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Machine Learning-based quadratic closures for non-intrusive Reduced Order Models

T0 review · 4 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read A neural-operator-parameterized quadratic correction re-introduces neglected POD modes and reduces prediction error by up to 90% over the linear ROM baseline on two fluid benchmarks.

desk verdict Useful moderate extension of quadratic closures; the 80% claim over Quad-LS rests on an underdetermined baseline, and discretization agnosticism is untested. read the letter →

arxiv 2506.09830 v2 pith:KBW6C5VR submitted 2025-06-11 math.NA cs.NA

Gabriele Codega , Anna Ivagnes , Nicola Demo , Gianluigi Rozza This is my paper · ORCID
classification math.NAcs.NA MSC 68T0735Q3565N99
keywords ReducedOrderModelsProperOrthogonalDecompositionQuadraticclosureNeuraloperatorsMIONetDeepONetNon-intrusiveROMFluiddynamics
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 tries to show that a quadratic closure for POD-based non-intrusive reduced-order models can be made substantially more accurate by learning the quadratic operator with a neural operator rather than fitting one global tensor by least squares. The proposed models, QuadNet and QuadNet-$\mu$, add a correction $a(\mu)^T C(\Phi_r,x,\mu)a(\mu)$ to the linear POD reconstruction, where $C$ is produced by a DeepONet or MIONet and is therefore continuous in space and, in the $\mu$ version, in the parameter. On two fluid benchmarks the corrected models cut test errors by up to 90 percent relative to the linear POD-RBF baseline and up to 80 percent relative to the least-squares quadratic closure. The method matters because it offers a route to more accurate ROMs in the under-resolved, scarce-data regime while preserving the interpretable POD-plus-quadratic structure.

What carries the argument

The load-bearing object is the quadratic closure operator $C(\Phi_r,x,\mu)$, which replaces the fixed tensor of the least-squares closure with the output of a neural operator. In QuadNet the operator is built with a DeepONet-like split, a branch network that reads the $r$ POD modes and a trunk network that reads spatial coordinates, followed by a small combining network that outputs the symmetric quadratic form; in QuadNet-$\mu$ a second branch network also reads the parameter $\mu$, giving a MIONet structure, a multi-input operator network with separate branch networks for each input type. This makes the correction continuous in space and parameter, so the trained operator can be evaluated at any mesh point and adapts locally in parameter space, while training with a relative-error loss keeps small-magnitude corrections from being swamped by scale differences.

What would settle it

Manufacture a test problem whose exact correction is cubic, such as $\tau_{\mathrm{exact}}=(a^T a)a$ scaled by a known field, train QuadNet-$\mu$ on abundant data, and check whether the learned correction can reduce the residual below the best quadratic approximation; if it cannot, the quadratic ansatz, not the network, is the limiting factor.

Watch

Extended reading notes

Core claim

The central claim is that the exact truncation error $\tau_{\mathrm{exact}}(\mu)=u(\hat{x},\mu)-\sum_i a_i(\mu)\phi_i(\hat{x})$ can be approximated by a quadratic form $a(\mu)^T C(\Phi_r,x,\mu)a(\mu)$ in the retained POD coefficients, with the operator $C$ emitted by a multi-input operator network. Because the network receives the POD modes, the spatial coordinate, and the parameter as inputs, the correction varies continuously in space and in parameter space, so it is independent of the discretization and acts as a local closure rather than one global least-squares tensor. In the backward-facing step and lid-driven cavity benchmarks, the parameter-dependent version, QuadNet-$\mu$, reports the lowest mean, median, and spread of errors, with the paper summarizing gains of up to 90 percent over POD-RBF and up to 80 percent over Quad-LS, including when training uses only a subset of spatial points or a small number of snapshots.

Load-bearing premise

The load-bearing premise is that the entire neglected part of the flow can be written as quadratic combinations of the retained POD coefficients; if the missing part depends on those coefficients in a cubic or higher-order way, the correction is misspecified and the reported gains may not transfer to harder problems.

Editorial extensions

If this is right

  • If the central claim holds, the neural-operator closure is the more reliable correction in the under-resolved regime: QuadNet-$\mu$ beats both POD-RBF and Quad-LS on the two tested flow problems, with the largest relative gains when few POD modes are kept.
  • Because the operator is space-continuous, a model trained on 10 to 50 percent of the mesh nodes keeps accuracy close to full-mesh training, lowering offline memory and training cost.
  • In scarce-data settings the neural closure is more robust than the least-squares tensor, which can produce errors larger than the uncorrected linear model when snapshots are few.
  • The same architecture can be adjusted to higher-order corrections by changing the network's output, so the framework is not limited to quadratic forms.

Reading between the lines

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

  • Beyond the paper, the quadratic ansatz is the real ceiling: on problems where the neglected-mode contribution has significant cubic or higher-order dependence on $a$, the same network would need a higher-degree output, and the reported gains should be expected to shrink.
  • Because $C$ is evaluated pointwise in space, a trained QuadNet-$\mu$ could plausibly be queried on a different or re-meshed domain without retraining; the paper demonstrates training on partial meshes but not cross-mesh transfer, so this is a testable extrapolation.
  • The Boltzmann-like collocation-point sampling could be turned into an active-learning loop that adds points where the current network's residual is largest; the paper introduces the sampling idea but does not explore that adaptive strategy.
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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 data-driven quadratic closure for non-intrusive ROMs built on POD. The correction term tau(x;mu) = a^T C(Phi_r,x,mu) a is added to the POD reconstruction, and the operator C is learned either by least squares (Quad-LS), by a DeepONet (QuadNet), or by a MIONet with parameter and space inputs (QuadNet-mu). The methods are tested on the backward-facing step and lid-driven cavity benchmarks, with fixed-mode comparisons, partial-observation experiments, and a scarce-data study over (r, N_mu) combinations. The central empirical claims are that QuadNet-mu improves over POD-RBF by up to 90% and over Quad-LS by up to 80% in relative error.

Significance. If the numerical claims are robust, the paper makes a useful contribution: the parameter- and space-continuous parametrization of the quadratic operator is a natural extension of quadratic-manifold ROMs, and the reported gains in the scarce-data regime would be practically relevant. The paper also gives a clear presentation of the loss function, the architecture, and the test cases, and the fixed-mode comparison in Section 3.4 is well-posed. The main weakness is that the headline comparison against Quad-LS relies on an unregularized, underdetermined baseline, and the neural results are reported without repeated training runs. These issues must be addressed before the central claims can be accepted as stated.

major comments (4)
  1. [Section 3.6, Eqs. (7) and (11), Figures 8-9] The Quad-LS baseline in the scarce-data comparison is unregularized. For r=7 (S=28) and r=9 (S=45), with N_mu=10,20,30,40, the per-node least-squares system has more unknowns than equations; the paper itself states that Quad-LS 'struggles in providing meaningful corrections' in this regime. The conclusion's 'up to 80% error reduction compared to Quad-LS' is therefore computed against a degenerate baseline. Please add a regularized Quad-LS variant (e.g., ridge or truncated SVD) and a Quad-LS trained with the same relative-error weighting as in Eq. (13), and recompute the reported improvements.
  2. [Tables 3-4 and Figures 8-9] All numerical results are reported for a single training run per configuration. Since QuadNet and QuadNet-mu are stochastic neural networks, and Section 3.6 acknowledges sensitivity to weight initialization and hyperparameters, the reported rankings may be seed artifacts. Please provide means and standard deviations over at least several random seeds, and state whether the observed differences are statistically significant.
  3. [Section 3.5 and abstract] The claim that the model is 'agnostic of the domain discretisation' is not supported by the experiments. Training on 10%, 20%, or 50% of the nodes of the same mesh and evaluating on the remaining nodes of that same mesh tests spatial subsampling, not independence from the discretization. Please either test on a genuinely different mesh or off-mesh evaluation points, or soften the claim to 'space-continuous' rather than discretization-agnostic.
  4. [Eq. (14)] The exact correction is assumed to be quadratic in the retained POD coefficients, but the manuscript does not provide any diagnostic of how well this ansatz represents tau_exact on the training data. Please report the relative residual of the best quadratic fit or an analysis of the leading omitted terms, so that the reader can assess the misspecification risk when the method is applied to flows with stronger nonlinearity.
minor comments (5)
  1. [Tables 1-2] The 'Min. loss 10^-2' column is unclear; please specify whether it is a stopping criterion, a target loss, or a lower bound for the learning rate.
  2. [Throughout] There are several typographical and formatting issues: 'F unding' in the funding line, 'Sentries' for 'S entries', inconsistent spacing in 'QuadNet -mu', and 'Ntrain_mu' spacing. Please proofread.
  3. [Eq. (19)] The normalization parameter tilde_tau is not precisely defined; please specify whether it is the mean or maximum of the correction norm over nodes or snapshots.
  4. [Section 3.4] The statement that 'QuadNet is not necessarily better than Quad-LS' is an understatement given Tables 3 and 4, where QuadNet is consistently worse; please rephrase to reflect the observed ranking.
  5. [Section 3.5] The reported approximate 10% training-time reduction is based on runs on a non-dedicated workstation; please report more robust timing statistics or acknowledge the noise in these measurements.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the neural closure operators are trained on source snapshots and evaluated on held-out configurations, and the quadratic ansatz is an explicit modeling assumption rather than a derived result.

full rationale

The paper's central derivation is a data-driven correction pipeline: the exact correction is defined in Eq. (5), a quadratic ansatz is introduced in Eqs. (6), (12), and (14), and the operator C is learned by minimizing the relative-error loss (13) on training snapshots. Predictions are then evaluated on held-out test snapshots in Tables 3-4 and Figures 8-9, so the reported errors are genuine out-of-sample quantities and no fitted parameter is relabeled as a prediction. The quadratic form a^T C(Phi_r, x, mu) a is explicitly presented as an ansatz ('all providing a quadratic ansatz in the reduced coefficients'), not as a theorem derived from Navier-Stokes, so adopting it is an assumption rather than a circular step. Self-citations such as Ref. [16] (DeepONet residual learning) and Refs. [22-23] (closure strategies) are used only as background pointers and do not carry the argument. The skeptical concern that the Quad-LS baseline is underdetermined for some (r, N_mu) combinations in Section 3.6 is an experimental-fairness and correctness question, not a circularity, because the comparison is between two independently trained models on test data. Therefore the derivation chain is self-contained with respect to the paper's claims, and no circular step can be exhibited from the text and equations.

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

The method is supervised learning: it does not introduce new physical entities. The core assumptions are the quadratic ansatz for the closure, the smoothness of the operator, the representativeness of the training manifold, and the accuracy of the RBF coefficient interpolation. These are stated explicitly in Sections 2 and 3.

free parameters (5)
  • r (number of retained POD modes) = 3 in main setting; 3, 5, 7, 9 in Section 3.6
    User-selected truncation that defines the under-resolved regime; all results depend on it.
  • RBF kernel type = linear for backstep; thin plate spline for cavity
    Chosen per test case in Section 3.3.1; affects baseline POD-RBF accuracy and the coefficients a(mu) used by all corrected models.
  • Network hyperparameters = p=S=6, hidden layers 7x20, Tanh, lr O(10^-3), max epochs 20000, min loss 10^-2
    Hand-tuned and reported as sensitive; Tables 1 and 2 give ranges rather than exact values, so the reported accuracy is conditional on these choices.
  • Training snapshot count N_mu = 400 (backstep), 100 (cavity)
    Fixed for the main comparison; varied in scarce-data experiments. Not fitted, but determines the amount of information the closure can learn.
  • Boltzmann sampling constants = tilde tau and epsilon unspecified
    Partial-data results in Section 3.5 depend on the sampling distribution in Equation (19); the normalization and epsilon are not given, which impacts reproducibility.
assumptions (5)
  • domain assumption The exact residual tau_exact = u minus the POD reconstruction is well approximated by a quadratic form a^T C(Phi_r, x, mu) a in the retained POD coefficients.
    Core ansatz in Equations (12) and (14), inherited from quadratic manifold literature; the paper does not test whether higher-order terms are needed.
  • domain assumption The map C is smooth and learnable by a fully connected DeepONet/MIONet so that it generalizes in space and in the parameter.
    Sections 2.2.2 and 2.2.3 assume neural approximation; no approximation guarantee or monotonicity condition is shown for this specific operator.
  • domain assumption POD modes from training snapshots represent the solution manifold for unseen parameters.
    Section 2.1 builds the basis from the snapshot matrix S; Section 3.6 admits that few snapshots may be non-representative, showing this assumption is load-bearing.
  • domain assumption RBF interpolation yields accurate reduced coefficients a(mu*) at test parameters.
    Equation (2) supplies all a values used in the corrections; its error is never separated from the closure error.
  • standard math FEniCS and OpenFOAM solutions are reliable ground truth.
    Sections 3.1 and 3.2 use these high-fidelity discretizations as reference; convergence or mesh-independence studies are not reported.

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Cite this review

Pith. "Pith review of Machine Learning-based quadratic closures for non-intrusive Reduced Order Models." pith.science (2026). https://pith.science/paper/KBW6C5VR

@misc{pith2026250609830,
  author       = {Pith},
  title        = {Pith review of: Machine Learning-based quadratic closures for non-intrusive Reduced Order Models},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KBW6C5VR}},
  note         = {Machine review of arXiv:2506.09830}
}
read the original abstract

In the present work, we introduce a data-driven approach to enhance the accuracy of non-intrusive Reduced Order Models (ROMs). In particular, we focus on ROMs built using Proper Orthogonal Decomposition (POD) in an under-resolved and marginally-resolved regime, i.e. when the number of modes employed is not enough to capture the system dynamics. We propose a method to re-introduce the contribution of neglected modes through a quadratic correction term, given by the action of a quadratic operator on the POD coefficients. Differently from the state-of-the-art methodologies, where the operator is learned via least-squares optimisation, we propose to parametrise the operator by a Multi-Input Operators Network (MIONet). This way, we are able to build models with higher generalisation capabilities, where the operator itself is continuous in space -- thus agnostic of the domain discretisation -- and parameter-dependent. We test our model on two standard benchmarks in fluid dynamics and show that the correction term improves the accuracy of standard POD-based ROMs.

Figures

Figures reproduced from arXiv: 2506.09830 by the authors.

Figure 1
Figure 1. Schematic representation of the DeepONet architecture employed in QuadNet. [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Schematic representation of the MIONet architecture employed in QuadNet - [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Domains and corresponding offline grid for the two test cases. [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: Decay of the snapshot matrix’s singular values for the two test cases. Values are [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]
Figure 5
Figure 5. Figure 5: Plots for a random test FOM solution and corresponding error fields for POD-RBF, [PITH_FULL_IMAGE:figures/full_fig_p012_5.png]
Figure 6
Figure 6. Figure 6: Comparison between samples of points drawn from different distribution. The blue [PITH_FULL_IMAGE:figures/full_fig_p014_6.png]
Figure 7
Figure 7. Figure 7: Boxplots comparing the accuracy of models. [PITH_FULL_IMAGE:figures/full_fig_p014_7.png]
Figure 8
Figure 8. Figure 8: Comparison of QuadNet -µ errors with POD-RBF and Quad-LS. The left half of the circles shows the relative difference in error between QuadNet -µ and POD-RBF, while the right half shows the difference between QuadNet -µ and Quad-LS. We find that QuadNet -µ performs bett…
Figure 9
Figure 9. Figure 9: Comparison between POD-RBF, Quad-LS and QuadNet - [PITH_FULL_IMAGE:figures/full_fig_p017_9.png]

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Reviewed August 7, 2026 · model on record in the stance chip above.