this as future work, and it is the natural next test of the framework.
  • The storage footprint of precomputed reduced operators grows with powers of n, so for very large reduced dimension the memory requirement, rather than arithmetic, may become the limiting factor, a trade-off the paper's experiments do not stress.
  • The lifting route trades a larger latent space for polynomial structure; a direct comparison at equal polynomial degree (cubic HRF versus lifted quadratic HRF) on a more complex benchmark would separate lifting-induced error from reduction error.
  • For legacy or black-box codes whose nonlinearities cannot be lifted exactly, the method does not apply; a data-driven operator-inference extension would be required to make the approach non-intrusive.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. This paper proposes an intrusive projection-based model-order reduction framework for semidiscrete dynamical systems of the form (1), where the nonlinearity is quadratic in the state and bilinear in state/input, with affine parameter dependence. The key idea is that all operators appearing in the projected residual and Jacobian of a reduced Newton solver can be precomputed offline using Kronecker identities; the online evaluation then costs only in the reduced dimension and reproduces the Galerkin or LSPG projection of the full-order residual exactly, avoiding the additional approximation layer of hyper-reduction. The HRF-G and HRF-LSPG schemes are tested on a 1D Burgers equation and on a heat equation with a cubic reaction term handled by the lifting w=q^2, with comparisons to ECSW-based Galerkin and LSPG ROMs. Numerical results show that the HRF schemes match the unreduced projected ROMs in accuracy and achieve speedups relative to the FOM, although HRF-LSPG is markedly slower than HRF-G.

Significance. The central algebraic observation is correct and clearly presented: for polynomial systems of the stated form, the projected residual and Jacobian terms factor into precomputable reduced operators. The exact-reproduction property is convincingly demonstrated by ROM evaluation errors that are orders of magnitude lower than those of ECSW. The lifting example extends the method to a cubic nonlinearity and shows that the lifting error is small. The availability of public code is a plus for reproducibility. The framework is incremental relative to existing quadratic-bilinear MOR and operator-inference literature, but it is a clean and potentially useful treatment of the Newton-solver setting, with honest experimental comparisons against ECSW. The main weaknesses are overbroad claims in the abstract and in Section 3 about independence from the full-order dimension and about the speedups obtained by HRF-LSPG.

major comments (2)
  1. [Section 3, Eqs. (14)-(15); Appendix A] The paper's central claim that online evaluation "does not depend on the dimension of the full-order model" is not fully supported. In Eq. (15), the bilinear-input term is [Phi^T N(I_Nu (x) Phi)](u_{m-j} (x) xhat), where the underbraced operator has dimension n x (N_u n); each evaluation costs O(N_u n^2). The Jacobian term in Eq. (14) has the same cost. If N_u is a fixed constant, the cost is independent of N, and the claim holds. However, the manuscript nowhere states the condition N_u=O(1). For distributed or spatially resolved inputs, N_u can scale with N, in which case the online cost scales linearly with N and no offline-online speedup is obtained. The two numerical experiments use N_u=1 and N_u=2, so they do not exercise this dependence. The condition should be stated explicitly in Section 2.1 and used to qualify the abstract and Section 5.
  2. [Abstract vs. Section 4.1, Figure 3 and Figure 5] The abstract states that HRF-G and HRF-LSPG achieve two and one order of magnitude speedup, respectively, with respect to the FOM. This is not uniformly supported. Section 4.1 explicitly reports that HRF-LSPG "becomes roughly as expensive as the FOM" at eps_POD=10^-4, and Figure 5 shows that HRF-LSPG generally attains the smallest speedup among the methods considered. The one-order speedup claim is therefore parameter-dependent and potentially misleading. The authors should report the range of observed speedups and state the conditions (modal energy, test parameter set) under which the one-order figure is achieved, or revise the abstract to avoid overclaiming.
minor comments (4)
  1. [Section 4.2, Figure 10 caption] The caption reads "ECSW-G, and ECSW-G" twice in the same phrase. One of these should presumably be ECSW-LSPG.
  2. [Section 4.2, Figure 9 and ROM evaluation error definition] For the lifted HRF schemes, it is unclear whether the reference Galerkin-ROM/LSPG-ROM used in Eq. (19) is the lifted or the non-lifted ROM. The text says all other methods use the non-lifted formulation, but the definition of ROM evaluation error in Eq. (19) requires the same projection scheme. If the reference is the non-lifted ROM, the metric conflates lifting error with ROM evaluation error. Please clarify.
  3. [Eq. (15)] The sentence "the superscript (k) from x_m^(k) has been dropped for notational convenience" is slightly confusing because the same notation is used for the previous time-step values. A brief notational clarification would help.
  4. [Section 4.1, Figure 3] The text states that ECSW results are presented for tolerances 10^-5 and 10^-9. In some panels the speedup values are difficult to read, especially where curves overlap. Consider adding a table with numerical speedup values for selected eps_POD values.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: HRF operators are derived algebraically from the stated polynomial FOM; no fitted inputs are relabeled as predictions.

full rationale

The paper's derivation chain is self-contained algebra. Starting from the polynomial FOM (1), substitution of x ≈ Φ xhat gives the approximate residual (4); differentiating gives the reduced Jacobian (9); the Kronecker identities (10) and (12) are exact algebraic decompositions of xhat⊗I_n and I_n⊗xhat. Substituting these into the Newton system (6) with Ψ=Φ yields the precomputable, n-dimensional Galerkin operators (14)-(15). The matched reduced operators are derived, not fitted: no parameter is calibrated to make HRF-G or HRF-LSPG match Galerkin-ROM/LSPG-ROM, and the paper verifies the match with the ROM evaluation error (19), which is a consistency check against the full projected solver rather than the prediction metric. The POD basis is extracted from training snapshots and then evaluated on unseen parameters, so the reported test-set errors are predictive rather than in-sample. The only self-citations, [23] for cubic-term operator details and [40] for block-diagonal structure preservation, are prior external results providing auxiliary algebraic facts; they are not used to define the central HRF operators and are not a uniqueness theorem or fitted input. The abstract's complexity claim that online cost does not depend on the full-order dimension is overbroad when the input dimension N_u grows with N, since Eq. (14) contains the term ΦᵀN(I_{N_u}⊗Φ)(u_m⊗I_n) costing O(N_u n²); however, this is a correctness/scope issue, not a circularity, because the derivation does not presuppose the conclusion.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

No physical entities are introduced. The lifting variable w=q^2 is an exact mathematical auxiliary, not a new physical degree of freedom. The listed free parameters are numerical tolerances and solver settings, not fitted model constants.

free parameters (4)
  • POD truncation tolerance eps_POD = 10^-1 to 10^-4 in experiments
    User-selected basis-size criterion in Eq. (16); not fitted to data, but it sets the reduced dimension n and hence the speed/accuracy trade-off.
  • ECSW tolerance eps_ecsw = 10^-5, 10^-7, 10^-9
    Baseline hyper-reduction hyperparameter; not part of the HRF method, but used in comparisons.
  • Newton convergence tolerance kappa_res = 10^-6
    User-set stopping criterion in Eq. (8); affects iteration counts.
  • Newton step length gamma = 1
    Full Newton step; fixed for all ROMs in the experiments.
assumptions (5)
  • standard math Standard Kronecker product identities and chain-rule differentiation (Eqs. 4, 9-12).
    These linear algebra facts are used to factor the full-order operators out of the reduced residual and Jacobian.
  • domain assumption The FOM has the affine-parametric polynomial form (1) with known, accessible operators C, A, F, B, N.
    Section 2.1 states this as the problem class; it is the core restriction of the method.
  • domain assumption Non-polynomial nonlinearities can be exactly recast as polynomial via a finite lifting transformation (e.g., w=q^2).
    Section 4.2 relies on this to make the cubic heat equation quadratic; exactness follows from the algebraic identity w=q^2.
  • domain assumption POD basis built from training snapshots spans the solution manifold for unseen parameters and extended time horizons.
    Section 4 constructs the trial basis from FOM snapshots and tests on held-out parameters; this is the standard ROM assumption, not proven here.
  • domain assumption LSPG test basis is the residual Jacobian (least-squares Petrov-Galerkin), following [28,34].
    Section 2.2 and Appendix A use this cited projection choice rather than re-deriving it.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Hyper-reduction-free reduced-order Newton solvers for projection-based model-order reduction of nonlinear dynamical systems." pith.science (2026). https://pith.science/paper/I474ZPDT

@misc{pith2026260303420,
  author       = {Pith},
  title        = {Pith review of: Hyper-reduction-free reduced-order Newton solvers for projection-based model-order reduction of nonlinear dynamical systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/I474ZPDT}},
  note         = {Machine review of arXiv:2603.03420}
}
read the original abstract

This study proposes an intrusive projection-based model-order reduction framework for nonlinear problems with a polynomial structure, solved iteratively using a Newton solver in the reduced space. It is demonstrated that for the targeted class of polynomial nonlinearities, all operators appearing in the projected approximate residual and Jacobian can be precomputed in the offline phase, eliminating the need for hyper-reduction. Additionally, the evaluation of both the projected approximate residual and its Jacobian scales only with the dimension of the reduced space, and does not depend on the dimension of the full-order model, enabling effective offline-online decomposition. The proposed hyper-reduction-free (HRF) framework is applied to both Galerkin (HRF-G) and least-squares Petrov-Galerkin (HRF-LSPG) projection schemes. The accuracy and computational efficiency of the proposed HRF schemes are evaluated in two numerical experiments and compared with a commonly used hyper-reduction scheme, namely the energy-conserving sampling and weighting method, for both the Galerkin and LSPG schemes. In the first numerical example, a parametric Burgers' equation is used to assess the predictive capabilities of the considered model reduction approaches on parameter sets not seen in the training snapshots. In the second example, a parametric heat equation with a cubic reaction term is studied, for which a lifting transformation is employed to expose the desired structure. The efficacy of the HRF methods in accurately reducing the dimensionality of the lifted formulation is investigated. For the studied problems, the results show that HRF-G and HRF-LSPG achieve two and one order of magnitude speedup, respectively, with respect to the full-order model while resulting in state prediction errors below O(10^-2).

Figures

Figures reproduced from arXiv: 2603.03420 by the authors.

Figure 1
Figure 1. Sample FOM solutions demonstrating the variation of the solution with respect to the parameters. The left and right columns present [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. State solutions for the test parameter set [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. State prediction errors (top row) and speedup factors (bottom row) versus truncated modal energy, [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figures from the paper (7 more)
Figure 4
Figure 4. Figure 4: ROM evaluation errors for methods employing Galerkin (left) and LSPG (right) projections, plotted with respect to truncated modal [PITH_FULL_IMAGE:figures/full_fig_p012_4.png]
Figure 5
Figure 5. Figure 5: State prediction errors versus the speedup factor for HRF-G, HRF-LSPG, ECSW-G, and ECSW-LSPG schemes, where ECSW results [PITH_FULL_IMAGE:figures/full_fig_p012_5.png]
Figure 6
Figure 6. Figure 6: Range of state prediction errors for HRF schemes (left panel), and ECSW schemes for tolerance values of [PITH_FULL_IMAGE:figures/full_fig_p013_6.png]
Figure 7
Figure 7. Figure 7: presents the solutions generated by the FOM (i.e., ground truth), lifted HRF-G, lifted HRF-LSPG, ECSW￾G, and ECSW-LSPG for the test parameter set 𝝁 = (𝑎, 𝑏) = (1.5, 0.5), using a truncated modal energy 𝜖POD = 10−3 . ECSW results are reported for the tolerance 𝜖ecsw = 1…
Figure 8
Figure 8. Figure 8: The top row presents state prediction errors, along with the corresponding projection errors, plotted against the truncated modal energy, [PITH_FULL_IMAGE:figures/full_fig_p016_8.png]
Figure 9
Figure 9. Figure 9: ROM evaluation errors plotted with respect to truncated modal energy, [PITH_FULL_IMAGE:figures/full_fig_p017_9.png]
Figure 10
Figure 10. Figure 10: State prediction error for lifted HRF-G, lifted HRF-LSPG, non-lifted HRF-G, ECSW-G, and ECSW-G with respect to the speedup [PITH_FULL_IMAGE:figures/full_fig_p017_10.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

43 extracted references

  1. [1]

    Benner, S

    P. Benner, S. Gugercin, K. Willcox, A survey of projection-based model reduction methods for parametric dynamical systems, SIAM Review 57 (4) (2015) 483–531

    2015

  2. [2]

    T. Lieu, C. Farhat, M. Lesoinne, Reduced-order fluid/structure modeling of a complete aircraft configuration, Computer Methods in Applied Mechanics and Engineering 195 (41) (2006) 5730–5742

    2006

  3. [3]

    Amsallem, M

    D. Amsallem, M. Zahr, Y. Choi, C. Farhat, Design optimization using hyper-reduced-order models, Structural and Multidisciplinary Optimization 51 (4) (2015) 919–940

    2015

  4. [4]

    L.Sirovich,TurbulenceandthedynamicsofcoherentstructurespartI:coherentstructures,QuarterlyofApplied Mathematics 45 (3) (1987) 561–571

    1987

  5. [5]

    K.Willcox,J.Peraire,Balancedmodelreductionviatheproperorthogonaldecomposition,AIAAJournal40(11) (2002) 2323–2330

    2002

  6. [6]

    O. T. Schmidt, T. Colonius, Guide to spectral proper orthogonal decomposition, AIAA Journal 58 (3) (2020) 1023–1033

    2020

  7. [7]

    Antoulas, Approximation of large-scale dynamical systems, Society for Industrial and Applied Mathematics, Philadelphia, PA, 2009

    A. Antoulas, Approximation of large-scale dynamical systems, Society for Industrial and Applied Mathematics, Philadelphia, PA, 2009. Distribution Statement A. Approved for public release. Distribution is unlimited.18

    2009

  8. [8]

    S.Chaturantabut,D.C.Sorensen,Nonlinearmodelreductionviadiscreteempiricalinterpolation,SIAMJournal on Scientific Computing 32 (5) (2010) 2737–2764

    2010

Show all 43 references
  1. [9]

    S.Bhattacharyya,J.Tao,E.Gildin,J.C.Ragusa,Hyper-reductiontechniquesforefficientsimulationoflarge-scale engineering systems, Archives of Computational Methods in Engineering 32 (2025) 5337 – 5379

    2025

  2. [10]

    M.Barrault, Y.Maday, N.C.Nguyen, A.T.Patera, An‘empiricalinterpolation’method: applicationtoefficient reduced-basis discretization of partial differential equations, Comptes Rendus Mathematique 339 (9) (2004) 667–672

    2004

  3. [11]

    Drmac, S

    Z. Drmac, 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

    2016

  4. [12]

    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

    2013

  5. [13]

    C.Farhat,T.Chapman,P.Avery,Structure-preserving,stability,andaccuracypropertiesoftheenergy-conserving samplingandweightingmethodforthehyperreductionofnonlinearfiniteelementdynamicmodels,International Journal for Numerical Methods in Engineering 102 (5) (2015) 1077–1110

    2015

  6. [14]

    Grimberg, C

    S. Grimberg, C. Farhat, R. Tezaur, C. Bou-Mosleh, Mesh sampling and weighting for the hyperreduction of nonlinear Petrov–Galerkin reduced-order models with local reduced-order bases, International Journal for Numerical Methods in Engineering 122 (7) (2021) 1846–1874

    2021

  7. [15]

    Nasika, P

    C. Nasika, P. Díez, P. Gerard, T. J. Massart, S. Zlotnik, Discrete empirical interpolation for hyper-reduction of hydro-mechanicalproblemsingroundwaterflowthroughsoil,InternationalJournalforNumericalandAnalytical Methods in Geomechanics 47 (5) (2023) 667–693

    2023

  8. [16]

    M. R. Chmiel, J. Barnett, C. Farhat, Unified LSPG model reduction framework and assessment for hypersonic computational fluid dynamics, AIAA Journal 63 (1) (2025) 72–90

    2025

  9. [17]

    P. J. Schmid, Dynamic mode decomposition of numerical and experimental data, Journal of Fluid Mechanics 656 (2010) 5–28

    2010

  10. [18]

    Peherstorfer, K

    B. Peherstorfer, K. Willcox, Data-driven operator inference for nonintrusive projection-based model reduction, Computer Methods in Applied Mechanics and Engineering 306 (2016) 196–215

    2016

  11. [19]

    C.Gu,Qlmor: Aprojection-basednonlinearmodelorderreductionapproachusingquadratic-linearrepresentation of nonlinear systems, IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 30 (9) (2011) 1307–1320

    2011

  12. [20]

    Kramer, K

    B. Kramer, K. E. Willcox, Nonlinear model order reduction via lifting transformations and proper orthogonal decomposition, AIAA journal 57 (6) (2019) 2297–2307

    2019

  13. [21]

    E.Qian,B.Kramer,B.Peherstorfer,K.Willcox,Lift&Learn: physics-informedmachinelearningforlarge-scale nonlinear dynamical systems, Physica D: Nonlinear Phenomena 406 (2020) 132401

    2020

  14. [22]

    S. A. McQuarrie, C. Huang, K. Willcox, Data-driven reduced-order models via regularised operator inference for a single-injector combustion process, Journal of the Royal Society of New Zealand 51 (2) (2021) 194–211

    2021

  15. [23]

    Khodabakhshi, K

    P. Khodabakhshi, K. Willcox, Non-intrusive data-driven model reduction for differential–algebraic equations derived from lifting transformations, Computer Methods in Applied Mechanics and Engineering 389 (2022) 114296

    2022

  16. [24]

    Barnett, C

    J. Barnett, C. Farhat, Quadratic approximation manifold for mitigating the Kolmogorov barrier in nonlinear projection-based model order reduction, Journal of Computational Physics (2022) 111348. Distribution Statement A. Approved for public release. Distribution is unlimited.19

    2022

  17. [25]

    R.Geelen,S.Wright,K.Willcox,Operatorinferencefornon-intrusivemodelreductionwithquadraticmanifolds, Computer Methods in Applied Mechanics and Engineering 403 (2023) 115717

    2023

  18. [26]

    K. Lee, K. Carlberg, Model reduction of dynamical systems on nonlinear manifolds using deep convolutional autoencoders, Journal of Computational Physics 404 (2020) 108973

    2020

  19. [27]

    Magargal, P

    L. Magargal, P. Khodabakhshi, S. Rodriguez, J. Jaworski, J. Michopoulos, Projection-based model-order reduc- tion via graph autoencoders suited for unstructured meshes, Data-Centric Engineering 6 (2025) e52

    2025

  20. [28]

    J.Barnett,C.Farhat,Y.Maday,Neural-network-augmentedprojection-basedmodelorderreductionformitigating the Kolmogorov barrier to reducibility, Journal of Computational Physics 492 (2023) 112420

    2023

  21. [29]

    Wiewel, M

    S. Wiewel, M. Becher, N. Thuerey, Latent space physics: Towards learning the temporal evolution of fluid flow, Computer Graphics Forum 38 (2) (2019) 71–82

    2019

  22. [30]

    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) 1–36

    2021

  23. [31]

    S. L. Brunton, J. L. Proctor, J. N. Kutz, Discovering governing equations from data by sparse identification of nonlinear dynamical systems, Proceedings of the national academy of sciences 113 (15) (2016) 3932–3937

    2016

  24. [32]

    W. D. Fries, X. He, Y. Choi, Lasdi: Parametric latent space dynamics identification, Computer Methods in Applied Mechanics and Engineering 399 (2022) 115436

    2022

  25. [33]

    X. He, Y. Choi, W. D. Fries, J. L. Belof, J.-S. Chen, gLaSDI: Parametric physics-informed greedy latent space dynamics identification, Journal of Computational Physics 489 (2023) 112267

    2023

  26. [34]

    K.Carlberg,C.Bou-Mosleh,C.Farhat,Efficientnon-linearmodelreductionviaaleast-squaresPetrov–Galerkin projection and compressive tensor approximations, International Journal for Numerical Methods in Engineering 86 (2) (2011) 155–181

    2011

  27. [35]

    Crank, P

    J. Crank, P. Nicolson, A practical method for numerical evaluation of solutions of partial differential equations of the heat-conduction type, Mathematical Proceedings of the Cambridge Philosophical Society 43 (1) (1947) 50–67

    1947

  28. [36]

    Magnus, H

    J. Magnus, H. Neudecker, Kronecker products, vec operator, and Moore-Penrose inverse, John Wiley & Sons, Ltd, 2019, Ch. 2, pp. 31–45

    2019

  29. [37]

    Grimberg, C

    S. Grimberg, C. Farhat, N. Youkilis, On the stability of projection-based model order reduction for convection- dominated laminar and turbulent flows, Journal of Computational Physics 419 (2020)

    2020

  30. [38]

    thesis, Massachusetts Institute of Technology (2003)

    M.J.Rewieński,Atrajectorypiecewise-linearapproachtomodelorderreductionofnonlineardynamicalsystems, Ph.D. thesis, Massachusetts Institute of Technology (2003)

    2003

  31. [39]

    S. A. McQuarrie, A. Chaudhuri, K. E. Willcox, M. Guo, Bayesian learning with Gaussian processes for low- dimensional representationsof time-dependentnonlinear systems, Physica D:Nonlinear Phenomena 475(2025) 134572

    2025

  32. [40]

    S. A. McQuarrie, P. Khodabakhshi, K. E. Willcox, Nonintrusive reduced-order models for parametric partial differential equations via data-driven operator inference, SIAM Journal on Scientific Computing 45 (4) (2023) A1917–A1946

    2023

  33. [41]

    L.Gkimisis,N.Aretz,M.Tezzele,T.Richter,P.Benner,K.E.Willcox,Non-intrusivereduced-ordermodelingfor dynamical systems with spatially localized features, Computer Methods in Applied Mechanics and Engineering 444 (2025) 118115

    2025

  34. [42]

    E.J.Parish,C.R.Wentland,K.Duraisamy,TheadjointPetrov–Galerkinmethodfornon-linearmodelreduction, Computer Methods in Applied Mechanics and Engineering 365 (2020) 112991

    2020

  35. [43]

    C. L. Lawson, R. J. Hanson, Solving Least Squares Problems, SIAM, 1995, Ch. 23, pp. 158–173. Distribution Statement A. Approved for public release. Distribution is unlimited.20 Appendix A. Hyper-reduction-free least-squares Petrov-Galerkin reduced-order Newton solver Thisappen...

    1995

Pith tools

Reviewed August 2, 2026 · model on record in the stance chip above.