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A single global radial-basis field, shaped by local attractor geometry, recovers the long-term statistics of chaotic flows without using the governing equations.

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2026-07-10 05:07 UTC pith:HPNGWJ4R

load-bearing objection Solid non-intrusive ROM that matches intrusive local Galerkin and neural baselines on invariant measures; the corrector is a real but honestly quantified limitation, not a fatal one. the 2 major comments →

arxiv 2607.08571 v1 pith:HPNGWJ4R submitted 2026-07-09 physics.flu-dyn nlin.CD

Manifold-adapted radial basis functions for reduced-order modelling of chaotic flows

Miguel P\'erez Cuadrado , Giorgio Maria Cavallazzi , Alfredo Pinelli This is my paper
classification physics.flu-dyn nlin.CD
keywords reduced-order modellingradial basis functionschaotic dynamicsinvariant measurenon-intrusive methodsdata-driven modellingmanifold-adapted kernelsLyapunov exponents
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

Chaotic flows often live on a thin, curved attractor whose local orientation changes from region to region. This paper shows that you can learn an explicit, differentiable reduced vector field for such systems from snapshots alone. The method clusters the reduced coordinates to read local geometry, then builds anisotropic radial-basis kernels that stretch along the attractor and stay thin across it. One global regularised least-squares fit of the reduced velocity onto that library produces a continuous field that, under long free integration, reproduces the invariant measure—energy densities, spectra, and marginals—as accurately as an intrusive local Galerkin model that has access to the equations, and better than a global Galerkin projection of the same dimension. Short-term forecast skill is competitive with tuned neural and reservoir forecasters, while the model remains open to inspection because the field is explicit. A kinematic corrector is added only to return rare escapes, and its magnitude is reported so the reader can see how much of each result rests on the learned field.

Core claim

A non-intrusive reduced-order model that shapes a radial-basis library by clustering the attractor and fitting the reduced velocity by one global regularised least-squares solve yields an explicit, differentiable vector field that reproduces the invariant measure of chaotic and quasiperiodic flows as faithfully as an intrusive quantised-local Galerkin model of the same dimension, without ever projecting the governing equations.

What carries the argument

Manifold-adapted anisotropic radial basis library: kernels inherit each cluster’s local principal directions and widths so they elongate along the attractor and stay thin normal to it; the reduced velocity is then recovered by a single global ridge-regularised regression onto this library, producing one continuous differentiable field rather than switched local models.

Load-bearing premise

That the kinematic corrector, needed because the pure radial-basis field under-recovers transverse contraction and decays away from the data, stays inactive on the attractor and does not bias the reported long-term statistics.

What would settle it

On a system whose excursions routinely leave the training support, integrate the bare learned field without the corrector for thousands of Lyapunov times and check whether the energy density, spectrum, or leading Lyapunov exponents systematically deviate from the full-order reference while the reported corrector magnitude remains large.

Watch this falsifier — get emailed when new claim-graph text bears on it.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. The paper proposes a non-intrusive reduced-order model for chaotic and quasiperiodic flows: reduced coordinates (global POD when needed) are clustered, local principal directions shape anisotropic Gaussian RBF kernels, and a single global regularised least-squares fit yields an explicit, differentiable reduced vector field. Integration is stabilised by a kinematic trust-region corrector whose relative magnitude is reported. On Lorenz-63 the model recovers the attractor, marginals, and the positive/neutral Lyapunov exponents (analytic Jacobian) while under-recovering transverse contraction; on Lorenz-96 short-term VPT is competitive with tuned RC/GRU/LSTM baselines and the invariant measure is reproduced; on chaotic Kuramoto–Sivashinsky and quasiperiodic Kolmogorov flow the energy density and spectrum match an intrusive quantised-local Galerkin model of the same dimension and improve on a global Galerkin projection, without projecting the governing equations.

Significance. If the results hold, the work supplies a practical middle path between opaque neural/reservoir forecasters and intrusive Galerkin ROMs: an explicit, differentiable, non-intrusive field that is competitive on short-term skill and matches equation-based local Galerkin models on long-term statistics. Strengths that raise the contribution above a pure methods note include (i) Lyapunov spectra read from the analytic Jacobian rather than trajectory estimates, (ii) systematic diagnosis of under-contraction via divergence and λ3, (iii) data-driven calibration of the corrector with on-attractor inactivity checks (KS, spectra, autocorrelation), (iv) head-to-head comparison against both published neural baselines and the ql-ROM/g-ROM of Colanera & Magri, and (v) open code and data. The manifold-adapted library (clustering shapes kernels, not switched local dynamics) is a clear design choice relative to cluster-based ROMs and sparse regression/operator inference.

major comments (2)
  1. [§3.1, Eq. (15), Appendix A] §3.1 (Fig. 7), §2.3.3 Eq. (15), and Appendix A: the pure RBF field systematically under-recovers transverse contraction (λ3 ≈ −12 vs −14.57; divergence sum ≈ −11 vs −13.67 on Lorenz-63), and kernels decay off the data so the learned Jacobian lacks restoring eigenvalues. Long-time invariant-measure results that underwrite the central claim (Figs. 10–12; KS distances, spectra, energy densities) are obtained from the composite field f+c. The paper is transparent and verifies on-attractor inactivity within tolerance, but the main text should state, for each system, whether pure f remains bounded under the reported integration length and should report the median corrector/RMS∥ȧ∥ ratio for the headline runs in the main results (currently largely confined to Appendix B density plots). Without that, readers cannot judge how far each statistic rests on the learned field versus the safeguard.
  2. [§2.2.3, Appendix B] §2.2.3 and Appendix B: selection of K is only partly automatic. BIC supplies a range; geometric/dynamical distinctness is then judged by eye (“rise clearly above baseline”). For Lorenz-96 the marginal BIC has no elbow and K=10 is fixed “by convention” (Fig. 16). Because the manifold-adapted claim rests on the clusters correctly reading local geometry, a short sensitivity study (invariant-measure diagnostics at neighbouring K, or at K=1 isotropic) for the headline systems would show that the reported statistics are not an artefact of a hand-chosen partition.
minor comments (5)
  1. [Abstract, §1] Abstract and §1: the phrase “reproduces the long-term statistics… without any use of the governing equations” is accurate for the fit, but a brief clause that long integrations use the reported kinematic corrector would align the abstract with the body and with the honesty of §2.3.3.
  2. [§3.2, Fig. 9] Fig. 9 caption and text: the violin comparison is carefully caveated (baselines span hyperparameters; RBF spans initial conditions), but the main-text sentence “mid-field” could explicitly point the reader to the best-tuned baseline medians so the figure is not misread as a hyperparameter-averaged contest.
  3. [§2.3.1–2.3.2] §2.3.1–2.3.2: the normal-width floor and the ridge λ (leave-one-out then long-term confirmation) are important conditioning choices; a one-line default recipe (or the values used per system) in a small table would aid reproduction beyond the GitHub release.
  4. [Figs. 1, 3] Fig. 1 and Fig. 3 are helpful; ensure axis labels and the distinction between time sampling vs arc-length sampling remain legible at journal column width.
  5. [§3.3, Refs.] References: the ql-ROM comparison is central; ensure the published Colanera & Magri (2025) citation and any shared data protocols are stated so the energy-density rescaling on Kolmogorov (Fig. 12) is fully reproducible.

Circularity Check

0 steps flagged

No load-bearing circularity: the RBF field is a standard least-squares fit to trajectory data, evaluated on independent long-term statistics, known Lyapunov spectra, and external published benchmarks; the corrector is an explicit, reported safeguard rather than a hidden redefinition of the claim.

full rationale

The paper is a methodological contribution that constructs an explicit vector field by global regularised least-squares regression of reduced velocities onto a geometry-adapted RBF library (eqs. 11–14), then integrates the composite field and compares the resulting invariant measure (energy densities, spectra, marginals) and short-term skill (VPT) against independent references: the known Lorenz-63 Lyapunov spectrum, the published neural/reservoir numbers of Vlachas et al., and the intrusive ql-ROM/g-ROM of Colanera & Magri. None of these comparisons is forced by construction; the fit is to instantaneous velocities on the training snapshots, while the reported quantities are long-horizon occupation measures and analytic Jacobian eigenvalues that a merely interpolating field need not recover. The kinematic corrector (eq. 15, App. A) is calibrated from residual outward drift on the same training cloud and is therefore data-dependent, yet the paper treats it as an external patch, reports its relative magnitude on every case, and verifies that on-attractor statistics remain unbiased within KS tolerance. This is transparent engineering, not a self-definitional loop or a fitted constant renamed as prediction. There are no uniqueness theorems, no load-bearing self-citations of prior author results, and no renaming of a known empirical pattern. The single minor note is that the corrector strength is itself a statistic of the training residual; because the paper never claims the pure RBF field alone produces the reported long-term statistics on systems that leave the support, this does not elevate the score above 1.

Axiom & Free-Parameter Ledger

6 free parameters · 4 axioms · 2 invented entities

The central claim rests on standard POD/RBF/K-means machinery plus several modelling choices (Markovian closure of the reduced dynamics, local-PCA geometry as a faithful manifold proxy, and the external kinematic corrector). Free parameters are numerous but mostly set by transparent heuristics or cross-validation rather than by fitting the final statistics. No new physical entities are postulated.

free parameters (6)
  • number of clusters K
    Chosen via BIC elbow plus geometric/dynamical distinctness tests; different systems use K=3 (L63), K=10 (L96/KS), K=8 (Kolmogorov).
  • tangent width multiplier α_t
    Controls kernel overlap; swept and selected for long-term statistics rather than one-step error.
  • ridge penalty λ
    Set by leave-one-out cross-validation then confirmed by stable integration; order-of-magnitude only.
  • number of RBF centres n_c
    Budget chosen for convergence of NRMSE and Lyapunov spectrum (typically 1000–2800).
  • corrector parameters (k, p, α, q)
    Strength k calibrated from residual outward drift; engagement fraction α=0.85, ramp p=2, quantile q=0.99 held universal.
  • POD energy threshold
    Fixed at 99 % retained fluctuation energy for all systems.
axioms (4)
  • domain assumption Reduced coordinates evolve under a closed Markovian velocity field f(a) that can be learned from the projected trajectory alone.
    Stated in §2.1; standard closure assumption of non-intrusive ROM.
  • domain assumption Local principal directions of K-means clusters furnish a faithful approximate tangent space of the attractor for shaping anisotropic kernels.
    Core modelling choice of §2.2–2.3; justified by geometry of dissipative attractors.
  • standard math A radial-basis sum is a universal approximator on compact sets and remains sufficiently accurate when kernels are elongated along the data cloud.
    Invoked via classical RBF theory (Park & Sandberg, etc.).
  • ad hoc to paper The kinematic corrector of eq. (15) can be made inactive on the attractor while restoring boundedness off it without biasing invariant measures.
    Introduced in §2.3.3 and calibrated in Appendix A; verified empirically but not derived from first principles.
invented entities (2)
  • manifold-adapted anisotropic RBF library independent evidence
    purpose: Provide a geometry-aware, non-parametric basis for the reduced velocity that adapts orientation and width region-by-region while remaining globally continuous.
    The library itself is the methodological invention; independent evidence is the successful recovery of known attractors and spectra on multiple systems.
  • kinematic trust-region corrector no independent evidence
    purpose: Supply the missing transverse contraction that pure decaying RBFs cannot provide once a trajectory leaves the training support.
    Explicitly labelled a patch rather than learned dynamics; magnitude is reported as a diagnostic.

pith-pipeline@v1.1.0-grok45 · 35328 in / 3177 out tokens · 32120 ms · 2026-07-10T05:07:54.297746+00:00 · methodology

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read the original abstract

Chaotic systems often evolve on a low-dimensional attractor whose geometry varies from one region to another. We propose a non-intrusive reduced-order model that reads this local geometry by clustering and uses it to shape a radial basis library whose kernels adapt to each region. Fitting the reduced velocity onto this library by one global regularised least-squares solve gives an explicit, differentiable vector field that reproduces the long-term statistics, that is, the invariant measure, without any use of the governing equations. Since a radial basis field decays away from the data and cannot by itself return an escaped state, the integration is stabilised by a kinematic corrector whose magnitude is reported as a measure of how far each result rests on the learned field rather than on the corrector. On Lorenz-63 the model recovers the attractor, its marginal densities, and the positive and neutral Lyapunov exponents, while under-recovering the strong transverse contraction. On Lorenz-96 its valid prediction time is competitive with tuned neural-network and reservoir-computing forecasters, and the invariant measure is reproduced on both the full state and a reduced observable. On the Kuramoto--Sivashinsky equation and the quasiperiodic Kolmogorov flow the model matches the energy distribution and spectrum of an intrusive quantised-local Galerkin model, and improves on a global Galerkin projection of the same dimension, without ever projecting the governing equations.

Figures

Figures reproduced from arXiv: 2607.08571 by Alfredo Pinelli, Giorgio Maria Cavallazzi, Miguel P\'erez Cuadrado.

Figure 1
Figure 1. Figure 1: Sampling and scaling in the clustering step. [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The radial basis library in one dimension. [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Centre placement and the anisotropic kernel widths. [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Clustering of the Lorenz-63 state and selection of the number of regions. [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: A-priori regression on Lorenz-63, against the number of radial basis functions. [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Recovery of the Lorenz-63 attractor and its marginals, integrated for [PITH_FULL_IMAGE:figures/full_fig_p012_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Lyapunov spectrum of Lorenz-63 from the analytic Jacobian, against the number of radial basis functions, for [PITH_FULL_IMAGE:figures/full_fig_p013_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Lorenz-96 forecasting against the data-driven baselines of [ [PITH_FULL_IMAGE:figures/full_fig_p014_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Distribution of the valid prediction time on Lorenz-96, by reduced-order dimension, for [PITH_FULL_IMAGE:figures/full_fig_p015_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Lorenz-96 invariant measure of the reduced coordinates, full-order system (FOM) against the radial-basis [PITH_FULL_IMAGE:figures/full_fig_p016_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Kuramoto–Sivashinsky in the chaotic regime ( [PITH_FULL_IMAGE:figures/full_fig_p017_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Kolmogorov flow at 𝑅𝑒 = 20 in the quasiperiodic regime. (a) projection onto the two leading reduced coordinates, full-order (black) and model (red), tracing the invariant torus. (b) isotropic energy spectrum ⟨𝐸(|𝐤|)⟩. (c) reduced kinetic energy 𝐸(𝑡) and its density 𝑃 (𝐸). The ql-ROM and g-ROM are the Galerkin models of [26]; their densities are rescaled to the variance of the present full-order data, for … view at source ↗
Figure 13
Figure 13. Figure 13: The radial velocity 𝑑̇ = 𝐧̂ ⋅ ( ̃𝐟 + 𝐜̃), whose sign is that of 𝑉̇ for 𝑉 = 𝑑 2 , against distance 𝑑, at strengths 𝑘 = 0, 1, 10, 100. At 𝑘 = 0 the bare field drives outward past the trust radius; as 𝑘 grows the balance turns inward past 𝛼𝑅, confining the trajectory. With no restoring force, in panel (a), a population of states keeps 𝑉 >̇ 0 well past the region, pulled outward even once already outside it, … view at source ↗
Figure 14
Figure 14. Figure 14: The two-step calibration on the headline model. [PITH_FULL_IMAGE:figures/full_fig_p022_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Long-time statistics of the guarded model against the full-order reference. [PITH_FULL_IMAGE:figures/full_fig_p023_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Cluster-number selection for Lorenz-96, for the four combinations of [PITH_FULL_IMAGE:figures/full_fig_p024_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Transition structure of the Lorenz-96 partition at [PITH_FULL_IMAGE:figures/full_fig_p024_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: Convergence of the RBF fit for Lorenz-96, one row per configuration (full and SVD-reduced coordinates, [PITH_FULL_IMAGE:figures/full_fig_p025_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: POD truncation for Kuramoto–Sivashinsky. [PITH_FULL_IMAGE:figures/full_fig_p026_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: Cluster-number selection for Kuramoto–Sivashinsky. [PITH_FULL_IMAGE:figures/full_fig_p026_20.png] view at source ↗
Figure 22
Figure 22. Figure 22: Convergence of the RBF fit for Kuramoto–Sivashinsky. [PITH_FULL_IMAGE:figures/full_fig_p027_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: fixes the reduced dimension at twenty-three POD modes, retaining 99.1% of the fluctuation energy; the leading modes are the characteristic Kolmogorov rolls, and the dominant wavenumber broadens with mode index [PITH_FULL_IMAGE:figures/full_fig_p027_23.png] view at source ↗
Figure 24
Figure 24. Figure 24: Cluster-number selection for Kolmogorov flow. [PITH_FULL_IMAGE:figures/full_fig_p028_24.png] view at source ↗
Figure 25
Figure 25. Figure 25: Energetic structure of the Kolmogorov partition. [PITH_FULL_IMAGE:figures/full_fig_p028_25.png] view at source ↗
Figure 26
Figure 26. Figure 26: Convergence of the RBF fit for Kolmogorov flow. [PITH_FULL_IMAGE:figures/full_fig_p029_26.png] view at source ↗

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Reference graph

Works this paper leans on

80 extracted references · 80 canonical work pages · 1 internal anchor

  1. [1]

    G. K. Vallis, Atmospheric and Oceanic Fluid Dynamics: Fundamentals and Large-Scale Circulation, 2nd ed., Cambridge University Press, 2017

    work page 2017

  2. [2]

    Kalnay, Atmospheric Modeling, Data Assimilation and Predictability, Cambridge University Press, 2003

    E. Kalnay, Atmospheric Modeling, Data Assimilation and Predictability, Cambridge University Press, 2003

    work page 2003

  3. [3]

    Bauer, A

    P. Bauer, A. Thorpe, G. Brunet, The quiet revolution of numerical weather prediction, Nature 525 (2015) 47–55. doi:10.1038/nature14956

    work page doi:10.1038/nature14956 2015

  4. [4]

    S. B. Pope, Turbulent Flows, Cambridge University Press, 2000. doi:10.1017/CBO9780511840531

    work page doi:10.1017/cbo9780511840531 2000

  5. [5]

    Frisch, Turbulence: The Legacy of A

    U. Frisch, Turbulence: The Legacy of A. N. Kolmogorov, Cambridge University Press, 1995

    work page 1995

  6. [6]

    E. N. Lorenz, Deterministic nonperiodic flow, Journal of the Atmospheric Sciences 20 (1963) 130–141. doi:10.1175/1520-0469(1963) 020<0130:DNF>2.0.CO;2

    work page doi:10.1175/1520-0469(1963 1963

  7. [7]

    S. H. Strogatz, Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering, 2nd ed., Westview Press, 2015

    work page 2015

  8. [8]

    P. Moin, K. Mahesh, Direct numerical simulation: a tool in turbulence research, Annual Review of Fluid Mechanics 30 (1998) 539–578. doi:10.1146/annurev.fluid.30.1.539

    work page doi:10.1146/annurev.fluid.30.1.539 1998

  9. [9]

    Ishihara, T

    T. Ishihara, T. Gotoh, Y. Kaneda, Study of high–reynolds number isotropic turbulence by direct numerical simulation, Annual Review of Fluid Mechanics 41 (2009) 165–180

    work page 2009

  10. [10]

    H. Choi, P. Moin, Grid-point requirements for large eddy simulation: Chapman’s estimates revisited, Physics of Fluids 24 (2012) 011702

    work page 2012

  11. [11]

    P. R. Spalart, Strategies for turbulence modelling and simulations, International Journal of Heat and Fluid Flow 21 (2000) 252–263

    work page 2000

  12. [12]

    S. L. Brunton, B. R. Noack, Closed-loop turbulence control: Progress and challenges, Applied Mechanics Reviews 67 (2015) 050801. doi:10.1115/1.4031175

    work page doi:10.1115/1.4031175 2015

  13. [13]

    Evensen, Data Assimilation: The Ensemble Kalman Filter, 2nd ed., Springer, 2009

    G. Evensen, Data Assimilation: The Ensemble Kalman Filter, 2nd ed., Springer, 2009

    work page 2009

  14. [14]

    Benner, S

    P. Benner, S. Gugercin, K. Willcox, A survey of projection-based model reduction methods for parametric dynamical systems, SIAM Review 57 (2015) 483–531. doi:10.1137/130932715

    work page doi:10.1137/130932715 2015

  15. [15]

    C. W. Rowley, S. T. M. Dawson, Model reduction for flow analysis and control, Annual Review of Fluid Mechanics 49 (2017) 387–417. doi:10.1146/annurev-fluid-010816-060042

    work page doi:10.1146/annurev-fluid-010816-060042 2017

  16. [16]

    Benettin, L

    G. Benettin, L. Galgani, A. Giorgilli, J.-M. Strelcyn, Lyapunov characteristic exponents for smooth dynamical systems and for hamiltonian systems; a method for computing all of them. part 1: Theory, Meccanica 15 (1980) 9–20. doi:10.1007/BF02128236

    work page doi:10.1007/bf02128236 1980

  17. [17]

    E. N. Lorenz, The predictability of a flow which possesses many scales of motion, Tellus 21 (1969) 289–307. doi:10.3402/tellusa.v21i3. 10086

    work page doi:10.3402/tellusa.v21i3 1969

  18. [18]

    P. R. Vlachas, W. Byeon, Z. Y. Wan, T. P. Sapsis, P. Koumoutsakos, Data-driven forecasting of high-dimensional chaotic systems with long short-term memory networks, Proceedings of the Royal Society A 474 (2018) 20170844. doi:10.1098/rspa.2017.0844

    work page doi:10.1098/rspa.2017.0844 2018

  19. [19]

    Eckmann, D

    J.-P. Eckmann, D. Ruelle, Ergodic theory of chaos and strange attractors, Reviews of Modern Physics 57 (1985) 617–656. doi:10.1103/ RevModPhys.57.617

    work page 1985

  20. [20]

    Schlegel, B

    M. Schlegel, B. R. Noack, On long-term boundedness of galerkin models, Journal of Fluid Mechanics 765 (2015) 325–352. doi:10.1017/jfm. 2014.736

    work page doi:10.1017/jfm 2015

  21. [21]

    Pathak, Z

    J. Pathak, Z. Lu, B. R. Hunt, M. Girvan, E. Ott, Using machine learning to replicate chaotic attractors and calculate lyapunov exponents from data, Chaos: An Interdisciplinary Journal of Nonlinear Science 27 (2017) 121102. doi:10.1063/1.5010300. M. Pérez Cuadrado et al.:Preprint submitted to ElsevierPage 29 of 31 Manifold-adapted RBFs for reduced-order mo...

    work page doi:10.1063/1.5010300 2017

  22. [22]

    P. R. Vlachas, J. Pathak, B. R. Hunt, T. P. Sapsis, M. Girvan, E. Ott, P. Koumoutsakos, Backpropagation algorithms and reservoir computing in recurrent neural networks for the forecasting of complex spatiotemporal dynamics, Neural Networks 126 (2020) 191–217. doi:10.1016/j.neunet.2020.02.016

    work page doi:10.1016/j.neunet.2020.02.016 2020

  23. [23]

    Pathak, B

    J. Pathak, B. Hunt, M. Girvan, Z. Lu, E. Ott, Model-free prediction of large spatiotemporally chaotic systems from data: A reservoir computing approach, Physical Review Letters 120 (2018) 024102. doi:10.1103/PhysRevLett.120.024102

    work page doi:10.1103/physrevlett.120.024102 2018

  24. [24]

    Sirovich, Turbulence and the dynamics of coherent structures

    L. Sirovich, Turbulence and the dynamics of coherent structures. I. Coherent structures, Quarterly of Applied Mathematics 45 (1987) 561–571. doi:10.1090/qam/910462

    work page doi:10.1090/qam/910462 1987

  25. [25]

    Holmes, J

    P. Holmes, J. L. Lumley, G. Berkooz, C. W. Rowley, Turbulence, Coherent Structures, Dynamical Systems and Symmetry, 2nd ed., Cambridge University Press, 2012

    work page 2012

  26. [26]

    Colanera, L

    A. Colanera, L. Magri, Quantized local reduced-order modeling in time (ql-rom), Computer Methods in Applied Mechanics and Engineering 447 (2025) 118393. doi:10.1016/j.cma.2025.118393

    work page doi:10.1016/j.cma.2025.118393 2025

  27. [27]

    Amsallem, M

    D. Amsallem, M. J. Zahr, C. Farhat, Nonlinear model order reduction based on local reduced-order bases, International Journal for Numerical Methods in Engineering 92 (2012) 891–916. doi:10.1002/nme.4371

    work page doi:10.1002/nme.4371 2012

  28. [28]

    K. Lee, K. T. Carlberg, Model reduction of dynamical systems on nonlinear manifolds using deep convolutional autoencoders, Journal of Computational Physics 404 (2020) 108973. doi:10.1016/j.jcp.2019.108973

    work page doi:10.1016/j.jcp.2019.108973 2020

  29. [29]

    S. L. Brunton, J. L. Proctor, J. N. Kutz, Discovering governing equations from data by sparse identification of nonlinear dynamical systems, Proceedings of the National Academy of Sciences 113 (2016) 3932–3937. doi:10.1073/pnas.1517384113

    work page doi:10.1073/pnas.1517384113 2016

  30. [30]

    Loiseau, S

    J.-C. Loiseau, S. L. Brunton, Constrained sparse galerkin regression, Journal of Fluid Mechanics 838 (2018) 42–67. doi:10.1017/jfm.2017. 823

    work page doi:10.1017/jfm.2017 2018

  31. [31]

    Peherstorfer, K

    B. Peherstorfer, K. Willcox, Data-driven operator inference for nonintrusive projection-based model reduction, Computer Methods in Applied Mechanics and Engineering 306 (2016) 196–215. doi:10.1016/j.cma.2016.03.025

    work page doi:10.1016/j.cma.2016.03.025 2016

  32. [32]

    D. S. Broomhead, D. Lowe, Multivariable functional interpolation and adaptive networks, Complex Systems 2 (1988) 321–355

    work page 1988

  33. [33]

    M.Casdagli, Nonlinearpredictionofchaotictimeseries, PhysicaD:NonlinearPhenomena35(1989)335–356.doi: 10.1016/0167-2789(89) 90074-2

    work page doi:10.1016/0167-2789(89 1989

  34. [34]

    Kaiser, B

    E. Kaiser, B. R. Noack, L. Cordier, A. Spohn, M. Segond, M. Abel, G. Daviller, J. Östh, S. Krajnović, R. K. Niven, Cluster-based reduced-order modelling of a mixing layer, Journal of Fluid Mechanics 754 (2014) 365–414. doi:10.1017/jfm.2014.355

    work page doi:10.1017/jfm.2014.355 2014

  35. [35]

    Fernex, B

    D. Fernex, B. R. Noack, R. Semaan, Cluster-based network modeling: From snapshots to complex dynamical systems, Science Advances 7 (2021) eabf5006. doi:10.1126/sciadv.abf5006

    work page doi:10.1126/sciadv.abf5006 2021

  36. [36]

    C.Eckart,G.Young, Theapproximationofonematrixbyanotheroflowerrank, Psychometrika1(1936)211–218.doi: 10.1007/BF02288367

    work page doi:10.1007/bf02288367 1936

  37. [37]

    Ginelli, P

    F. Ginelli, P. Poggi, A. Turchi, H. Chaté, R. Livi, A. Politi, Characterizing dynamics with covariant Lyapunov vectors, Physical Review Letters 99 (2007) 130601. doi:10.1103/PhysRevLett.99.130601

    work page doi:10.1103/physrevlett.99.130601 2007

  38. [38]

    Q. L. Li-Hu, G. Y. Cornejo Maceda, A. Ianiro, S. Discetti, Divide and conquer: Cluster and manifold-based interpretation of complex flows, arXiv preprint arXiv:2601.05117 (2026).arXiv:2601.05117

    work page internal anchor Pith review arXiv 2026

  39. [39]

    S. P. Lloyd, Least squares quantization in PCM, IEEE Transactions on Information Theory 28 (1982) 129–137. doi:10.1109/TIT.1982. 1056489

    work page doi:10.1109/tit.1982 1982

  40. [40]

    Arthur, S

    D. Arthur, S. Vassilvitskii, k-means++: The advantages of careful seeding, in: Proceedings of the Eighteenth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), 2007, pp. 1027–1035

    work page 2007

  41. [41]

    J. L. Lumley, The structure of inhomogeneous turbulent flows, in: A. M. Yaglom, V. I. Tatarski (Eds.), Atmospheric Turbulence and Radio Wave Propagation, Nauka, Moscow, 1967, pp. 166–178

    work page 1967

  42. [42]

    Berkooz, P

    G. Berkooz, P. Holmes, J. L. Lumley, The proper orthogonal decomposition in the analysis of turbulent flows, Annual Review of Fluid Mechanics 25 (1993) 539–575. doi:10.1146/annurev.fl.25.010193.002543

    work page doi:10.1146/annurev.fl.25.010193.002543 1993

  43. [43]

    Burkardt, M

    J. Burkardt, M. Gunzburger, H.-C. Lee, POD and CVT-based reduced-order modeling of Navier–Stokes flows, Computer Methods in Applied Mechanics and Engineering 196 (2006) 337–355. doi:10.1016/j.cma.2006.04.004

    work page doi:10.1016/j.cma.2006.04.004 2006

  44. [44]

    P. L. Zador, Asymptotic quantization error of continuous signals and the quantization dimension, IEEE Transactions on Information Theory 28 (1982) 139–149. doi:10.1109/TIT.1982.1056490

    work page doi:10.1109/tit.1982.1056490 1982

  45. [45]

    Gersho, Asymptotically optimal block quantization, IEEE Transactions on Information Theory 25 (1979) 373–380

    A. Gersho, Asymptotically optimal block quantization, IEEE Transactions on Information Theory 25 (1979) 373–380. doi:10.1109/TIT. 1979.1056067

    work page doi:10.1109/tit 1979

  46. [46]

    S. Graf, H. Luschgy, Foundations of Quantization for Probability Distributions, volume 1730 ofLecture Notes in Mathematics, Springer, 2000. doi:10.1007/BFb0103945

    work page doi:10.1007/bfb0103945 2000

  47. [47]

    Young, What are SRB measures, and which dynamical systems have them?, Journal of Statistical Physics 108 (2002) 733–754

    L.-S. Young, What are SRB measures, and which dynamical systems have them?, Journal of Statistical Physics 108 (2002) 733–754. doi:10.1023/A:1019762724717

    work page doi:10.1023/a:1019762724717 2002

  48. [48]

    Schwarz,Estimating the dimension of a model, The annals of statistics, 6 (1978), pp

    G. Schwarz, Estimating the dimension of a model, The Annals of Statistics 6 (1978) 461–464. doi:10.1214/aos/1176344136

    work page doi:10.1214/aos/1176344136 1978

  49. [49]

    Pelleg, A

    D. Pelleg, A. W. Moore, X-means: Extending k-means with efficient estimation of the number of clusters, in: Proceedings of the Seventeenth International Conference on Machine Learning (ICML), Morgan Kaufmann, 2000, pp. 727–734

    work page 2000

  50. [50]

    Satopaa, J

    V. Satopaa, J. Albrecht, D. Irwin, B. Raghavan, Finding a “kneedle” in a haystack: Detecting knee points in system behavior, in: 2011 31st International Conference on Distributed Computing Systems Workshops (ICDCSW), IEEE, 2011, pp. 166–171. doi:10.1109/ICDCSW.2011. 20

    work page doi:10.1109/icdcsw.2011 2011

  51. [51]

    Kambhatla, T

    N. Kambhatla, T. K. Leen, Dimension reduction by local principal component analysis, Neural Computation 9 (1997) 1493–1516. doi:10.1162/neco.1997.9.7.1493

    work page doi:10.1162/neco.1997.9.7.1493 1997

  52. [52]

    Björck, G

    Å. Björck, G. H. Golub, Numerical methods for computing angles between linear subspaces, Mathematics of Computation 27 (1973) 579–594. doi:10.1090/S0025-5718-1973-0348991-3

    work page doi:10.1090/s0025-5718-1973-0348991-3 1973

  53. [53]

    R. J. Bell, P. Dean, Atomic vibrations in vitreous silica, Discussions of the Faraday Society 50 (1970) 55–61. doi:10.1039/DF9705000055. M. Pérez Cuadrado et al.:Preprint submitted to ElsevierPage 30 of 31 Manifold-adapted RBFs for reduced-order modelling of chaotic flows

    work page doi:10.1039/df9705000055 1970

  54. [54]

    C. A. Micchelli, Interpolation of scattered data: Distance matrices and conditionally positive definite functions, Constructive Approximation 2 (1986) 11–22. doi:10.1007/BF01893414

    work page doi:10.1007/bf01893414 1986

  55. [55]

    M.J.D.Powell, Radialbasisfunctionsformultivariableinterpolation:areview, in:J.C.Mason,M.G.Cox(Eds.),AlgorithmsforApproximation, Clarendon Press, Oxford, 1987, pp. 143–167

    work page 1987

  56. [56]

    J. Park, I. W. Sandberg, Universal approximation using radial-basis-function networks, Neural Computation 3 (1991) 246–257. doi:10.1162/neco.1991.3.2.246

    work page doi:10.1162/neco.1991.3.2.246 1991

  57. [57]

    E. J. Hartman, J. D. Keeler, J. M. Kowalski, Layered neural networks with Gaussian hidden units as universal approximations, Neural Computation 2 (1990) 210–215. doi:10.1162/neco.1990.2.2.210

    work page doi:10.1162/neco.1990.2.2.210 1990

  58. [58]

    P. C. Mahalanobis, On the generalised distance in statistics, Proceedings of the National Institute of Sciences of India 2 (1936) 49–55

    work page 1936

  59. [59]

    M. P. Wand, M. C. Jones, Kernel Smoothing, number 60 in Monographs on Statistics and Applied Probability, Chapman & Hall/CRC, London, 1995

    work page 1995

  60. [60]

    Lipiecki, K

    T. Berry, J. Harlim, Variable bandwidth diffusion kernels, Applied and Computational Harmonic Analysis 40 (2016) 68–96. doi:10.1016/j. acha.2015.01.001

    work page doi:10.1016/j 2016

  61. [61]

    T. F. Gonzalez, Clustering to minimize the maximum intercluster distance, Theoretical Computer Science 38 (1985) 293–306. doi:10.1016/ 0304-3975(85)90224-5

    work page 1985

  62. [62]

    Schaback, Error estimates and condition numbers for radial basis function interpolation, Advances in Computational Mathematics 3 (1995) 251–264

    R. Schaback, Error estimates and condition numbers for radial basis function interpolation, Advances in Computational Mathematics 3 (1995) 251–264. doi:10.1007/BF02432002

    work page doi:10.1007/bf02432002 1995

  63. [63]

    G. E. Fasshauer, Meshfree Approximation Methods with MATLAB, volume 6 ofInterdisciplinary Mathematical Sciences, World Scientific, Singapore, 2007

    work page 2007

  64. [64]

    Poggio, F

    T. Poggio, F. Girosi, Networks for approximation and learning, Proceedings of the IEEE 78 (1990) 1481–1497. doi:10.1109/5.58326

    work page doi:10.1109/5.58326 1990

  65. [65]

    M. J. L. Orr, Regularization in the selection of radial basis function centers, Neural Computation 7 (1995) 606–623. doi:10.1162/neco. 1995.7.3.606

    work page doi:10.1162/neco 1995

  66. [66]

    A. E. Hoerl, R. W. Kennard, Ridge regression: biased estimation for nonorthogonal problems, Technometrics 12 (1970) 55–67. doi:10.1080/00401706.1970.10488634

    work page doi:10.1080/00401706.1970.10488634 1970

  67. [67]

    Hastie, R

    T. Hastie, R. Tibshirani, J. Friedman, The Elements of Statistical Learning: Data Mining, Inference, and Prediction, 2 ed., Springer, New York,

    work page

  68. [68]

    doi:10.1007/978-0-387-84858-7

    work page doi:10.1007/978-0-387-84858-7

  69. [69]

    P. C. Hansen, Rank-Deficient and Discrete Ill-Posed Problems: Numerical Aspects of Linear Inversion, SIAM Monographs on Mathematical Modeling and Computation, Society for Industrial and Applied Mathematics, Philadelphia, 1998. doi:10.1137/1.9780898719697

    work page doi:10.1137/1.9780898719697 1998

  70. [70]

    doi:10.1080/00401706.1974.10489157

    D.M.Allen, Therelationshipbetweenvariableselectionanddataaugmentationandamethodforprediction, Technometrics16(1974)125–127. doi:10.1080/00401706.1974.10489157

    work page doi:10.1080/00401706.1974.10489157 1974

  71. [71]

    G. H. Golub, M. Heath, G. Wahba, Generalized cross-validation as a method for choosing a good ridge parameter, Technometrics 21 (1979) 215–223. doi:10.1080/00401706.1979.10489751

    work page doi:10.1080/00401706.1979.10489751 1979

  72. [72]

    A. A. Kaptanoglu, J. L. Callaham, A. Aravkin, C. J. Hansen, S. L. Brunton, Promoting global stability in data-driven models of quadratic nonlinear dynamics, Physical Review Fluids 6 (2021) 094401. doi:10.1103/PhysRevFluids.6.094401

    work page doi:10.1103/physrevfluids.6.094401 2021

  73. [73]

    E. N. Lorenz, Predictability: a problem partly solved, in: Proceedings of the Seminar on Predictability, volume 1, ECMWF, Reading, UK, 1996, pp. 1–18

    work page 1996

  74. [74]

    doi:10.1175/1520-0469(1998)055<0399:OSFSWO>2.0.CO;2

    E.N.Lorenz,K.A.Emanuel, Optimalsitesforsupplementaryweatherobservations:Simulationwithasmallmodel, JournaloftheAtmospheric Sciences 55 (1998) 399–414. doi:10.1175/1520-0469(1998)055<0399:OSFSWO>2.0.CO;2

    work page doi:10.1175/1520-0469(1998)055 1998

  75. [75]

    Kuramoto, T

    Y. Kuramoto, T. Tsuzuki, Persistent propagation of concentration waves in dissipative media far from thermal equilibrium, Progress of Theoretical Physics 55 (1976) 356–369. doi:10.1143/PTP.55.356

    work page doi:10.1143/ptp.55.356 1976

  76. [76]

    G. I. Sivashinsky, Nonlinear analysis of hydrodynamic instability in laminar flames—I. Derivation of basic equations, Acta Astronautica 4 (1977) 1177–1206. doi:10.1016/0094-5765(77)90096-0

    work page doi:10.1016/0094-5765(77)90096-0 1977

  77. [77]

    Ghrist, Barcodes: the persistent topology of data, Bulletin of the American Mathematical Society 45 (2008) 61–75

    R. Ghrist, Barcodes: the persistent topology of data, Bulletin of the American Mathematical Society 45 (2008) 61–75. doi:10.1090/ S0273-0979-07-01191-3

    work page 2008

  78. [78]

    G.Carlsson, Topologyanddata, BulletinoftheAmericanMathematicalSociety46(2009)255–308.doi: 10.1090/S0273-0979-09-01249-X

    work page doi:10.1090/s0273-0979-09-01249-x 2009

  79. [79]

    Schreiber, Measuring information transfer, Physical Review Letters 85 (2000) 461–464

    T. Schreiber, Measuring information transfer, Physical Review Letters 85 (2000) 461–464. doi:10.1103/PhysRevLett.85.461

    work page doi:10.1103/physrevlett.85.461 2000

  80. [80]

    H. K. Khalil, Nonlinear Systems, 3 ed., Prentice Hall, Upper Saddle River, NJ, 2002. M. Pérez Cuadrado et al.:Preprint submitted to ElsevierPage 31 of 31

    work page 2002