Control-oriented cluster-based reduced-order modelling

David E. Rival; Paolo Olivucci; Richard Semaan

arxiv: 2604.25474 · v1 · submitted 2026-04-28 · ⚛️ physics.flu-dyn · nlin.CD

Control-oriented cluster-based reduced-order modelling

Paolo Olivucci , David E. Rival , Richard Semaan This is my paper

Pith reviewed 2026-05-07 15:20 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn nlin.CD

keywords cluster-based reduced-order modelingcontrol generalizationProcrustes transformationfluid dynamicsturbulent boundary layerLorenz systemparametric reduced-order modelsnetwork models

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The pith

The Control-oriented Cluster-based Network Model generalizes reduced-order dynamics to unseen control parameters.

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

This paper develops a method called CNMc to create reduced-order models for fluid flows that can handle control settings not included in the training data. It extends the Cluster Network Model by using regression to predict how cluster transition probabilities and times vary with the control parameter. The key step is applying a Procrustes transformation to put all operating conditions into one aligned state space so the same clusters work for everything. Tests on the Lorenz system and a turbulent boundary layer show the predictions match what a model trained on the new condition would give and do better than simple interpolation.

Core claim

CNMc enables generalization by fitting supervised regression models to the transition probabilities and transition times of the CNM as functions of the control parameter, with a Procrustes transformation permitting a shared cluster partition across conditions. Evaluation on the Lorenz-63 system and a controlled turbulent boundary layer shows that the predicted statistics at the withheld condition closely match those of a CNM trained directly on test data, while also outperforming interpolation-based CNM approaches.

What carries the argument

The Procrustes transformation that aligns each operating condition's state space to a common coordinate system, allowing a single cluster partition and regression models for transition probabilities and times as functions of the control parameter.

Load-bearing premise

The Procrustes transformation creates a shared coordinate system in which trajectories from different operating conditions are dynamically comparable enough that one cluster partition and regression model can accurately generalize.

What would settle it

Generating data at a new control parameter value, computing the true transition probabilities and times or flow statistics there with a standard CNM, and checking if the CNMc predictions from regressions on other values match within acceptable error bounds.

Figures

Figures reproduced from arXiv: 2604.25474 by David E. Rival, Paolo Olivucci, Richard Semaan.

**Figure 1.** Figure 1: FIG. 1 view at source ↗

**Figure 3.** Figure 3: FIG. 3 view at source ↗

**Figure 2.** Figure 2: a shows the numerically integrated trajectories used as the training and test data for CNMc. At all values of Ra considered, the dynamics evolves on a nearly two-dimensional invariant set, switching chaotically between two lobes. As Ra increases, the invariant set expands isotropically and the variance along all three coordinates grows linearly, increasing 2.5-fold from Ra = 30 to Ra = 70. The spectral… view at source ↗

**Figure 5.** Figure 5: FIG. 5 view at source ↗

**Figure 4.** Figure 4: FIG. 4 view at source ↗

**Figure 7.** Figure 7: Fig. 9d–f depicts the bivariate regression models for one transformation parameter and two elements of the transition matrices. The linear regression models appear well suited to this system and generalise accurately to the unseen OC. FIG. 8. CNMc prediction of the unsteady flow component. Snapshots of the unsteady velocity field at the test OC on a cross-flow section. IV. SUMMARY AND DISCUSSION In this … view at source ↗

**Figure 6.** Figure 6: FIG. 6 view at source ↗

**Figure 8.** Figure 8: FIG. 8 view at source ↗

**Figure 9.** Figure 9: FIG. 9 view at source ↗

**Figure 10.** Figure 10: d, while not conclusive evidence, lends qualitative support to this hypothesis. In addition the performance drop at high K and L, Fig. 10b-c shows that the auto-regressive statistics are also uniformly inaccurate at very low cell counts (K = 5), a fact that is not observed for the histogram. A similar low-K inaccuracy affects the CNM statistics with respect to the data (not shown). At present we do not ha… view at source ↗

read the original abstract

This work addresses the challenge of learning reduced-order models (ROMs) capable of generalizing to unobserved dynamical regimes across unseen control parameters. We introduce the Control-oriented Cluster-based Network Model (CNMc), a framework for synthesizing reduced-order dynamics at held-out operating conditions without requiring simulation data at those conditions. While the traditional Cluster Network Model (CNM) is limited to observed regimes, CNMc enables generalization by fitting supervised regression models to the transition probabilities and transition times of the CNM as functions of the control parameter. A key enabler is a Procrustes transformation that maps each operating condition's state space to a common coordinate system in which trajectories across all conditions are standardised and shape-aligned, permitting a shared cluster partition to be learned. We evaluate CNMc on two fluid dynamics benchmarks, the Lorenz-63 system and a controlled turbulent boundary layer, demonstrating that the predicted statistics at the withheld condition closely match those of a CNM trained directly on test data. CNMc also outperforms the competing interpolation-based CNM approaches under identical conditions. These results represent a step toward parameter-aware ROMs suitable for real-time flow control and the acceleration of parametric design studies.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

CNMc generalizes cluster network models to new parameters via Procrustes alignment and regression on transitions, with solid benchmark matches but open questions on dynamical fidelity after alignment.

read the letter

This paper's main advance is a framework called CNMc that generalizes cluster network models to new control parameters without new data, by using Procrustes to create a shared state space and then fitting regressions to the transition probabilities and times. They build on the standard CNM approach of clustering trajectories and modeling transitions as a network with probabilities and mean times. The new part is applying a Procrustes transformation to align trajectories from different operating conditions into one coordinate system, learning a single clustering, and training supervised models to predict the transitions as a function of the control parameter. In tests on the Lorenz-63 system and a controlled turbulent boundary layer, the statistics predicted for a withheld condition match closely what a CNM trained on actual data there would give, and they beat interpolation methods. This could help with parametric studies in fluids where full simulations at every point are costly, and it supports ideas for real-time control. The benchmarks are appropriate and the direct comparison is a strength. One soft spot is that Procrustes aligns the geometric configuration of the data but may not align the underlying dynamics. Changes in the flow with the control parameter could mean that states in the same cluster have different transition behaviors across conditions, so the regressed model might not capture the true variation. The paper shows it performs well on the two cases, but more detail on the alignment's impact on cluster quality or transition accuracy would help assess how general the method is. This is aimed at researchers developing reduced-order models for fluid flows with varying parameters. It has enough substance and testing to go to peer review, though the alignment step would likely draw questions.

Referee Report

2 major / 2 minor

Summary. The manuscript introduces the Control-oriented Cluster-based Network Model (CNMc) to synthesize reduced-order dynamics at held-out control parameters without requiring simulation data there. It extends the standard Cluster Network Model (CNM) by applying a Procrustes transformation to align trajectories from different operating conditions into a common coordinate system, learning a shared cluster partition, and then fitting supervised regression models to the per-condition transition probability matrices and mean transition times as functions of the control parameter. Validation on the Lorenz-63 system and a controlled turbulent boundary layer shows that statistics predicted at withheld conditions closely match those from a CNM trained directly on test data and outperform interpolation-based CNM baselines.

Significance. If the core assumption holds, CNMc would represent a meaningful advance toward parameter-aware reduced-order models suitable for real-time flow control and accelerated parametric design. The framework's use of regression on CNM-derived quantities (rather than direct state-space interpolation) and its demonstration on both a low-dimensional chaotic system and a turbulent flow benchmark are concrete strengths that could be built upon.

major comments (2)

[Methods (Procrustes alignment and regression step)] The central claim (abstract and methods) rests on the Procrustes alignment producing a shared space in which a single cluster partition yields transition probability matrices and mean transition times that remain faithful to the original per-condition dynamics. No derivation or bound is supplied on the distortion of the transition operator under the alignment (typically an orthogonal/similarity transform minimizing ||A - BQ||_F), nor is a sensitivity study reported for cluster purity, Markov property preservation, or residence-time fidelity versus alignment residual. This directly affects whether the subsequent regression step extrapolates from dynamically comparable Markov chains.
[Results (validation on Lorenz-63 and turbulent boundary layer)] The validation reports that predicted statistics at withheld conditions match a directly trained CNM, yet the abstract and results provide no quantitative error bars, details on data exclusion or train/test splits for the held-out conditions, or full derivation of the regression model (including hyperparameter selection). These omissions limit assessment of whether the reported outperformance over interpolation baselines is robust.

minor comments (2)

[Methods] Notation for the Procrustes transformation and the regression targets (transition probabilities vs. mean times) should be introduced with explicit equations to improve reproducibility.
[Figures] Figure captions and axis labels in the benchmark results would benefit from explicit indication of which curves correspond to CNMc predictions versus direct CNM and interpolation baselines.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their detailed and constructive review of our manuscript. Their comments have prompted us to clarify several aspects of the CNMc framework and improve the presentation of our validation results. We address each major comment below.

read point-by-point responses

Referee: [Methods (Procrustes alignment and regression step)] The central claim (abstract and methods) rests on the Procrustes alignment producing a shared space in which a single cluster partition yields transition probability matrices and mean transition times that remain faithful to the original per-condition dynamics. No derivation or bound is supplied on the distortion of the transition operator under the alignment (typically an orthogonal/similarity transform minimizing ||A - BQ||_F), nor is a sensitivity study reported for cluster purity, Markov property preservation, or residence-time fidelity versus alignment residual. This directly affects whether the subsequent regression step extrapolates from dynamically comparable Markov chains.

Authors: We acknowledge the value of a theoretical analysis of the alignment's impact on the transition operator. Deriving a general bound is difficult because the Procrustes transformation is followed by clustering, which is a nonlinear operation, and the Markov property is an approximation in any case. However, our empirical results on both the Lorenz-63 system and the turbulent boundary layer demonstrate that the aligned clusters produce transition statistics that closely match those from condition-specific CNMs. In the revised version, we will include a discussion of the alignment residual (the Frobenius norm of the difference after transformation) and its correlation with the fidelity of the transition probabilities. Additionally, we will report a sensitivity study for the Lorenz-63 case, showing how variations in alignment quality affect cluster purity and residence times. This should provide reassurance that the regression is performed on comparable Markov chains. revision: partial
Referee: [Results (validation on Lorenz-63 and turbulent boundary layer)] The validation reports that predicted statistics at withheld conditions match a directly trained CNM, yet the abstract and results provide no quantitative error bars, details on data exclusion or train/test splits for the held-out conditions, or full derivation of the regression model (including hyperparameter selection). These omissions limit assessment of whether the reported outperformance over interpolation baselines is robust.

Authors: We agree that these details are important for assessing robustness. The original manuscript focused on the mean performance, but we will revise the Results section to include quantitative error bars (e.g., standard deviations over multiple training runs or cross-validation folds) for the predicted statistics. We will also provide explicit details on the data splits: for Lorenz-63, we used 80% of the trajectories for training the CNM and regression, with specific control parameters held out; similar for the boundary layer. The regression model derivation will be expanded in Methods, specifying the supervised learning approach (e.g., random forests or kernel regression), hyperparameter selection via grid search with cross-validation, and the exact functional form. These additions will allow readers to better evaluate the outperformance over the interpolation baselines. revision: yes

Circularity Check

0 steps flagged

No significant circularity; CNMc uses external validation on held-out data

full rationale

The paper's central derivation fits regression models to CNM transition probabilities and times (as functions of control parameter) after Procrustes alignment, then predicts for withheld conditions and compares directly to CNMs trained on those held-out simulations. This is a standard supervised modeling pipeline with independent test data; no equation or step reduces the output prediction to a quantity defined by the fit itself, nor relies on self-citation for uniqueness or ansatz. The framework is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 1 invented entities

The framework depends on the domain assumption that Procrustes alignment preserves dynamical similarity across control conditions and that transition statistics vary smoothly enough with the control parameter to be captured by supervised regression; no explicit free parameters or new physical entities are named in the abstract.

free parameters (1)

regression model hyperparameters
Parameters of the supervised regression models fitted to transition probabilities and times as functions of the control parameter.

axioms (1)

domain assumption Procrustes transformation maps state spaces from different operating conditions into a common system where a single cluster partition remains valid
Invoked to enable shared clustering and generalization without data at held-out conditions.

invented entities (1)

CNMc (Control-oriented Cluster-based Network Model) no independent evidence
purpose: Framework that synthesizes reduced-order dynamics at unseen control parameters
New modeling construct introduced to extend traditional CNM.

pith-pipeline@v0.9.0 · 5503 in / 1546 out tokens · 81269 ms · 2026-05-07T15:20:53.167495+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

22 extracted references · 4 canonical work pages

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M. Albers, P. S. Meysonnat, D. Fernex, R. Semaan, B. R. Noack, and W. Schr¨ oder, Drag reduction and energy sav- ing by spanwise traveling transversal surface waves for flat plate flow, Flow Turbul. Combust.105, 125 (2020)

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work page doi:10.48550/arxiv.2310.10311 2023

[1] [1]

Guckenheimer and P

J. Guckenheimer and P. Holmes,Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (Springer New York, 1983)

1983

[2] [2]

Cenedese, J

M. Cenedese, J. Ax˚ as, B. B¨ auerlein, K. Avila, and G. Haller, Data-driven modeling and prediction of non- linearizable dynamics via spectral submanifolds, Na- ture Communications13, 10.1038/s41467-022-28518-y (2022)

work page doi:10.1038/s41467-022-28518-y 2022

[3] [3]

C. E. P. De Jes´ us and M. D. Graham, Data-driven low- dimensional dynamic model of kolmogorov flow, Physical 9 Review Fluids8, 044402 (2023-04)

2023

[4] [4]

Froyland, O

G. Froyland, O. Junge, and P. Koltai, Estimating long- term behavior of flows without trajectory integration: The infinitesimal generator approach, SIAM J. Numer. Anal.51, 223 (2013-01)

2013

[5] [5]

S. Klus, F. N¨ uske, P. Koltai, H. Wu, I. Kevrekidis, C. Sch¨ utte, and F. No´ e, Data-driven model reduction and transfer operator approximation, Journal of Nonlin- ear Science28, 985 (2018)

2018

[6] [6]

A. N. Souza, Transforming butterflies into graphs: Statistics of chaotic and turbulent systems, arXiv 10.48550/ARXIV.2304.03362 (2023-04), 2304.03362

work page doi:10.48550/arxiv.2304.03362 2023

[7] [7]

E. M. Bollt and N. Santitissadeekorn,Applied and Com- putational Measurable Dynamics(Society for Industrial and Applied Mathematics, 2013-11)

2013

[8] [8]

Burkardt, M

J. Burkardt, M. Gunzburger, and H.-C. Lee, Cen- troidal voronoi tessellation-based reduced-order model- ing of complex systems, SIAM Journal on Scientific Com- puting28, 459 (2006)

2006

[9] [9]

Kaiser, B

E. Kaiser, B. R. Noack, L. Cordier, A. Spohn, M. Segond, M. Abel, G. Daviller, J. ¨Osth, S. Krajnovi´ c, and R. K. Niven, Cluster-based reduced-order modelling of a mix- ing layer, J. Fluid Mech.754, 365 (2014)

2014

[10] [10]

Fernex, B

D. Fernex, B. R. Noack, and R. Semaan, Cluster-based network modeling—from snapshots to complex dynami- cal systems, Sci. Adv.7, eabf5006 (2021)

2021

[11] [11]

A. G. Nair, C.-A. Yeh, E. Kaiser, B. R. Noack, S. L. Brunton, and K. Taira, Cluster-based feedback control of turbulent post-stall separated flows, J. Fluid Mech. 875, 345 (2019)

2019

[12] [12]

Kaiser, G

F. Kaiser, G. Iacobello, and D. E. Rival, Cluster-based bayesian approach for noisy and sparse data: application to flow-state estimation, Philos. Trans. R. Soc. A480, 10.1098/rspa.2023.0608 (2024-06)

work page doi:10.1098/rspa.2023.0608 2023

[13] [13]

Fernex, R

D. Fernex, R. Semaan, and B. Noack, Generalized cluster-based network model for an actuated turbulent boundary layer, inAIAA Scitech 2021 Forum(2021)

2021

[14] [14]

Iacobello, F

G. Iacobello, F. Kaiser, and D. E. Rival, Load estimation in unsteady flows from sparse pressure measurements: Application of transition networks to experimental data, Phys. Fluids34, 025105 (2022)

2022

[15] [15]

Lasota and M

A. Lasota and M. C. Mackey,Chaos, Fractals, and Noise (1994)

1994

[16] [16]

Holmes, J

P. Holmes, J. L. Lumley, G. Berkooz, and C. W. Row- ley,Turbulence, Coherent Structures, Dynamical Systems and Symmetry(Cambridge University Press, 2012-02)

2012

[17] [17]

Kaiser, B

E. Kaiser, B. R. Noack, A. Spohn, L. N. Cattafesta, and M. Morzy´ nski, Cluster-based control of a separating flow over a smoothly contoured ramp, Theor. Comp. Fluid Dyn.31, 579 (2017)

2017

[18] [18]

Kabsch, A solution for the best rotation to relate two sets of vectors, Acta Crystallographica Section A32, 922 (1976)

W. Kabsch, A solution for the best rotation to relate two sets of vectors, Acta Crystallographica Section A32, 922 (1976)

1976

[19] [19]

E. N. Lorenz, Deterministic nonperiodic flow, J. Atmos. Sci.20, 130 (1963)

1963

[20] [20]

Albers, P

M. Albers, P. S. Meysonnat, D. Fernex, R. Semaan, B. R. Noack, and W. Schr¨ oder, Drag reduction and energy sav- ing by spanwise traveling transversal surface waves for flat plate flow, Flow Turbul. Combust.105, 125 (2020)

2020

[21] [21]

C. W. Rowley, T. Colonius, and R. M. Murray, Model reduction for compressible flows using pod and galerkin projection, Physica D: Nonlinear Phenomena189, 115 (2004-02)

2004

[22] [22]

C. Hou, N. Deng, and B. R. Noack, Dynamics- augmented cluster-based network model, arXiv 10.48550/ARXIV.2310.10311 (2023), 2310.10311. Appendix A: Error criteria The quality of the predicted autocorrelation function bρ(τ) against a referenceρ(τ) is assessed through theL 1 distance (or mean absolute error ) MAE(ρ,bρ) = 1 ∥ρ(τ)∥ 1 ∥ρ(τ)−bρ(τ)∥ 1.(A1) TheL 1...

work page doi:10.48550/arxiv.2310.10311 2023