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Operator Inference Aware Quadratic Manifolds with Isotropic Reduced Coordinates for Nonintrusive Model Reduction

T0 review · 4 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read Training quadratic manifolds with a greedy criterion that includes the reduced model's prediction error, not just reconstruction error, produces nonintrusive reduced models that are up to two orders of magnitude more accurate.

desk verdict Promising extension of greedy quadratic manifolds, but the headline accuracy gain is not causally isolated from the new isotropic scaling, and the turbine test is interleaved. read the letter →

arxiv 2507.20463 v1 pith:BFLWJKH7 submitted 2025-07-28 math.DS cs.LGcs.NAmath.NA

Paul Schwerdtner , Prakash Mohan , Julie Bessac , Marc T. Henry de Frahan , Benjamin Peherstorfer This is my paper · ORCID
classification math.DScs.LGcs.NAmath.NA MSC 35A0165L1065L1265L2065L70
keywords scientificmachinelearningoperatorinferencemodelreductionquadraticmanifoldsgreedybasisselectionisotropicreducedcoordinatespredictionerror
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 claims that quadratic manifolds—reduced-state spaces formed by a linear subspace plus a quadratic correction term—should be trained not only to reproduce snapshot data, but to make the downstream reduced model accurate. It proposes a greedy column-selection procedure whose objective combines the manifold's reconstruction error with the rollout prediction error of an operator-inference model, a low-dimensional quadratic dynamical system fitted to the encoded training trajectory. On a linear transport problem and a large-eddy simulation of flow around a wind turbine, the resulting reduced models achieve up to two orders of magnitude lower prediction error than manifolds trained on reconstruction error alone. The accompanying isotropic scaling of the basis, dividing each selected singular vector by its singular value, keeps reduced coordinates on a common scale and thereby avoids the oscillatory embeddings that reconstruction-only greedy training produces.

What carries the argument

The central mechanism is the operator-inference-aware greedy selection criterion used together with isotropic reduced coordinates. At each greedy step the next column of the reduced basis is chosen as the scaled left-singular vector $\phi^{(j)}_{\sigma^{-1}} = \phi^{(j)}/\sigma_j$ that minimizes $J'_{\mathrm{tot}}$, the manifold reconstruction error plus $\gamma_{\mathrm{op}}\|\Psi_I^\top - \hat{e}S\|_F^2$, the rollout error of an operator-inference model trained on the encoded trajectory. Division by $\sigma_j$ gives every reduced coordinate unit $\ell^2$ norm over the training snapshots, so no coordinate is dominated by scale, and the trajectory-level error then guides selection toward embeddings that remain stable under the fitted quadratic model. This combination is what prevents the oscillatory, non-smooth reduced coordinates that make reconstruction-only greedy manifolds hard to model.

What would settle it

A decisive check is to evaluate the method on a test set that is temporally separated from the training set by more than one time step (e.g., a later, non-interleaved time window), at matched reduced dimension, and see whether the operator-inference-aware greedy manifolds still outperform reconstruction-only greedy manifolds by the claimed margin; if the advantage shrinks or disappears, the training-rollout proxy is the weak link.

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Extended reading notes

Core claim

The central discovery is that the way a quadratic manifold embeds snapshot data determines how well a reduced dynamical model can be fitted to it, and this dependence can be exploited in training. The paper defines a greedy selection criterion $J'_{\mathrm{tot}}$ that scores each candidate singular vector by the sum of the manifold reconstruction error and a penalty proportional to the squared Frobenius distance between the encoded training trajectory and the trajectory predicted by an operator-inference model—a quadratic low-dimensional dynamical system—fitted to that encoding. Because the criterion penalizes trajectory-level rollout error rather than the one-step least-squares residual, it prefers embeddings that a first-order quadratic model can track over many time steps. Scaling the basis columns by the inverse singular values $\sigma_j^{-1}$ makes the reduced coordinates isotropic, so coordinates from later singular vectors are not tiny and fast-oscillating; this avoids the non-smooth embeddings that reconstruction-only greedy training can produce. On the transport and turbulent-flow test problems, this changes the reduced-model prediction error by up to two orders of magnitude.

Load-bearing premise

The method depends on the assumption that the operator-inference rollout error on the training trajectory is a faithful stand-in for prediction error on unseen data, an assumption that the turbulent-flow test (Section 4.2.1) weakens by placing test snapshots only one time step away from training snapshots.

Editorial extensions

If this is right

  • On the transport and turbulent-flow test sets, operator-inference models built on the proposed greedy quadratic manifolds achieve up to two orders of magnitude lower prediction error than manifolds trained on reconstruction error alone.
  • The greedy procedure avoids oscillatory reduced coordinates, so the fitted operator-inference models remain stable over the test horizon rather than producing divergent or artifact-ridden predictions.
  • The reconstruction error of the proposed manifolds stays comparable to reconstruction-only greedy manifolds, so the accuracy gain in predictions does not come at the cost of data fidelity.
  • Because the selection criterion is evaluated without differentiating through the time stepper, the greedy search has per-evaluation cost $O(r^3 k)$ and can reuse a precomputed singular value decomposition of the snapshot matrix.

Reading between the lines

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

  • The same trajectory-rollout criterion could in principle be applied to other learned encoders and decoders, such as polynomial manifolds or autoencoders, wherever a downstream reduced dynamical model is fitted to the latent trajectory; the paper's argument is not tied to the specific quadratic feature map.
  • The turbulent-flow experiment interleaves test snapshots with training snapshots (odd versus even time indices), so the claimed gain on unseen data should be re-tested on a temporally disjoint window before generalizing it to longer forecasting horizons.
  • The isotropic-scaling step is separable from the greedy selection and could improve operator inference even with the leading-singular-vector basis whenever the singular values decay steeply.
  • The scalar $\gamma_{\mathrm{op}}$ controls the trade-off between reconstruction fidelity and model learnability; the experiments use fixed values (1.0 and $10^2$), so tuning it per problem is a natural next step.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. The paper proposes a greedy training procedure for quadratic manifolds in nonintrusive model reduction. The key idea is to augment the reconstruction-error-based greedy selection of [11] with a term that measures the rollout error of an operator-inference model fit to the reduced trajectory, and to use isotropically scaled reduced coordinates. Numerical experiments on a linear transport problem and a large-eddy simulation of a wind-turbine flow demonstrate that the proposed approach achieves lower operator-inference test errors than the compared baselines, in some cases by up to two orders of magnitude.

Significance. If the central causal claim were established, this would be a useful advance: the greedy selection objective explicitly accounts for downstream model prediction error, which is a natural and potentially important idea for nonlinear nonintrusive model reduction. The paper is clearly written, discusses the computational complexity of the greedy objective, and includes a realistic wind-turbine test case. However, the current experimental evidence does not isolate the proposed selection criterion from the isotropic scaling, and the turbine experiment's train/test split is not a genuine temporal holdout. The ideas are promising and the issues appear addressable within the manuscript's scope.

major comments (4)
  1. [Section 3.2, Eq. (10)] The definition of the prediction-error term is inconsistent. The text states that the operator-inference model is trained once per greedy iteration on the reduced trajectory eS_{i-1}=f_{V^{σ^{-1}}_{i-1}}(S), which has dimension i−1, and that \hat{eS} is the corresponding prediction. Equation (10) then defines eS = f_{[V^{σ^{-1}},v]}(S), which has dimension i, so the Frobenius norm ||eS − \hat{eS}||_F is not well-defined. Equation (15) and the cost discussion in Section 3.3 imply that an operator-inference model is refit for the candidate-augmented trajectory, but this is not what Section 3.2 describes. This is central to the method and must be clarified.
  2. [Section 4, Figures 2–6] The proposed method differs from the 'QM, greedy [11]' baseline in two respects: it uses the isotropic scaling of Section 3.1 and the operator-inference-aware criterion of Eq. (10). No experiment is reported with isotropic scaling but with the prediction-error term disabled (γ_op=0), so the observed gains cannot be attributed to the new criterion; they may be due to scaling alone, which changes the conditioning of the operator-inference least-squares problem in Eq. (8). Please add an ablation with γ_op=0 and isotropic scaling to isolate the effect of the OI-aware criterion.
  3. [Section 4.2.1] The training and test sets for the turbine experiment are defined as S_train = [s(0),s(2),...,s(998)] and S_test = [s(1),s(3),...,s(999)], i.e., even-odd interleaving of the same trajectory with a 0.1 s time step. The test snapshots are thus one time step from training states and do not represent an unseen trajectory or a temporally disjoint test condition. A hold-out interval (e.g., early vs. late time) or a different inflow condition should be used to support the generalization claim.
  4. [Sections 4.1.2 and 4.2.2] The text says that for the operator-inference models 'we show results for the regularization parameters γA and γH with the lowest error.' If this selection is performed on the test set, the reported model prediction errors are optimistically biased. Please specify how these parameters were selected (e.g., using a separate validation set) and apply the same rule to all methods. In addition, γop is set to 1.0 (transport) and 10^2 (turbine) without a stated validation procedure; provide a selection rule or sensitivity analysis for these values.
minor comments (5)
  1. [Eq. (10)] The argument \tilde{v} in J_rec(\tilde{v}, V^{σ^{-1}}) is not defined; it should presumably be the candidate vector v.
  2. [Eq. (14) and Algorithm 1] Equation (14) defines W' ∈ R^{n×r(r+1)/2}, while Algorithm 1 and the surrounding text use W' ∈ R^{(k-i)×p}; the dimensions should be made consistent.
  3. [Figure 1] The caption uses 'relative model prediction error' without defining the normalization; please state the precise formula.
  4. [Section 4.1.1] The grid search for the reconstruction regularization parameter γ is described, but it is not stated whether γ is chosen on a training or validation set; this should be stated explicitly to avoid ambiguity.
  5. [Section 3.3] The complexity statement says the model prediction error costs O(r^3 k) per candidate; multiplying by q candidates and r iterations gives O(q r^4 k), which could be significant, and a brief discussion of this total cost would be helpful.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular derivation found: the greedy objective is a training criterion evaluated on the training trajectory, and the reported accuracy is measured on held-out test snapshots; self-citations are transparent and not load-bearing.

full rationale

The paper's derivation chain is not circular. The novel selection criterion in Eq. (10) minimizes a sum of the reconstruction error and the rollout error of an operator-inference model fitted to the current reduced training trajectory. This is a training loss, not a prediction: the final claims are evaluated on held-out snapshots via Eq. (18), so the loop is broken. The operator-inference model used in the criterion is fitted to training data only, and no parameter is fitted to the test error. The isotropic scaling of Section 3.1 is a reparameterization defined from the SVD of the training snapshots and is not defined in terms of the test target. The only self-citation of note is [11], the authors' prior greedy quadratic-manifold construction, which is used transparently as a baseline and as the starting point for the new criterion; the central claim does not reduce to any unverified assertion in [11]. The main internal-validity concern is experimental rather than circular: no ablation isolates the operator-inference-aware term from the isotropic scaling, and the turbulent-flow test snapshots are interleaved with training snapshots (even vs. odd time indices). These are design limitations that could affect whether the reported gains are attributable to the proposed criterion, but they are not instances of a result being equivalent to its inputs by construction.

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

No new physical entities or conserved quantities are introduced. The method relies on standard SVD, a quadratic feature map, and a linear-quadratic reduced model. The main additional burden is the reliance on training-rollout error as a selection signal and the undocumented choice of q and gamma_op.

free parameters (4)
  • gamma (reconstruction regularization) = grid search over {1e-6, 1e-4, 1e-2} for transport; {1e-4, 1e-3, 1e-2} for turbulent flow
    Appears in Eq. (5) for computing W; chosen by grid search without a stated validation split.
  • gamma_A and gamma_H (operator-inference regularization) = grid search over {1e-4, 1e-3, 1e-2, 1e-1}
    Appear in Eq. (8) when fitting the reduced linear-quadratic model; selected per experiment without a documented validation set.
  • gamma_op (prediction-error weight in greedy objective) = 1.0 for transport, 1e2 for turbulent flow
    Appears in Eq. (10) and Eq. (15); set to a fixed value per problem with no sensitivity analysis or systematic tuning protocol.
  • q (number of candidate singular vectors in greedy search) = not specified
    Algorithm 1 takes q as input but the experiments never state its value; the greedy selection and therefore the final manifold depend on this choice.
assumptions (5)
  • domain assumption The quadratic feature map h, the condensed Kronecker product, is a sufficient nonlinearity for the decoder.
    Adopted from prior work [46, 48] in Section 2.1, Eq. (3); no derivation or justification beyond precedent.
  • domain assumption The reduced dynamics can be represented by a first-order linear-quadratic model of the form A e + H h(e).
    Central to operator inference in Section 2.3, Eq. (9); the paper relies on test error to justify this for the given problems.
  • standard math The snapshot matrix has full column rank so that the right singular vectors are orthonormal rows of unit norm, making the isotropic scaling well-defined.
    Section 3.1 uses the inverse singular values and assumes the scaled reduced coordinates have unit row norms; this holds if Psi is square orthogonal, which requires full column rank of S.
  • domain assumption The operator-inference rollout error on the training trajectory is a reliable proxy for prediction error on unseen data.
    Used in Eq. (10) as the selection criterion; the paper does not analyze overfitting or validate on a separate trajectory during selection.
  • ad hoc to paper The candidate pool of the first q singular vectors is sufficient to contain a good basis for the reduced manifold.
    The greedy search in Algorithm 1 is restricted to q candidates, but q is never specified or justified in the experiments.

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Cite this review

Pith. "Pith review of Operator Inference Aware Quadratic Manifolds with Isotropic Reduced Coordinates for Nonintrusive Model Reduction." pith.science (2026). https://pith.science/paper/BFLWJKH7

@misc{pith2026250720463,
  author       = {Pith},
  title        = {Pith review of: Operator Inference Aware Quadratic Manifolds with Isotropic Reduced Coordinates for Nonintrusive Model Reduction},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BFLWJKH7}},
  note         = {Machine review of arXiv:2507.20463}
}
read the original abstract

Quadratic manifolds for nonintrusive reduced modeling are typically trained to minimize the reconstruction error on snapshot data, which means that the error of models fitted to the embedded data in downstream learning steps is ignored. In contrast, we propose a greedy training procedure that takes into account both the reconstruction error on the snapshot data and the prediction error of reduced models fitted to the data. Because our procedure learns quadratic manifolds with the objective of achieving accurate reduced models, it avoids oscillatory and other non-smooth embeddings that can hinder learning accurate reduced models. Numerical experiments on transport and turbulent flow problems show that quadratic manifolds trained with the proposed greedy approach lead to reduced models with up to two orders of magnitude higher accuracy than quadratic manifolds trained with respect to the reconstruction error alone.

Figures

Figures reproduced from arXiv: 2507.20463 by the authors.

Figure 1
Figure 1. Transport problem: The proposed operator inference-aware greedy approach (last row) considers both the reconstruction and prediction errors when train￾ing encoder-decoder pairs, which results in quadratic-manifold embeddings that are well-suited for nonintrusive model reduction with operator inference. fV (S) = V ⊤ΦΣΨ⊤ = Σ[:r,:r]Ψ⊤ [:r,:]. Notice that Σ[:r,:r] denotes the upper left r×r block of the matrix Σ and Ψ⊤ … view at source ↗
Figure 2
Figure 2. Transport problem: The greedy methods for constructing quadratic manifolds achieve the lowest reconstruction error among all approaches in this example. Furthermore, the error corresponding to the manifold trained with the pro￾posed operator inference-aware (OI-aware) greedy approach is comparable to the error achieved with the regular greedy approach from [11], even though our operator inference-aware approach bala… view at source ↗
Figure 3
Figure 3. Transport problem: The proposed operator inference-aware (OI-aware) greedy approach leads to operator inference models that achieve errors that are more than one order of magnitude lower than the error achieved with other reduction methods in this example. and a test data set S (test) = [s (1001) , . . . , s (2000)] . We then train quadratic manifolds with three different techniques. First, we follow [46, 50] and us… view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: Transport problem: The operator inference models based on linear approxima￾tions and on quadratic manifolds trained with the first r leading left-singular vectors lead to visible artifacts in the predictions. In contrast, the operator in￾ference model obtained with the…
Figure 5
Figure 5. Figure 5: Turbulent flow: The proposed operator inference-aware (OI-aware) greedy approach achieves a comparable reconstruction error as the regular greedy approach even though it balances reconstruction error and model prediction error. Notice that both greedy methods achieve o…
Figure 6
Figure 6. Figure 6: Turbulent flow: Operator inference models based on the proposed operator￾inference aware (OI-aware) greedy approach achieve orders of magnitude lower errors than models learned with other reduction methods in this example. the operator-inference model. The greedy metho…
Figure 7
Figure 7. Figure 7: Turbulent flow: The errors shown in the bottom row demonstrate that the proposed operator inference-aware greedy approach achieves the lowest error in this example. 16 [PITH_FULL_IMAGE:figures/full_fig_p016_7.png]
Figure 8
Figure 8. Figure 8: Turbulent flow: The prediction of the wind velocity vx after the turbine at x = 720m shows that operator inference models based on linear approximations lead to a smoothing in the prediction that misses the oscillatory behavior of the quantity of interest. The model tr…

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Works this paper leans on

64 extracted references · 60 canonical work pages

  1. [11]

    arXiv 2403.06732 (2024)

    Schwerdtner, P., Peherstorfer, B.: Greedy construction of quadratic manifolds for nonlinear dimensionality reduction and nonlinear model reduction. arXiv 2403.06732 (2024)

    arXiv 2024

  2. [1]

    Acta Numerica 30, 445–554 (2021)

    Ghattas, O., Willcox, K.: Learning physics-based models from data: perspectives from inverse problems and model reduction. Acta Numerica 30, 445–554 (2021)

    2021

  3. [2]

    Society for Industrial and Applied Mathematics, Philadelphia, PA (2020)

    Antoulas, A.C., Beattie, C.A., G¨ u˘ gercin, S.: Interpolatory Methods for Model Re- duction. Society for Industrial and Applied Mathematics, Philadelphia, PA (2020)

    work page 2020

  4. [3]

    Annual Review of Fluid Mechanics 56, 521–548 (2024) 18

    Kramer, B., Peherstorfer, B., Willcox, K.E.: Learning nonlinear reduced models from data with operator inference. Annual Review of Fluid Mechanics 56, 521–548 (2024) 18

    work page 2024

  5. [4]

    In: Guyon, I., Luxburg, U.V., Bengio, S., Wallach, H., Fergus, R., Vishwanathan, S., Garnett, R

    Takeishi, N., Kawahara, Y., Yairi, T.: Learning Koopman invariant subspaces for dynamic mode decomposition. In: Guyon, I., Luxburg, U.V., Bengio, S., Wallach, H., Fergus, R., Vishwanathan, S., Garnett, R. (eds.) Advances in Neural Information Processing Systems, vol. 30. Curran Associates, Inc., Long Beach (2017)

    work page 2017

  6. [5]

    Nature Communications 9(1), 4950 (2018)

    Lusch, B., Kutz, J.N., Brunton, S.L.: Deep learning for universal linear embeddings of nonlinear dynamics. Nature Communications 9(1), 4950 (2018)

    work page 2018

  7. [6]

    SIAM Journal on Applied Dynamical Systems 18(1), 558–593 (2019)

    Otto, S.E., Rowley, C.W.: Linearly recurrent autoencoder networks for learning dynamics. SIAM Journal on Applied Dynamical Systems 18(1), 558–593 (2019)

    work page 2019

  8. [7]

    Proceedings of the National Academy of Sciences 116(45), 22445–22451 (2019)

    Champion, K., Lusch, B., Kutz, J.N., Brunton, S.L.: Data-driven discovery of coor- dinates and governing equations. Proceedings of the National Academy of Sciences 116(45), 22445–22451 (2019)

    work page 2019

Show all 64 references
  1. [8]

    In: American Control Conference, ACC 2006,

    Borggaard, J.: Optimal reduced-order modeling for nonlinear distributed parameter systems. In: American Control Conference, ACC 2006,. IEEE, Minneapolis, MN, USA (2006)

    2006

  2. [9]

    Jour- nal of Computational Physics 224(2), 880–896 (2007)

    Bui-Thanh, T., Willcox, K., Ghattas, O., van Bloemen Waanders, B.: Goal- oriented, model-constrained optimization for reduction of large-scale systems. Jour- nal of Computational Physics 224(2), 880–896 (2007)

    2007

  3. [10]

    Journal of Computational Physics 228(6), 1932–1946 (2009)

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

    2009

  4. [12]

    Peherstorfer, B., Willcox, K.: Data-driven operator inference for nonintrusive projection-based model reduction. Comput. Methods Appl. Mech. Engrg. 306, 196– 215 (2016)

    2016

  5. [13]

    Antoulas, A.C.: Approximation of Large-Scale Dynamical Systems. Adv. Des. Con- trol, vol. 6. SIAM, Philadelphia, PA, USA (2005)

    2005

  6. [14]

    Rozza, G., Huynh, D.B.P., Patera, A.T.: Reduced basis approximation and a pos- teriori error estimation for affinely parametrized elliptic coercive partial differential equations. Arch. Comput. Methods Eng. 15(3), 229–275 (2008)

    2008

  7. [15]

    SIAM Rev

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

    2015

  8. [16]

    Hesthaven, J.S., Rozza, G., Stamm, B.: Certified Reduced Basis Methods for Parametrized Partial Differential Equations vol. 590. Springer, New York (2016)

    2016

  9. [17]

    Journal of Fluid Mechanics 641, 115–127 (2009)

    Rowley, C.W., Mezi´ c, I., Bagheri, S., Schlatter, P., Henningson, D.S.: Spectral analysis of nonlinear flows. Journal of Fluid Mechanics 641, 115–127 (2009)

    2009

  10. [18]

    Fluid Mech

    Schmid, P.J.: Dynamic mode decomposition of numerical and experimental data. Fluid Mech. 656, 5–28 (2010) 19

    2010

  11. [19]

    Society for Industrial and Applied Mathematics, Philadelphia, PA (2016)

    Kutz, J.N., Brunton, S.L., Brunton, B.W., Proctor, J.L.: Dynamic Mode Decom- position. Society for Industrial and Applied Mathematics, Philadelphia, PA (2016)

    2016

  12. [20]

    PLOS One 11(2), 0150171 (2016)

    Brunton, S.L., Brunton, B.W., Proctor, J.L., Kutz, J.N.: Koopman invariant sub- spaces and finite linear representations of nonlinear dynamical systems for control. PLOS One 11(2), 0150171 (2016)

    2016

  13. [21]

    In: Benner, P., Cohen, A., Ohlberger, M., Willcox, K

    Antoulas, A.C., Lefteriu, S., Ionita, A.C.: A tutorial introduction to the Loewner framework for model reduction. In: Benner, P., Cohen, A., Ohlberger, M., Willcox, K. (eds.) Model Reduction and Approximation: Theory and Algorithms, pp. 335–

  14. [22]

    Linear Algebra Appl

    Mayo, A.J., Antoulas, A.C.: A framework for the solution of the generalized real- ization problem. Linear Algebra Appl. 425(4), 634–662 (2007)

    2007

  15. [23]

    SIAM, Philadelphia, PA, USA (2020)

    Antoulas, A.C., Beattie, C., Gugercin, S.: Interpolatory Methods for Model Reduc- tion. SIAM, Philadelphia, PA, USA (2020)

    2020

  16. [24]

    Schwerdtner, P., Voigt, M.: SOBMOR: Structured optimization-based model order reduction. SIAM J. Sci. Comput. 45(2), 502–529 (2023)

    2023

  17. [25]

    Schwerdtner, P., Schaller, M.: Structured optimization-based model order reduction for parametric systems. SIAM J. Sci. Comput. 47(1), 72–101 (2025)

    2025

  18. [26]

    SIAM Journal on Scientific Computing 44(1), 554–582 (2022)

    Gosea, I.V., Gugercin, S., Beattie, C.: Data-driven balancing of linear dynamical systems. SIAM Journal on Scientific Computing 44(1), 554–582 (2022)

    2022

  19. [27]

    arXiv 2312.12561 (2023)

    Reiter, S., Gosea, I.V., Gugercin, S.: Generalizations of data-driven balancing: what to sample for different balancing-based reduced models. arXiv 2312.12561 (2023)

    2023 arXiv

  20. [28]

    Journal of the Royal Society of New Zealand 51(2), 194–211 (2021)

    McQuarrie, S.A., Huang, C., Willcox, K.E.: 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), 194–211 (2021)

    2021

  21. [29]

    Benner, P., Goyal, P., Kramer, B., Peherstorfer, B., Willcox, K.: Operator inference for non-intrusive model reduction of systems with non-polynomial nonlinear terms. Comput. Methods Appl. Mech. Engrg. 372, 113433 (2020)

    2020

  22. [30]

    Physica D: Nonlinear Phenomena 406, 132401 (2020)

    Qian, E., Kramer, B., Peherstorfer, B., Willcox, K.: Lift & learn: Physics-informed machine learning for large-scale nonlinear dynamical systems. Physica D: Nonlinear Phenomena 406, 132401 (2020)

    2020

  23. [31]

    Computer Methods in Applied Mechanics and En- gineering 402, 115336 (2022)

    Guo, M., McQuarrie, S.A., Willcox, K.E.: Bayesian operator inference for data- driven reduced-order modeling. Computer Methods in Applied Mechanics and En- gineering 402, 115336 (2022)

    2022

  24. [32]

    SIAM Journal on Scientific Computing 45(4), 1917–1946 (2023)

    McQuarrie, S.A., Khodabakhshi, P., Willcox, K.E.: Nonintrusive reduced-order models for parametric partial differential equations via data-driven operator infer- ence. SIAM Journal on Scientific Computing 45(4), 1917–1946 (2023)

    2023

  25. [33]

    Peherstorfer, B.: Sampling low-dimensional Markovian dynamics for preasymptot- ically recovering reduced models from data with operator inference. SIAM J. Sci. Comp. 42(5), 3489–3515 (2020) 20

    2020

  26. [34]

    Electronic Transactions on Numerical Analysis 56 (2022)

    Benner, P., Goyal, P., Heiland, J., Duff, I.P.: Operator inference and physics- informed learning of low-dimensional models for incompressible flows. Electronic Transactions on Numerical Analysis 56 (2022)

    2022

  27. [35]

    Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380(2229), 20210206 (2022)

    Geelen, R., Willcox, K.: Localized non-intrusive reduced-order modelling in the operator inference framework. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380(2229), 20210206 (2022)

    2022

  28. [36]

    SIAM Journal on Scientific Computing 45(4), 1462–1490 (2023)

    Uy, W.I.T., Wang, Y., Wen, Y., Peherstorfer, B.: Active operator inference for learning low-dimensional dynamical-system models from noisy data. SIAM Journal on Scientific Computing 45(4), 1462–1490 (2023)

    2023

  29. [37]

    Uy, W.I.T., Hartmann, D., Peherstorfer, B.: Operator inference with roll outs for learning reduced models from scarce and low-quality data. Comput. Math. Appl. 145, 224–239 (2023)

    2023

  30. [38]

    Sharma, H., Wang, Z., Kramer, B.: Hamiltonian operator inference: Physics- preserving learning of reduced-order models for canonical Hamiltonian systems. Phys. D 431, 133122 (2022)

    2022

  31. [39]

    Preprint (arXiv:2304.06262) (2023)

    Gruber, A., Tezaur, I.: Canonical and noncanonical Hamiltonian operator inference. Preprint (arXiv:2304.06262) (2023)

    2023 arXiv

  32. [40]

    Physica D: Nonlinear Phenomena 462, 134128 (2024)

    Sharma, H., Kramer, B.: Preserving Lagrangian structure in data-driven reduced- order modeling of large-scale dynamical systems. Physica D: Nonlinear Phenomena 462, 134128 (2024)

    2024

  33. [41]

    Computer Methods in Applied Mechanics and Engineering 404, 115836 (2023)

    Sawant, N., Kramer, B., Peherstorfer, B.: Physics-informed regularization and structure preservation for learning stable reduced models from data with operator inference. Computer Methods in Applied Mechanics and Engineering 404, 115836 (2023)

    2023

  34. [42]

    AIAA Journal 62(11), 4071–4086 (2024)

    Farcas, I.-G., Gundevia, R.P., Munipalli, R., Willcox, K.E.: Domain decomposition for data-driven reduced modeling of large-scale systems. AIAA Journal 62(11), 4071–4086 (2024)

    2024

  35. [43]

    Computer Physics Communications 313, 109619 (2025)

    Farca¸ s, I.-G., Gundevia, R.P., Munipalli, R., Willcox, K.E.: Distributed computing for physics-based data-driven reduced modeling at scale: Application to a rotating detonation rocket engine. Computer Physics Communications 313, 109619 (2025)

    2025

  36. [44]

    Physics of Plasmas 31(11), 113904 (2024)

    Gahr, C., Farca¸ s, I.-G., Jenko, F.: Scientific machine learning based reduced- order models for plasma turbulence simulations. Physics of Plasmas 31(11), 113904 (2024)

    2024

  37. [45]

    International Journal for Numerical Methods in Fluids 93, 2803–2821 (2021)

    Yıldız, S., Goyal, P., Benner, P., Karas¨ ozen, B.: Learning reduced-order dynam- ics for parametrized shallow water equations from data. International Journal for Numerical Methods in Fluids 93, 2803–2821 (2021)

    2021

  38. [46]

    Geelen, R., Wright, S., Willcox, K.: Operator inference for non-intrusive model reduction with quadratic manifolds. Comput. Methods Appl. Mech. Engrg. 403, 115717 (2023)

    2023

  39. [47]

    Notices of the American Mathematical Society 69, 725–733 (2022) 21

    Peherstorfer, B.: Breaking the kolmogorov barrier with nonlinear model reduction. Notices of the American Mathematical Society 69, 725–733 (2022) 21

    2022

  40. [48]

    Jain, S., Tiso, P., Rutzmoser, J.B., Rixen, D.J.: A quadratic manifold for model or- der reduction of nonlinear structural dynamics. Comput. Struct. 188, 80–94 (2017)

    2017

  41. [49]

    Rutzmoser, J.B., Rixen, D.J., Tiso, P., Jain, S.: Generalization of quadratic mani- folds for reduced order modeling of nonlinear structural dynamics. Comput. Struct. 192, 196–209 (2017)

    2017

  42. [50]

    Barnett, J., Farhat, C.: Quadratic approximation manifold for mitigating the Kol- mogorov barrier in nonlinear projection-based model order reduction. J. Comput. Phys. 464, 111348 (2022)

    2022

  43. [51]

    Comptes Rendus

    Cohen, A., Farhat, C., Maday, Y., Somacal, A.: Nonlinear compressive reduced ba- sis approximation for PDE’s. Comptes Rendus. M´ ecanique351(S1), 357–374 (2023)

    2023

  44. [52]

    Chaos: An Interdisciplinary Journal of Nonlinear Science 34(3), 033122 (2024)

    Geelen, R., Balzano, L., Wright, S., Willcox, K.: Learning physics-based reduced- order models from data using nonlinear manifolds. Chaos: An Interdisciplinary Journal of Nonlinear Science 34(3), 033122 (2024)

    2024

  45. [53]

    In: 2023 62nd IEEE Conference on Decision and Control (CDC), pp

    Geelen, R., Balzano, L., Willcox, K.: Learning latent representations in high- dimensional state spaces using polynomial manifold constructions. In: 2023 62nd IEEE Conference on Decision and Control (CDC), pp. 4960–4965 (2023)

    2023

  46. [54]

    PAMM 23(1), 202200049 (2023)

    Benner, P., Goyal, P., Heiland, J., Pontes Duff, I.: A quadratic decoder approach to nonintrusive reduced-order modeling of nonlinear dynamical systems. PAMM 23(1), 202200049 (2023)

    2023

  47. [55]

    Sharma, H., Mu, H., Buchfink, P., Geelen, R., Glas, S., Kramer, B.: Symplec- tic model reduction of hamiltonian systems using data-driven quadratic manifolds. Comput. Methods Appl. Mech. Engrg. 417, 116402 (2023)

    2023

  48. [57]

    arXiv 2412.09746 (2024)

    Weder, P., Schwerdtner, P., Peherstorfer, B.: Nonlinear model reduction with Neu- ral Galerkin schemes on quadratic manifolds. arXiv 2412.09746 (2024)

    2024 arXiv

  49. [58]

    Ballout, H., Maday, Y., Prud’Homme, C.: Combined Galerkin and regression based algorithm for parameter dependant PDEs (2025)

    2025

  50. [59]

    Diaz, A.N., McQuarrie, S.A., Tencer, J.T., Blonigan, P.J.: Interpretable and flex- ible non-intrusive reduced-order models using reproducing kernel Hilbert spaces arXiv:2506.10224 (2025)

    2025

  51. [60]

    Wind Energy 28(5), 70010 (2025)

    Kuhn, M.B., Henry de Frahan, M.T., Mohan, P., Deskos, G., Churchfield, M., Che- ung, L., Sharma, A., Almgren, A., Ananthan, S., Brazell, M.J., A., M.L., Thedin, R., Rood, J., Sakievich, P., Vijayakumar, G., Zhang, W., Sprague, M.A.: AMR- wind: A performance-portable, high-fide...

    2025

  52. [61]

    Wind Energy 27(3), 225– 257 (2024) 22

    Sharma, A., Brazell, M.J., Vijayakumar, G., Ananthan, S., Cheung, L., deVelder, N., Frahan, M.T., Matula, N., Mullowney, P., Rood, J., Sakievich, P., Almgren, A., Crozier, P.S., Sprague, M.: Exawind: Open-source cfd for hybrid-rans/les geometry- resolved wind turbine simulatio...

    2024

  53. [62]

    In: Journal of Physics: Conference Series, vol

    Sprague, M.A., Ananthan, S., Vijayakumar, G., Robinson, M.: Exawind: A multifi- delity modeling and simulation environment for wind energy. In: Journal of Physics: Conference Series, vol. 1452, p. 012071 (2020). IOP Publishing

    2020

  54. [63]

    Renewable Energy 147, 2259–2271 (2020)

    Sørensen, J.N., Nilsson, K., Ivanell, S., Asmuth, H., Mikkelsen, R.F.: Analytical body forces in numerical actuator disc model of wind turbines. Renewable Energy 147, 2259–2271 (2020)

    2020

  55. [64]

    Wind Energy Science 8(6), 1017–1027 (2023) 23

    Sørensen, J.N.: Generalized analytical body force model for actuator disc compu- tations of wind turbines. Wind Energy Science 8(6), 1017–1027 (2023) 23

    2023

  56. [376]

    SIAM, Philadelphia, PA, USA (2017). Chap. 8

    2017

Pith tools

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