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Data-Driven Model Order Reduction with pyMOR

T0 review · 4 major / 5 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read The paper claims that one open-source Python library, pyMOR, is the only MOR framework that unifies an extensive set of model-based and data-driven reduction algorithms for both parametric PDEs and control systems, and supports hybrid pipel

desk verdict A solid, thorough software-integration report for pyMOR's data-driven MOR suite; the architecture and code are the contribution, but the numerical comparisons have reporting gaps that need fixing before you rely on the performance claims. read the letter →

arxiv 2608.00082 v1 pith:XDVTJSHW submitted 2026-07-29 cs.MS cs.NAmath.NA

Hendrik Kleikamp , Petar Mlinarić , Stephan Rave , Felix Schindler This is my paper · ORCID
classification cs.MScs.NAmath.NA MSC 41A2065-0465D1565M6065N3093C1593C20
keywords pyMORmodelorderreductiondata-drivenparametricPDEsLTIsystemsdynamicmodedecompositionLoewnerframeworksurrogatemodeling
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

This paper describes recent additions of data-driven model order reduction (MOR) algorithms to the open-source library pyMOR and argues that pyMOR is the only software library offering such an extensive selection of model-based and data-driven MOR algorithms in a unified framework. It demonstrates that a user can expose a full-order model once through pyMOR's abstract interfaces and then choose among projection-based, data-only, and hybrid strategies. In the Navier-Stokes benchmark, projection-based ROMs run 47-74x faster than the full-order model, kernel-based machine-learning surrogates reach speedups around 1.8 million for direct output mapping, and using a model-based ROM to generate training data for a data-driven ROM cuts snapshot costs from about 9.4 hours to about 1.2 hours and improves neural-network accuracy. On a mass-spring-damper control system, the paper compares data-driven transfer-function methods (Loewner, AAA, ERA) with model-based balanced truncation and IRKA, finding the data-driven ROMs about an order of magnitude less accurate at the same reduced order.

What carries the argument

The organizing mechanism is pyMOR's interface layer: all equations appear as Model objects built from Operator and VectorArray objects, so projection-based reductors and data-driven reductors consume the same data structures and a model-based ROM can be reused as the data source for a data-driven ROM. Supporting machinery includes HAPOD for memory-limited snapshot compression (a three-level POD tree in the Navier-Stokes example), DEIM with least-squares residual minimization for hyperreduction, kernel interpolation (VKOGA) for low-data parameter-to-coefficient maps, and transfer-function interpolation via Loewner matrices, barycentric AAA, and ERA from impulse-response data.

What would settle it

Run the POD-ML and DMD pipelines on a convection-dominated parametric flow with viscosity below 1e-2, or on a non-periodic unsteady flow, and compare against the reported relative errors; if kernel surrogates degrade below roughly 1e-3 or DMD requires far more than five modes, the practical breadth claim is weakened. A survey-level check also settles the uniqueness claim: any actively maintained open-source package that offers transfer-function interpolation, state-space identification, and parametric projection methods through one model interface would directly refute the 'only library' asser

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Extended reading notes

Core claim

The paper's central claim is that a single software framework can meaningfully span both classical model-based model order reduction and recent data-driven approaches by expressing every full-order model through the same abstract Operator, VectorArray, and Model interfaces. It establishes this by adding data-driven reductors for POD-ML, DMD, Loewner, AAA, and ERA to the existing projection-based toolbox, and by showing a hybrid pipeline in which a projection-based POD-DEIM ROM generates training data for a kernel- or neural-network surrogate. The numerical evidence is presented as supporting evidence: model-based ROMs reach 47-74x speedups with relative state errors around 1e-4, kernel-based

Load-bearing premise

The demonstration depends on the favorable structure of the benchmarks: the Navier-Stokes parameter range is restricted to high viscosities (1e-2 to 1) to avoid slow Kolmogorov N-width decay, and DMD is tested only in a periodic vortex-shedding window; if the advertised real-world problems are not similarly compressible or periodic, the shown accuracy and speedups are not guaranteed to carry over.

Editorial extensions

If this is right

  • With a full-order model exposed through pyMOR's interfaces, comparing projection, data-only, and hybrid reduction methods becomes a matter of selecting a reductor rather than re-implementing a workflow.
  • A model-based ROM can act as a cheap surrogate for generating training data for a data-driven ROM: the paper shows 200 training parameters generated from the POD-DEIM ROM cost about 1.2 hours versus about 9.4 hours from the full-order model, and the neural-network surrogate improves by almost an order of magnitude.
  • Interpolation-based kernels (VKOGA, GPR) are the better default in low-data regimes; deep neural networks catch up only when hundreds of training parameters are available.
  • For periodic unsteady flows, five DMD modes reproduce vortex-shedding velocity fields to visual accuracy, with error decaying as the number of modes increases.
  • For LTI control systems, data-driven interpolation methods (Loewner, AAA, ERA) produce ROMs within roughly one order of magnitude of model-based balanced truncation and IRKA at reduced order 20, and p-AAA matches parametric BT/IRKA to about 1e-2 error on the tested damping range.

Reading between the lines

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

  • Editorial inference: the same interface abstraction that enables model/data hybrid pipelines also enables per-parameter method selection—for example, using DMD in periodic parameter regimes and POD-ML in transient regimes—which the paper mentions as future work through an adaptive hierarchy.
  • Editorial inference: the VKOGA advantage over neural networks at 30 training parameters suggests that greedy kernel centers may remain the data-efficient choice as parameter dimension grows, but the paper's parametric control example is only two-dimensional and does not test this directly.
  • Editorial inference: because a single 71,352-DoF Navier-Stokes trajectory already exceeds 8 GB of memory, HAPOD-style hierarchical compression is likely required for any high-dimensional unsteady output; whether the same speedups hold after accounting for external-solver I/O and wrapper overhead is not measured in the paper.
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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 presents the data-driven model order reduction (MOR) capabilities added to pyMOR, including POD-based methods (POD-ML with VKOGA, DNNs, and GPR), DMD, the Loewner framework, AAA/p-AAA, and ERA. It describes how these methods fit into pyMOR's Operator/Model abstraction and demonstrates them on two test cases: a parametrized incompressible Navier-Stokes problem, where POD-DEIM, POD-ML, a hierarchical POD-DEIM-to-POD-ML pipeline, and DMD are compared; and a mass-spring-damper chain, where model-based (BT, IRKA, SOBTp, PH-IRKA) and data-driven (Loewner, AAA, ERA, p-AAA) system-theoretic methods are compared. The paper claims that pyMOR is the only software library providing such an extensive selection of model-based and data-driven MOR algorithms in a unified framework for both control systems and parameterized PDEs in weak formulation.

Significance. If the framework claim holds, the paper is a valuable contribution to the MOR software ecosystem. It provides an open-source implementation, companion code with a DOI, and extensive numerical experiments spanning both PDE-based and control-system problems. The unified interface-based design, which allows model-based and data-driven methods to be combined in hierarchical pipelines, is a genuine strength. The main caveat is that the numerical evidence for the data-driven methods is obtained in deliberately favorable regimes, and several reporting gaps affect the strength of the stated comparisons. These issues are fixable and do not invalidate the architecture claim, but they are central to the paper's demonstration that the unified framework is practically useful across the advertised scope.

major comments (4)
  1. [Section 6.1.4, Fig. 10] The DMD errors reported in Fig. 10 are reconstruction errors computed on the same time interval [6,8] that is used to build the DMD model. The text states that the solution is 'restricted to the time interval [6,8]', and the DMD code in Section 5.3 fits W and omegas from the same snapshot sequence X that is later compared against. Thus the reported 'relative errors' measure fit quality, not predictive accuracy. This is especially consequential because the eigenvalues are shown to lie on the unit circle, confirming that the test case is periodic. Please relabel these as training/reconstruction errors, or preferably evaluate on a holdout interval (e.g., t in [8,10]) or an unseen initial condition and report generalization error.
  2. [Section 6.1.3, Table 2] The machine learning results in Table 2 are single runs. The NeuralNetworkRegressor uses random initialization, mini-batching, and early stopping, so the reported errors and the relative ranking of VKOGA, DNN, and GPR may depend on the random seed. The statement 'restarting the training with different random initial conditions is not necessary' is not a substitute for reporting variance. Please provide means and standard deviations over at least 5--10 independent runs, or fixed seeds for all methods, and state the number of runs. This is needed to support conclusions such as the DNN benefiting from 200 training points in the POD-DEIM-ML setting.
  3. [Section 6.1] The numerical evidence for data-driven MOR is confined to two favorable regimes. The parametric Navier-Stokes study restricts the viscosity to [10^-2, 10^0] explicitly to avoid slow Kolmogorov N-width decay [58], and the only DMD experiment uses a periodic vortex-shedding regime at nu=10^-3 on t in [6,8]. These restrictions are acknowledged in the text, but the abstract and central claim (Sec. 1) advertise data-driven methods for a broad class of parameterized PDE problems. As presented, the demonstrated accuracy and speedups for POD-ML and DMD may not transfer to advection-dominated, non-periodic, or transport-dominated problems. Please either add at least one experiment outside these favorable regimes, or explicitly scope the claims in the abstract and conclusion to the tested parameter and time ranges.
  4. [Section 6.1.3, Table 2] The hierarchical POD-DEIM-ML pipeline is presented as enabling efficient training-data generation, but the numerical results show a mixed picture. For VKOGA and GPR, the POD-DEIM-ML state errors (2.41e-04) are an order of magnitude larger than the corresponding POD-ML errors (3.21e-05), and the speedups are comparable. The paper notes the state-error increase but does not discuss its structural reason: the hierarchical ROM inherits the error of its teacher ROM, so the extra training data did not compensate for the reduced fidelity of the teacher in the kernel-based methods. Please add a discussion of this accuracy/cost trade-off, since it directly bears on the claim that pyMOR enables 'flexible and efficient hierarchical model reduction pipelines.'
minor comments (5)
  1. [Section 5.2.1] The heading contains a typo: 'algortihms' should be 'algorithms'.
  2. [Section 5.6] There is a typo in the first sentence: 'recoverd form' should be 'recovered from'.
  3. [Section 3.3] The custom IMEXModel assumes that E, F_i, and g are time-independent, as noted in the code comment, but this assumption is not stated in the main text. Please make it explicit as a limitation of the current implementation.
  4. [Section 1] The claim that pyMOR is 'the only software library' with this combination is supported only by a narrative survey and 'to the best of our knowledge.' A feature-comparison table against the cited libraries (e.g., libROM, PyDMD, MORLAB, Pressio) would make the claim easier to verify.
  5. [Section 6.1.4, Fig. 12] The visual comparison between the FOM and DMD solutions at the final time is qualitative. Since the DMD errors in Fig. 10 are reconstruction errors on the training interval, the figure caption should specify that this is a reconstruction, not a forecast.

Circularity Check

1 steps flagged · score 3.0 of 10

DMD error curve is an in-sample fit; the central unified-framework claim remains independently grounded.

  1. fitted input called prediction [Section 6.1.4 (DMD, Fig. 10); DMD usage in Section 5.3]
    "To observe a time-periodic pattern, we set the inlet velocity to ... and restrict the solution to the time interval [6,8]. We use values up to N=15 for the number of retained DMD modes in the reduced model and investigate the approximation error for different truncation ranks. // W, omegas = dmd(X, modes=num_dmd_modes, order='phase', cont_time_dt=fom.dt) ... b = lstsq_solver.solve(VectorArrayOperator(W), X[0]) ... t = np.arange(len(X)) * fom.dt ... X_dmd = W.lincomb(coeffs)"

    The DMD model is built from the same snapshot sequence X that is later used as the reconstruction target: the Sec. 5.3 code computes the modes from X, fits the coefficient vector b to X[0], and evaluates the DMD reconstruction at t = np.arange(len(X)) * fom.dt, i.e., exactly the training times. The errors plotted in Fig. 10 are therefore in-sample reconstruction errors over the interval [6,8] that generated the snapshots, not holdout or predictive MOR errors. Reporting these errors as evidence of DMD performance reduces to measuring how well the rank-N DMD fit reproduces its own inputs; no unseen time interval, parameter, or input is used to test generalization.

full rationale

The central claim of the paper is a software-architecture claim: pyMOR uniquely integrates model-based and data-driven MOR in one framework. That claim is supported by the public code, the interface design, the code excerpts, and the numerical comparisons, not by a chain of equations that reduces to its own inputs. The POD-ML and POD-DEIM experiments use test parameters held out from training (the 20 log-uniform test set and error measure (24)), so those accuracy numbers are genuine holdout evaluations. The system-theoretic comparisons (BT, IRKA, Loewner, AAA, ERA) also report convergence curves over model orders and data amounts rather than renaming fitted parameters as predictions. There is no load-bearing self-citation chain: citations to the authors' prior pyMOR/HAPOD work are normal references to independently published algorithms and code; the central claim does not rest on an unverified uniqueness theorem or ansatz smuggled in by citation. The main circularity-relevant flaw is the DMD experiment in Sec. 6.1.4, where the approximation error is computed on the same trajectory used to build the DMD modes and coefficients, making Fig. 10 a fit-quality measure. This inflates the apparent DMD performance but does not undermine the framework claim itself. The post-hoc choice of 80/70 truncation ranks in Sec. 6.2.2 is a model-selection concern rather than circularity. Overall, the derivation chain is largely self-contained, with one in-sample evaluation that warrants a moderate score.

Assumptions & free parameters 7 free parameters · 6 assumptions · 0 invented entities

The capability claim rests on standard MOR mathematics plus benchmark-specific modeling choices: a favorable high-viscosity Navier-Stokes regime, a periodic DMD test, and hand-tuned ML hyperparameters (VKOGA/GPR/DNN) together with post-hoc truncation ranks for parametric BT/IRKA. No new physical entities are introduced; all free parameters are algorithm-tuning or experimental-design choices reported in the text.

free parameters (7)
  • HAPOD POD truncation tolerance = 10^-4 (mean ℓ2 error)
    Hand-chosen tolerance controls reduced-basis dimension (34–66) in the Navier-Stokes experiment; lower tolerances would change all reported errors and timings.
  • VKOGA kernel length scale = 0.3 (scaled parameter domain)
    Chosen by hand for the Gaussian kernel; no tuning procedure or sensitivity study is reported.
  • VKOGA greedy tolerance and regularization = tol=1e-6, reg=1e-12
    Hand-set to stop center selection and avoid overfitting; affects number of centers and hence the reported speedups.
  • DNN architecture and training hyperparameters = 3 layers × 128 neurons; Adam lr=1e-3; 1000 epochs; batch 2048; patience 250
    Hand-chosen for the Navier-Stokes experiments; the authors state restarting was found unnecessary, implying some tuning occurred.
  • GPR hyperparameters = Gaussian kernel shape 0.3, reg=1e-12, no tuning
    Deliberately matched to VKOGA for a fair comparison, but this choice is arbitrary and may handicap GPR.
  • Parametric BT/IRKA truncation ranks = 80 for BT, 70 for IRKA
    The paper states "We found that truncating to 80 and 70 columns respectively ... gave similarly good results" — a post-hoc selection on the test problem.
  • Data grids for Loewner/AAA/p-AAA and ERA = 50 frequencies logspace(-5,3); p-AAA 100×100 grid; ERA horizon [0,100], sample Δt=0.05
    Hand-chosen sampling designs; denser or different grids would change reported errors.
assumptions (6)
  • standard math Standard results in linear systems theory and approximation: SVD/POD optimality, Loewner interpolation, AAA barycentric rational approximation, ERA realization, balanced truncation H∞ bound, IRKA H2-optimality conditions.
    Invoked throughout Sections 4–5; all are established results cited from the literature.
  • domain assumption The LTI systems under reduction are asymptotically stable (E^{-1}A Hurwitz) and the port-Hamiltonian structure (6) holds for the spring-damper example.
    Stated in Sec. 3.2 and used for Gramian-based methods and PH-IRKA; the mass-spring-damper chain is constructed to satisfy this.
  • domain assumption The parametric Navier-Stokes tests are restricted to a relatively high-viscosity regime ν ∈ [10^-2, 10^0] to avoid slow Kolmogorov N-width decay.
    Stated in Sec. 6.1 with citation [58]; this limits the external validity of the demonstrated parametric MOR performance.
  • ad hoc to paper In the custom IMEXModel, E, F_i and g are assumed time-independent so they can be pre-assembled before time stepping.
    Assumption in the _compute implementation, Sec. 3.3; it holds for the test problems but is not guaranteed by the general system class (1).
  • ad hoc to paper The DMD example selects a periodic vortex-shedding regime (ν=10^-3, t ∈ [6,8]) so that DMD eigenvalues lie on the unit circle.
    The regime is chosen to make DMD succeed; the paper states this choice explicitly in Sec. 6.1.4.
  • ad hoc to paper Neural-network training uses subsampled time trajectories (every 10th step) and the random-access-in-time formulation due to limited training parameters.
    Described in Sec. 6.1.3 as a design decision to balance parameter vs time dimensions in the training data.

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Cite this review

Pith. "Pith review of Data-Driven Model Order Reduction with pyMOR." pith.science (2026). https://pith.science/paper/XDVTJSHW

@misc{pith2026260800082,
  author       = {Pith},
  title        = {Pith review of: Data-Driven Model Order Reduction with pyMOR},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XDVTJSHW}},
  note         = {Machine review of arXiv:2608.00082}
}
read the original abstract

pyMOR is a free and open-source software library of model order reduction algorithms for the Python programming language. Designed with classical model-based reduction methods for large-scale parametric partial differential equation problems in mind, algorithms in pyMOR are implemented in terms of operations on abstract VectorArray, Operator and Model interfaces, allowing for a seamless integration with external solver codes implementing the full-order model. For cases where a tight integration with the full-order model code is not feasible, data-driven model order reduction algorithms, which only require simulation or measurement data of the full-order model, are an attractive alternative. In this work we discuss the data-driven methods that have been recently added to pyMOR, show practical examples of their application using pyMOR and compare their performance with classical model-based methods. We show that pyMOR serves as a unified framework for combining model-based and data-driven methods, enabling the construction of flexible and efficient hierarchical model reduction pipelines.

Figures

Figures reproduced from arXiv: 2608.00082 by the authors.

Figure 1
Figure 1. Interplay of selected pyMOR components (boxes), algorithms (dashed boxes) and user code. [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. HAPOD tree for the example in Section 6.1. The training parameters are denoted [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗
Figure 3
Figure 3. Navier-Stokes example: Computational domain Ω and finite element grid. [PITH_FULL_IMAGE:figures/full_fig_p018_3.png] view at source ↗
Figures from the paper (14 more)
Figure 4
Figure 4. Figure 4: Navier-Stokes example: FOM simulations for different kinematic viscosities [PITH_FULL_IMAGE:figures/full_fig_p019_4.png]
Figure 5
Figure 5. Figure 5: Navier-Stokes example: Drag and lift vs. time of FOM simulations for different viscosities [PITH_FULL_IMAGE:figures/full_fig_p019_5.png]
Figure 6
Figure 6. Figure 6: Navier-Stokes example: POD singular values computed by HAPOD (top left) and relative [PITH_FULL_IMAGE:figures/full_fig_p021_6.png]
Figure 7
Figure 7. Figure 7: Navier-Stokes example: Error distribution in parameter space for the final models in all three [PITH_FULL_IMAGE:figures/full_fig_p025_7.png]
Figure 8
Figure 8. Figure 8: Navier-Stokes example: Statistics of relative error decay (drag for [PITH_FULL_IMAGE:figures/full_fig_p026_8.png]
Figure 9
Figure 9. Figure 9: Navier-Stokes example: Speedup and number of selected centers with respect to the number of [PITH_FULL_IMAGE:figures/full_fig_p026_9.png]
Figure 10
Figure 10. Figure 10: Navier-Stokes example: Discrete-time DMD eigenvalues (left) and error decays in state recon [PITH_FULL_IMAGE:figures/full_fig_p027_10.png]
Figure 11
Figure 11. Figure 11: Navier-Stokes example: Magnitude of the first 4 velocity DMD modes (there is one real mode [PITH_FULL_IMAGE:figures/full_fig_p028_11.png]
Figure 12
Figure 12. Figure 12: Navier-Stokes example: Magnitude of FOM solution (left) and DMD solution (right) velocity [PITH_FULL_IMAGE:figures/full_fig_p028_12.png]
Figure 13
Figure 13. Figure 13: Mass-spring-damper chain example: Magnitude plots of the mass-spring-damper chain FOM [PITH_FULL_IMAGE:figures/full_fig_p029_13.png]
Figure 14
Figure 14. Figure 14: Mass-spring-damper chain example: Relative [PITH_FULL_IMAGE:figures/full_fig_p029_14.png]
Figure 15
Figure 15. Figure 15: Mass-spring-damper chain example: Error system magnitudes for three data-driven ROMs of [PITH_FULL_IMAGE:figures/full_fig_p030_15.png]
Figure 16
Figure 16. Figure 16: Mass-spring-damper chain example: Relative [PITH_FULL_IMAGE:figures/full_fig_p031_16.png]
Figure 17
Figure 17. Figure 17: Mass-spring-damper chain example: Magnitude plots with respect to frequency and parameter [PITH_FULL_IMAGE:figures/full_fig_p031_17.png]

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Pith tools

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