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REVIEW 3 major objections 5 minor 62 references

Data-driven Closure Strategies for Parametrized Reduced Order Models via Deep Operator Networks

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper claims that appending data-driven closure terms, learned by deep operator networks, to the reduced momentum and pressure equations considerably improves pressure and velocity accuracy of parametric turbulent ROMs relative to…

desk verdict Solid incremental ROM-closure paper whose main caveat is the unverified transfer of learned corrections from projected to online coefficients; deserves review with a request for a gap measurement. read the letter →

arxiv 2505.17305 v1 pith:CR6EUTIP submitted 2025-05-22 math.NA cs.NA

Anna Ivagnes , Giovanni Stabile , Gianluigi Rozza This is my paper · ORCID
classification math.NAcs.NA
keywords reducedordermodelsPOD-GalerkinclosuremodelingdeepoperatornetworksMIONetturbulentflowsparametrizedPDEsmachinelearningcorrection
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 claims that a reduced-order model for parametrized turbulent flows can be made substantially more accurate by adding data-driven correction terms, learned with deep operator networks, directly into the reduced momentum and pressure equations. The corrections are meant to re-introduce the effect of POD modes that were discarded when the full system was projected onto a low-dimensional space. The authors extend an earlier quadratic-closure strategy to a nonlinear, parameter-aware setting, with the eddy-viscosity coefficients also produced by a learned operator network. Across three turbulent test cases (periodic flow past a cylinder, a channel-driven cavity, and a geometrically parametrized backward-facing step), the corrected ROMs report lower velocity and especially pressure errors than the baseline POD-Galerkin eddy-viscosity ROM, in both training and unseen test configurations.

What carries the argument

The central object is the closure-correction term $\tau_{\text{exact}} = \bigl(C(\hat a)\bigr)_{N_u+N_p} - C(a_{\text{proj}})$, where $C$ collects the nonlinear convective and turbulence contributions of the reduced equations, $\hat a$ and $a_{\text{proj}}$ are velocity coefficients projected onto a large and a small POD subspace, and the large-mode result is truncated to the small dimension. What carries the argument is a MIONet (multi-input deep operator network) trained to approximate this exact correction as $\tau_{\text{approx}} = M(a,g,\mu)$, together with a DeepONet that models the eddy-viscosity coefficients $g = G(a,\mu)$; the trained corrections are then added as source terms in the reduced momentum and pressure equations.

What would settle it

On a new parametrized turbulent test case with an ill-conditioned reduced system, record $\|a_{\text{sol}} - a_{\text{proj}}\|$ during online runs and compare DD-EV-ROM pressure error with EV-ROM: if negative gains appear whenever this gap exceeds a small threshold, the transfer assumption behind the closure is violated and the method should be considered tied to the tested regimes.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is a recipe: compute the exact correction term as the difference between the nonlinear reduced operators evaluated with a large set of POD modes and the same operators evaluated with the small retained set; train a Multi-Input Deep Operator Network to output that correction as a function of the reduced velocity coefficients, the eddy-viscosity coefficients, and the problem parameters; and append the correction to the reduced momentum and pressure residuals. A second network supplies the eddy-viscosity coefficients on demand. The numerical evidence indicates that the corrected ROMs improve on the baseline EV-ROM in almost all modal regimes considered, with the largest gains in pressure, and that a coupled training variant, called DD-EV-ROM*, is more stable when the reduced system is ill-conditioned, even in time-extrapolation settings.

Load-bearing premise

The load-bearing assumption is that the networks trained on exact projected coefficients still provide accurate corrections when evaluated online at the actual reduced solution, which can differ from the projection because the reduced system is ill-conditioned; the paper itself shows this transfer can fail for the standard training, and the coupled-training fix is validated only on the three test cases.

Editorial extensions

If this is right

  • Adding a learned pressure correction to the PPE-based ROM consistently reduces pressure error more than velocity error, because pressure is harder for the baseline to approximate.
  • The coupled training variant DD-EV-ROM* extends the method to ill-conditioned online systems where standard training gives negative gains, so the closure acts as a stabilizer as well as an accuracy enhancer.
  • Because the closure map takes the parameters as inputs, the same trained model applies across a parameter range rather than requiring retraining for each configuration.
  • The method remains usable in a purely online setting once the networks are trained, with the reduced operators assembled for each new parameter value in the geometric case.

Reading between the lines

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

  • The projection-vs-solution gap identified in Appendix A.1 suggests a cheap online diagnostic: track $\|a_{\text{sol}} - a_{\text{proj}}\|$ and flag configurations where the closure is likely to degrade; this could be used to switch off the closure or refine the reduced basis.
  • The same correction idea has a natural testbed in nonlinear-manifold ROMs (for example, autoencoder-based latent spaces), where the discarded-mode contribution is harder to define but a learned correction added to the latent equations could play the same role.
  • If the method generalizes, it would lower the barrier to using POD-Galerkin ROMs in industrial parametrized turbulent settings, since it relaxes the need for a large number of modes and for hand-tuned eddy-viscosity closures.
  • A straightforward stress test is to vary the Reynolds number range beyond the training window; the authors report time extrapolation, but Reynolds extrapolation is not separately tested, so the closure network's behavior there is an open empirical question.
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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

3 major / 5 minor

Summary. The paper proposes a data-driven closure strategy for parametric POD-Galerkin reduced-order models (ROMs) for turbulent incompressible flows. Starting from an eddy-viscosity ROM (EV-ROM), the authors append a correction term to the reduced momentum and pressure equations. The correction is modeled by a MIONet that takes the reduced velocity coefficients, the reduced eddy-viscosity coefficients, and the physical/geometrical parameters as inputs, while the eddy-viscosity coefficients themselves are produced by a DeepONet. Two training protocols are compared: a standard one (DD-EV-ROM) and a coupled one (DD-EV-ROM*) in which the two networks are trained together with an additional loss term. The method is evaluated on three turbulent test cases: periodic flow past a cylinder, unsteady channel-driven cavity flow, and steady geometrically parametrized backward-facing step flow. For each case, the authors report relative L2 error gains with respect to the EV-ROM baseline across several modal regimes, for both training and held-out test parameters, together with time trends, statistical box plots, and graphical error fields. The central claim is that the machine-learning-enhanced ROM considerably improves pressure and velocity accuracy compared with the standard POD-Galerkin/EV-ROM approach.

Significance. If the empirical claims hold, the paper provides a useful and fairly general recipe for parametric closure learning in projection-based ROMs, and it goes beyond earlier non-parametric closure models by using deep operator networks that take the parameters as explicit inputs. The study is unusually thorough in its experimental design: three different turbulent regimes, multiple modal combinations, held-out test parameters, time-extrapolation settings, comparison of two training strategies, and explicit discussion of ill-conditioning and of the discrepancy between projected and online reduced coefficients (Appendix A.1). The pressure-correction component, enabled by the PPE formulation, is a distinctive and plausible source of the observed gains. At the same time, the manuscript has notable reproducibility gaps: no code or data are released, no repeated-seed variability is reported for the neural-network training, and the central online/offline transfer issue identified by the authors is not quantitatively characterized. These gaps matter because the reported gains are not uniformly positive and the robustness of the proposed methods is a key part of the contribution.

major comments (3)
  1. [§2.4, Eq. (21), Algorithms 1–2, Appendix A.1] The closure map M is trained on pairs (a_proj, g_proj) obtained by direct projection of FOM fields, but at runtime it is evaluated at Newton iterates a_i and at g_i = G(a_i, μ), which can differ from a_proj. Appendix A.1 correctly identifies this exposure-bias problem and splits the resulting error into a generalization term of M and a term involving G through τ_exact in Eq. (25). However, the coupled loss L_MG in Eq. (21) still evaluates M and G only at a_proj; it never samples a_sol or closed-loop trajectories. Since the paper reports no measurement of ||a_sol − a_proj|| nor of the two terms in Eq. (25), the claim that DD-EV-ROM* resolves the exposure bias is not quantitatively established. I request a diagnostic showing the magnitude of the two terms in Eq. (25) across test cases and modal regimes, or an ablation demonstrating that the coupled training changes the online behavior beyond acting as a regularizer.
  2. [Section 3, reproducibility] The empirical conclusions rest on trained neural networks, but the paper reports no variability across random initializations or training seeds and releases no code or data. This is a load-bearing point because the observed gains are moderate in some regimes and even negative in others (e.g., negative velocity gain for the standard DD-EV-ROM in Figure 12 for test case (b), and negative velocity gain at (N_u, N_p, N_νt) = (3,4,14) in Figure 17 for test case (c)). Without seed-over-seed statistics, a reader cannot assess whether the reported improvements are robust or could be the result of a favorable training run. Please report at least mean and standard deviation over a few seeds, or explicitly state that fixed random seeds were used, and clarify the availability of code and data.
  3. [Section 3.2.2 vs. Appendix B.1] The text contains a direct contradiction about generalization behavior. Section 3.2.2 states that for test case (b) 'Train and test results are always similar', while Appendix B.1, discussing the statistical plots in Figures 21 and 22, states: 'In all cases a large difference between train and test errors for all the methods.' These statements cannot both be true unless one refers to gains and the other to absolute errors. Since the similarity of train and test performance is used in §3.1.2 as evidence that the closure mapping generalizes well, this inconsistency needs to be resolved and the text corrected.
minor comments (5)
  1. [§2.3, step 2] The condition 'ˆNp ą N u' appears to be a typo for 'ˆNp ą N p' (or 'ˆNp ą N_p'); as written, the two dimensions are treated asymmetrically without justification.
  2. [Algorithm 1, line 3] The initialization line 'b 0 = a proj 0 = [b proj i]^Np' incorrectly assigns the pressure coefficients to 'a proj 0'; it should read 'b_0 = b_proj_0 = [b_proj_i]_{i=1}^{N_p}' or simply 'b_0 = [b_proj_i]_{i=1}^{N_p}'.
  3. [§3.1.3, last sentence] The sentence 'the DD-EV-ROM* is significantly more accurate than the DD-EV-ROM*' is self-comparison; it should compare DD-EV-ROM* with DD-EV-ROM.
  4. [§2.3, before Eq. (17)] The notation 'τ = M(a,b,μ)' conflicts with the definition of M in §2.4 as M(a,g,μ); since the network takes the eddy-viscosity coefficients g as input, the formula should use g, not b.
  5. [Eq. (22)] The definition of the time-dependent relative error would be clearer if the FOM reference field and the ROM field were explicitly named, and if the superscript/subscript notation were defined consistently before its first use in Section 3.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the data-driven closure claim is benchmarked on held-out parameters against FOM references and does not reduce to its training inputs.

full rationale

The paper's central claim is that adding a learned closure term to the POD-Galerkin eddy-viscosity ROM improves pressure and velocity accuracy relative to the baseline EV-ROM. The closure target tau_exact is defined in Eq. (12) as the difference between the truncated high-dimensional projected operators and the reduced operators, and the networks M and G are trained on the resulting projected coefficients via the losses in Eqs. (18)-(19). This is an in-sample supervised fitting of the closure map, but the paper does not present that fit itself as the prediction. The reported gains are computed with Eqs. (22)-(24) against full-order-model references, separately for train and test parameter sets, and the test parameters are explicitly held out. The authors also identify and analyze the train/online mismatch in Appendix A.1, Eq. (25), where a_sol can differ from a_proj, and they introduce the coupled loss in Eqs. (20)-(21) to mitigate it. This exposure-bias concern is a correctness or robustness limitation, not a circular step: the method's success is not guaranteed by construction, and indeed test case (b) shows negative gains for the standard DD-EV-ROM in some regimes. Self-citations to prior closure work such as [23], [24], and [37] are used to motivate the approach, but the numerical validation is self-contained against external benchmarks, and no load-bearing argument reduces to a self-citation or to a uniqueness theorem. The derivation chain is therefore not circular.

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

The method pulls from prior literature the POD-Galerkin framework, the EV-ROM baseline, the kappa-omega SST FOM, and the DeepONet/MIONet architectures. The new content is the parametric closure map M and the coupled training procedure; these are fitted to data from the same snapshots used to build the POD basis, and their generalization is tested on held-out parameters.

free parameters (4)
  • Penalty coefficient tau = 10^3 (cylinder), 1 (cavity, backstep)
    Tuned through sensitivity analysis for each problem; affects boundary condition enforcement and condition number of the reduced system (Section 2.2).
  • Neural network weights of closure map M = Not listed (trained with loss LM or L*M)
    Fitted to exact closure terms from training snapshots; core data-driven component (Section 2.4).
  • Neural network weights of eddy viscosity map G = Not listed (trained with loss LG)
    Fitted to projected eddy viscosity coefficients (Section 2.4).
  • Mode counts (Nu, Np, N_nu_t) = Vary by regime, Tables 1-3
    Selected via retained energy thresholds; the regime determines ROM dimension and closure difficulty.
assumptions (5)
  • domain assumption RANS/URANS equations with kappa-omega SST turbulence model are an adequate high-fidelity description of the flows.
    The FOM in Section 2.1 is treated as ground truth for all snapshot computations.
  • domain assumption The solution field lies close to a low-dimensional linear POD subspace.
    Used in Eq. (4); the paper acknowledges in Section 4 that this linearity limits accuracy.
  • domain assumption The exact closure term depends only on the reduced coefficients (a,g) and problem parameters mu, and is learnable from data.
    Central modeling assumption of Section 2.3; determines the input-output structure of M.
  • standard math Empirical risk minimization over training parameters generalizes to unseen parameters.
    Standard supervised learning assumption; validated on held-out test parameters in Section 3.
  • domain assumption Boussinesq eddy viscosity hypothesis holds in the FOM.
    Used to define eddy viscosity snapshots nu_t in Section 2.1.

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Cite this review

Pith. "Pith review of Data-driven Closure Strategies for Parametrized Reduced Order Models via Deep Operator Networks." pith.science (2026). https://pith.science/paper/CR6EUTIP

@misc{pith2026250517305,
  author       = {Pith},
  title        = {Pith review of: Data-driven Closure Strategies for Parametrized Reduced Order Models via Deep Operator Networks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CR6EUTIP}},
  note         = {Machine review of arXiv:2505.17305}
}
read the original abstract

In this paper, we propose an equation-based parametric Reduced Order Model (ROM), whose accuracy is improved with data-driven terms added into the reduced equations. These additions have the aim of reintroducing contributions that in standard reduced-order approaches are not taken into account. In particular, in this work we focus on a Proper Orthogonal Decomposition (POD)-based formulation and our goal is to build a closure or correction model, aimed to re-introduce the contribution of the discarded modes. The approach has been investigated in previous works, and the goal of this manuscript is to extend the model to a parametric setting making use of machine learning procedures, and, in particular, of deep operator networks. More in detail, we model the closure terms through a deep operator network taking as input the reduced variables and the parameters of the problem. We tested the methods on three test cases with different behaviors: the periodic turbulent flow past a circular cylinder, the unsteady turbulent flow in a channel-driven cavity, and the geometrically-parametrized backstep flow. The performance of the machine learning-enhanced ROM is deeply studied in different modal regimes, and considerably improved the pressure and velocity accuracy with respect to the standard POD-Galerkin approach.

Figures

Figures reproduced from arXiv: 2505.17305 by the authors.

Figure 1
Figure 1. Standard version of the DeepONet, as in [33]. Such class of neural networks may also have more than two sub-networks, allowing to separately handle more than two inputs. We indeed distinguish two type of deep operator networks, one for the turbulence map and one for the closure map. For the turbulence map Gpa, µq we consider a modified version of the DeepONet, whose structure is detailed in [PITH_FULL_IMAGE:figures… view at source ↗
Figure 2
Figure 2. DeepONet architecture used for the eddy viscosity coefficients modeling G [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. MIONet architecture used for the closure modeling M. 11 [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figures from the paper (29 more)
Figure 4
Figure 4. Figure 4: The domain and full order mesh considered for the periodic flow around a circular cylinder. 20 21 22 23 24 25 26 t [s] 8.333e-05 0.0001 0.000125 0.00016667 0.00025 2 × 10−4 3 × 10−4 ν (train) [m 2/s] 7.69e-05 0.000115 0.000333 2 × 10−4 3 × 10−4 ν (test) [m 2/s] Test da…
Figure 5
Figure 5. Figure 5: Graphical representations of the sets of parameters used in the offline stage for test case (a), in green, and in the online stage, in red. The green shadow describes the accessible offline/training area. The time and parameter ranges considered to run the FOM and coll…
Figure 6
Figure 6. Figure 6: Cumulative eigenvalues and eigenvalues decay for the test case a [PITH_FULL_IMAGE:figures/full_fig_p015_6.png]
Figure 7
Figure 7. Figure 7: Comparison in the average performance of the DD-EV-ROM and DD-EV-ROM‹ with respect to the baseline EV-ROM, for test case a. The metric of interest is the time average gain in relative error for train and test viscosities (lower/upper triangles). The error is represente…
Figure 8
Figure 8. Figure 8: Graphical pressure absolute error at ν “ 1.15e ´ 4 m2 s at the final time instance (t “ 8), of EV-ROM, DD-EV-ROM, DD-EV-ROM‹ , and for the projected field. The modal regime is pNu, Np, Nνt q “ p10, 24, 30q. 18 [PITH_FULL_IMAGE:figures/full_fig_p018_8.png]
Figure 9
Figure 9. Figure 9: The domain and full order mesh considered for the channel-driven flow test case. 5.0 7.5 10.0 12.5 15.0 17.5 20.0 t [s] 5e-06 1e-05 2e-05 4e-05 0.0001 ν (train) [m 2/s] 4e-06 7e-06 0.000125 ν (test) [m 2/s] Test data Training data [PITH_FULL_IMAGE:figures/full_fig_p01…
Figure 10
Figure 10. Figure 10: Graphical representations of the sets of parameters used in the offline stage for test case (b), in green, and in the online stage, in red. The green shadow describes the accessible offline/training area. For the POD we consider the viscosity range specified in [PITH…
Figure 11
Figure 11. Figure 11: Cumulative eigenvalues and eigenvalues decay for test case b. The following part is dedicated to a quantitative error analysis, comparing the results obtained with different modal combinations, select through energy criteria and specified in [PITH_FULL_IMAGE:figures/…
Figure 12
Figure 12. Figure 12: Comparison in the average performance of the DD-EV-ROM and DD-EV-ROM‹ with respect to the baseline EV-ROM, for test case (b). The metric of interest is the time average gain in relative error for train and test viscosities (lower/upper triangles). The error is represe…
Figure 13
Figure 13. Figure 13: Graphical pressure absolute error at ν “ 7e ´ 6 m2 s at the final time instance (t “ 15), of EV-ROM, DD-EV-ROM, DD-EV-ROM‹ , and for the projected field. The modal regime is pNu, Np, Nνt q “ p2, 5, 1q. The Figure shows only a “zoom” close to the cavity, since the re￾m…
Figure 14
Figure 14. Figure 14: An example of the domain and full order mesh considered for the backstep flow, for one set of parameters. We consider a specific set of training parameters for the POD tµiu Nµ i“1 , where µi “ pαi , h1, h2q. Nµ “ 70 is the number of simulations with deformed mesh, whi…
Figure 15
Figure 15. Figure 15: Graphical representations of the sets of parameters used in the offline stage for test case (c), in green, and in the online stage, in red. The green shadow describes the accessible offline/training area, while the black square is the mid-configuration, where the mode…
Figure 16
Figure 16. Figure 16: Cumulative eigenvalues and eigenvalues decay for test case (c) [PITH_FULL_IMAGE:figures/full_fig_p024_16.png]
Figure 17
Figure 17. Figure 17: Comparison in the average performance of the DD-EV-ROM and DD-EV-ROM‹ with respect to the baseline EV-ROM, for test case (c). The metric of interest is the average gain in relative error for train and test parameters (lower/upper triangles). The error is represented f…
Figure 18
Figure 18. Figure 18: displays the errors for the pressure field, and highlight the enhancement obtained with the DD-EV-ROM and DD-EV-ROM‹ , with respect to the baseline EV-ROM. The velocity and eddy viscosity error fields are showed in B.3. However, we stress the fact that the improvement…
Figure 19
Figure 19. Figure 19: Time trends of the relative L 2 errors for the three fields of interest u, p, and νt, for one test viscosity ν “ 1.15e ´ 4 m2 s . The modal regime is pNu, Np, Nνt q “ p3, 6, 5q. Similar considerations on the comparison of different regimes arise from Figures 21 and 22…
Figure 20
Figure 20. Figure 20: Time trends of the relative L 2 errors for the three fields of interest u, p, and νt, for one test viscosity ν “ 1.15e ´ 4 m2 s . The modal regime is pNu, Np, Nνt q “ p10, 24, 30q. • The DD-EV-ROM‹ outperforms the DD-EV-ROM method when considering a larger number of m…
Figure 21
Figure 21. Figure 21: Statistical performance for train and test viscosities in the time average of the relative L 2 error with respect to the high-fidelity solutions. The Figure represents the median and the error bounds for the three fields of interest in the case pNu, Np, Nνt q “ p3, 6,…
Figure 22
Figure 22. Figure 22: Statistical performance for train and test viscosities in the time average of the relative L 2 error with respect to the high-fidelity solutions. The Figure represents the median and the error bounds for the three fields of interest in the case pNu, Np, Nνt q “ p10, 2…
Figure 23
Figure 23. Figure 23: Graphical absolute error for the velocity magnitude at ν “ 1.15e ´ 4 m2 s at the final time instance (t “ 8), of EV-ROM, DD-EV-ROM, DD-EV-ROM‹ , and for the projected field. The modal regime is pNu, Np, Nνt q “ p10, 24, 30q. 37 [PITH_FULL_IMAGE:figures/full_fig_p037_…
Figure 24
Figure 24. Figure 24: Graphical eddy viscosity absolute error at ν “ 1.15e ´ 4 m2 s at the final time instance (t “ 8), of EV-ROM, DD-EV-ROM, DD-EV-ROM‹ , and for the projected field. The modal regime is pNu, Np, Nνt q “ p10, 24, 30q. 38 [PITH_FULL_IMAGE:figures/full_fig_p038_24.png]
Figure 25
Figure 25. Figure 25: Time trends of the relative L 2 errors for the three fields of interest u, p, and νt, for one test viscosity ν “ 7e ´ 6 m2 s . The modal regime is pNu, Np, Nνt q “ p1, 3, 1q. 0 5 10 15 Time [s] 10−2 2 × 10−2 3 × 10−2 4 × 10−2 Relative error Field u 0 5 10 15 Time [s] …
Figure 26
Figure 26. Figure 26: Time trends of the relative L 2 errors for the three fields of interest u, p, and νt, for one test viscosity ν “ 7e ´ 6 m2 s . The modal regime is pNu, Np, Nνt q “ p2, 5, 1q. 39 [PITH_FULL_IMAGE:figures/full_fig_p039_26.png]
Figure 27
Figure 27. Figure 27: Statistical performance for train and test viscosities in the time average of the relative L 2 error with respect to the high-fidelity solutions. The Figure represents the median and the error bounds for the three fields of interest in the case pNu, Np, Nνt q “ p1, 3,…
Figure 28
Figure 28. Figure 28: Statistical performance for train and test viscosities in the time average of the relative L 2 error with respect to the high-fidelity solutions. The Figure represents the median and the error bounds for the three fields of interest in the case pNu, Np, Nνt q “ p2, 5,…
Figure 29
Figure 29. Figure 29: Statistical performance for train and test parameters of the relative L 2 error with respect to the high-fidelity solutions. The Figure represents the median and the error bounds for the three fields of interest in the case pNu, Np, Nνt q “ p4, 5, 20q. EV-ROM DD-EV-RO…
Figure 30
Figure 30. Figure 30: Statistical performance for train and test parameters of the relative L 2 error with respect to the high-fidelity solutions. The Figure represents the median and the error bounds for the three fields of interest in the case pNu, Np, Nνt q “ p3, 4, 12q. The overall imp…
Figure 31
Figure 31. Figure 31: Graphical absolute error for velocity magnitude field at parameters pα, h1, h2q “ p6.3 ˝ , 0.99 m, 1.58 mq of EV-ROM, DD-EV-ROM, DD-EV-ROM‹ , and for the projected field. The modal regime is pNu, Np, Nνt q “ p3, 4, 12q. (a) (b) (c) (d) [PITH_FULL_IMAGE:figures/full_f…
Figure 32
Figure 32. Figure 32: Graphical absolute error for eddy viscosity field at the test parameters pα, h1, h2q “ p6.3 ˝ , 0.99 m, 1.58 mq of EV-ROM, DD-EV-ROM, DD-EV-ROM‹ , and for the projected field. The modal regime is pNu, Np, Nνt q “ p3, 4, 12q. 42 [PITH_FULL_IMAGE:figures/full_fig_p042_…

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