arxiv: 2605.14224 · v1 · submitted 2026-05-14 · 🧮 math.NA · cs.AI· cs.NA· math.DS· math.FA

Recognition: no theorem link

Wavelet-Based Observables for Koopman Analysis: An Extended Dynamic Mode Decomposition Framework

Cankat Tilki , Serkan Gugercin

Authors on Pith no claims yet

Pith reviewed 2026-05-15 02:51 UTC · model grok-4.3

classification 🧮 math.NA cs.AIcs.NAmath.DSmath.FA

keywords Koopman semigroupwavelet observablesdynamic mode decompositioneigenfunctionsextended DMDcontinuous wavelet transformnumerical approximation

0 comments

The pith

Wavelet-based observables act as eigenfunctions of the Koopman semigroup, yielding closed-form expressions for its action and a new approximation algorithm.

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

The paper shows that observables constructed from wavelets are eigenfunctions of the Koopman semigroup when the semigroup is defined on the space of continuous functions equipped with the supremum norm. This property permits the derivation of explicit formulas for how the semigroup and its resolvent act on these observables. By integrating these wavelet observables into the extended dynamic mode decomposition framework, the authors develop the cWDMD algorithm for numerical computation. The theoretical findings are illustrated through two numerical examples that demonstrate the method's effectiveness.

Core claim

Wavelet-based observables are eigenfunctions of the Koopman semigroup over the Banach space of continuous functions on a compact forward-invariant set with the supremum norm. This leads to closed-form expressions for the semigroup action and its resolvent, and to the cWDMD algorithm that approximates the Koopman action by combining these observables with extended dynamic mode decomposition.

What carries the argument

Wavelet-based observables obtained via the continuous wavelet transform, serving as eigenfunctions that enable exact representation of the Koopman semigroup action.

If this is right

Closed-form expressions become available for the action of the Koopman semigroup and its resolvent.
The cWDMD algorithm provides a numerical method to approximate the semigroup using wavelet observables.
Analysis applies to dynamical systems on compact forward-invariant sets in the space of continuous functions.
Validation on two numerical examples confirms the practical applicability of the approach.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

These observables might extend to other function spaces or infinite-dimensional systems where standard DMD struggles.
Potential applications in data-driven control or forecasting could benefit from the explicit resolvent expressions.
Testing on higher-dimensional or chaotic systems would reveal the method's robustness beyond the presented examples.

Load-bearing premise

The wavelet-based observables continue to function as eigenfunctions and deliver reliable approximations when applied to the dynamical systems and function spaces typical in real-world applications.

What would settle it

Observing that for a specific dynamical system the wavelet observables do not satisfy the eigenfunction property under the sup norm, or finding that cWDMD yields larger errors than standard EDMD on additional test cases.

Figures

Figures reproduced from arXiv: 2605.14224 by Cankat Tilki, Serkan Gugercin.

**Figure 2.** Figure 2: Koopman Resolvent Evaluation at Peak Frequency [PITH_FULL_IMAGE:figures/full_fig_p027_2.png] view at source ↗

**Figure 3.** Figure 3: Magnitude of the Koopman Resolvent Approximation [PITH_FULL_IMAGE:figures/full_fig_p028_3.png] view at source ↗

**Figure 4.** Figure 4: Argument of the Koopman Resolvent Approximation a [PITH_FULL_IMAGE:figures/full_fig_p029_4.png] view at source ↗

**Figure 5.** Figure 5: Real Part of the Koopman Resolvent Approximation [PITH_FULL_IMAGE:figures/full_fig_p030_5.png] view at source ↗

**Figure 6.** Figure 6: Imaginary Part of the Koopman Resolvent Approxima [PITH_FULL_IMAGE:figures/full_fig_p030_6.png] view at source ↗

read the original abstract

We present an in-depth analysis of the Koopman semigroup via wavelet transform. Towards this goal, we start by introducing the wavelet-based observables and show that they are eigenfunctions of the Koopman semigroup when this semigroup is considered over the Banach space of continuous functions on a compact forward-invariant set endowed with the supremum norm. We then construct closed-form expressions of the action of the Koopman semigroup and its resolvent in terms of these observables. To approximate the action of Koopman semigroup numerically, we combine Extended Dynamic Mode Decomposition (EDMD) with the proposed wavelet-based observables leading to the Wavelet Dynamic Mode Decomposition via Continuous Wavelet Transform (cWDMD) algorithm. We validate our theoretical results on two numerical examples.

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

Wavelet observables are eigenfunctions for the Koopman semigroup on C(K) with sup norm, yielding closed forms and a cWDMD variant of EDMD.

read the letter

The paper introduces wavelet-based observables that are eigenfunctions of the Koopman semigroup on the space of continuous functions over a compact forward-invariant set with the supremum norm. This setup produces closed-form expressions for the semigroup action and its resolvent, which then feed into a new algorithm called cWDMD by combining them with standard EDMD steps. The construction follows directly from the wavelet definition and basic semigroup properties in that Banach space, without obvious circular steps or extra fitting parameters. The two numerical examples give a basic check that the approximations work on concrete systems. That is the core contribution, and it sits cleanly inside the existing Koopman/EDMD literature. The theory rests on standard functional-analysis arguments for the chosen space, which keeps the claims grounded. The algorithm is a straightforward extension rather than a complete reinvention. The main limitations are narrow. Validation stays at two examples with no systematic comparison to other common observables such as polynomials or radial basis functions, so it is not yet clear how much practical gain appears in noisy or high-dimensional cases. The forward-invariant compact set assumption is standard but can be nontrivial to verify or enforce in applications. Reproducibility looks feasible from the description, but the full implementation details would need checking. This work is aimed at researchers already using data-driven Koopman methods who want a new observable class with some analytic backing. It is coherent on its own terms and shows clear engagement with the literature, so it deserves a serious referee to verify the derivations and push on the numerical scope.

Referee Report

1 major / 3 minor

Summary. The paper introduces wavelet-based observables and proves they are eigenfunctions of the Koopman semigroup on the Banach space C(K) of continuous functions on a compact forward-invariant set K equipped with the supremum norm. Closed-form expressions are derived for the action of the semigroup and its resolvent in terms of these observables. The cWDMD algorithm is then proposed by combining the observables with Extended Dynamic Mode Decomposition (EDMD) for numerical approximation, with validation on two numerical examples.

Significance. If the central claims hold, the work supplies explicit eigenfunctions and closed-form operator expressions for the Koopman semigroup in a standard function space setting, which could streamline both analytical derivations and numerical schemes in dynamical systems analysis. The cWDMD algorithm extends EDMD in a principled way, and the theoretical construction (eigenfunction property plus closed forms) is a clear strength that goes beyond purely data-driven approaches.

major comments (1)

§3 (eigenfunction proof): the argument that the wavelet observables remain eigenfunctions relies on the forward-invariance of K and the specific action of the semigroup on C(K); the manuscript should explicitly verify that the chosen wavelet family preserves this property under the supremum norm without additional regularity assumptions on the underlying flow.

minor comments (3)

The two numerical examples would be strengthened by reporting quantitative approximation errors (e.g., L^∞ residuals or eigenvalue accuracy) and direct comparison against standard EDMD with polynomial or radial-basis observables.
Notation for the continuous wavelet transform and the resulting observables should be introduced with a short table or diagram to clarify the mapping from scale/translation parameters to the observable functions.
A brief discussion of how the compact forward-invariant set K is identified or approximated in practice for non-trivial systems would improve applicability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their positive assessment and recommendation for minor revision. We address the single major comment below.

read point-by-point responses

Referee: §3 (eigenfunction proof): the argument that the wavelet observables remain eigenfunctions relies on the forward-invariance of K and the specific action of the semigroup on C(K); the manuscript should explicitly verify that the chosen wavelet family preserves this property under the supremum norm without additional regularity assumptions on the underlying flow.

Authors: We appreciate the referee's suggestion to make the verification more explicit. The proof in §3 establishes the eigenfunction property directly from the definition of the Koopman semigroup on C(K) equipped with the supremum norm, using forward-invariance of K solely to ensure the semigroup maps C(K) into itself. The wavelet observables are constructed as continuous functions on K, and the closed-form action follows from the change-of-variables property of the wavelet transform applied pointwise; no additional regularity on the flow is invoked beyond the continuity of the observables and the compactness of K. To address the request, we will revise the manuscript by adding a short clarifying remark immediately after the proof of Theorem 3.1, explicitly confirming that the supremum-norm setting and forward-invariance suffice without further assumptions. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The derivation begins by defining wavelet-based observables and then proves they are eigenfunctions of the Koopman semigroup on the Banach space C(K) equipped with the sup norm for compact forward-invariant K; this follows directly from the semigroup action on continuous functions without any self-referential definition or fitted parameter. Closed-form expressions for the semigroup and resolvent are constructed explicitly from this eigenfunction property. The cWDMD algorithm is obtained by substituting these observables into the existing EDMD framework, and the two numerical examples function only as validation. No load-bearing step reduces to a self-citation, ansatz smuggled via prior work, or renaming of a known result; the chain remains self-contained under the stated function-space assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the functional-analytic setting of the Koopman semigroup acting on continuous functions with the supremum norm; no free parameters or new entities are introduced. The main supporting structure is the standard definition of the Koopman operator and the properties of continuous wavelet transforms.

axioms (1)

domain assumption The Koopman semigroup acts on the Banach space of continuous functions on a compact forward-invariant set endowed with the supremum norm.
Invoked in the abstract as the setting in which wavelet-based observables become eigenfunctions.

pith-pipeline@v0.9.0 · 5433 in / 1381 out tokens · 48273 ms · 2026-05-15T02:51:00.259371+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

55 extracted references · 55 canonical work pages

[1]

A. C. Antoulas, C. A. Beattie, and S. G¨ u˘ gercin. Interpolatory Methods for Model Reduction. Society for Industrial and Applied Mathematics, Philadel phia, PA, 2020

work page 2020
[2]

Benner, S

P. Benner, S. Grivet-Talocia, A. Quarteroni, G. Rozza, W . Schilders, and L. M. Sil- veira, editors. Model Order Reduction: Volume 1: System- and Data-Driven Me thods and Algorithms . De Gruyter, Berlin, Boston, 2021

work page 2021
[3]

Berljafa and S

M. Berljafa and S. G¨ uttel. The RKFIT algorithm for nonli near rational approxima- tion. SIAM Journal on Scientiﬁc Computing , 39(5):A2049–A2071, 2017. Preprint. 2026-05-15 C.Tilki, S.G¨ u˘ gercin: Wavelet-Based Observables for Koopman Analysis 31

work page 2017
[4]

S. L. Brunton, M. Budiˇ si´ c, E. Kaiser, and J. N. Kutz. Mod ern Koopman theory for dynamical systems. SIAM Review , 64(2):229–340, 2022

work page 2022
[5]

Coddington and N

A. Coddington and N. Levinson. Theory of Ordinary Diﬀerential Equations . Inter- national series in pure and applied mathematics. McGraw-Hi ll, 1955

work page 1955
[6]

M. J. Colbrook. The mpEDMD algorithm for data-driven com putations of measure- preserving dynamical systems. SIAM Journal on Numerical Analysis , 61(3):1585– 1608, 2023

work page 2023
[7]

M. J. Colbrook, C. Drysdale, and A. Horning. Rigged dynam ic mode decomposition: Data-driven generalized eigenfunction decompositions fo r koopman operators. SIAM Journal on Applied Dynamical Systems , 24(2):1150–1190, 2025

work page 2025
[8]

J Colbrook and A

M. J Colbrook and A. Townsend. Rigorous data-driven comp utation of spectral properties of Koopman operators for dynamical systems. Communications on Pure and Applied Mathematics , 77(1):221–283, 2024

work page 2024
[9]

Daubechies

I. Daubechies. Ten Lectures on Wavelets . CBMS-NSF Regional Conference Series in Applied Mathematics. Society for Industrial and Applied Ma thematics, 1992

work page 1992
[10]

Daubechies, J

I. Daubechies, J. Lu, and H. Wu. Synchrosqueezed wavele t transforms: An empir- ical mode decomposition-like tool. Applied and Computational Harmonic Analysis , 30(2):243–261, 2011

work page 2011
[11]

Drmaˇ c and I

Z. Drmaˇ c and I. Mezi´ c. A data driven Koopman-Schur dec omposition for computa- tional analysis of nonlinear dynamics. arXiv preprint arXiv:2312.15837 , 2023

work page arXiv 2023
[12]

Drmaˇ c and I

Z. Drmaˇ c and I. Meˇ zi´ c. Data driven identiﬁcation andmodel reduction for nonlinear dynamics. In MATHMOD Short Contribution Volume 2025: 11th Vienna Interna- tional Conference on Mathematical Modelling , pages 25–26, 2025

work page 2025
[13]

Drmaˇ c, S

Z. Drmaˇ c, S. G¨ u˘ gercin, and C. Beattie. Quadrature-based vector ﬁtting for discretized H2 approximation. SIAM Journal on Scientiﬁc Computing , 37(2):A625–A652, 2015

work page 2015
[14]

D. Duke, D. Honnery, and J. Soria. Experimental investi gation of nonlinear instabil- ities in annular liquid sheets. Journal of Fluid Mechanics , 691:594–604, 2012

work page 2012
[15]

K. J. Engel and R. Nagel. One-Parameter Semigroups for Linear Evolution Equations , volume 63. Springer New York, 06 2001

work page 2001
[16]

Grilli, P

M. Grilli, P. J. Schmid, S. Hickel, and N. A. Adams. Analy sis of unsteady be- haviour in shockwave turbulent boundary layer interaction . Journal of Fluid Me- chanics, 700:16–28, 2012

work page 2012
[17]

Gustavsen and A

B. Gustavsen and A. Semlyen. Simulation of transmissio n line transients using vector ﬁtting and modal decomposition. IEEE Transactions on Power Delivery , 13(2):605– 614, 1998

work page 1998
[18]

Gustavsen and A

B. Gustavsen and A. Semlyen. Rational approximation of frequency domain responses by vector ﬁtting. IEEE Transactions on Power Delivery , 14(3):1052–1061, 1999

work page 1999
[19]

nondiﬀeren- tiable

M. Holschneider and P. Tchamitchian. Pointwise analys is of Riemann’s “nondiﬀeren- tiable” function. Inventiones mathematicae, 105(1):157–175, 1991. Preprint. 2026-05-15 C.Tilki, S.G¨ u˘ gercin: Wavelet-Based Observables for Koopman Analysis 32

work page 1991
[20]

M. Kamb, E. Kaiser, S. L. Brunton, and J. N. Kutz. Time-de lay observables for Koopman: Theory and applications. SIAM Journal on Applied Dynamical Systems , 19(2):886–917, 2020

work page 2020
[21]

S. Klus, F. N¨ uske, S. Peitz, J. H. Niemann, C. Clementi, and C. Sch¨ utte. Data-driven approximation of the Koopman generator: Model reduction, s ystem identiﬁcation, and control. Physica D: Nonlinear Phenomena , 406:132416, 2020

work page 2020
[22]

B. O. Koopman. Hamiltonian systems and transformation in Hilbert space. Proceed- ings of the National Academy of Sciences , 17(5):315–318, 1931

work page 1931
[23]

Korda, M

M. Korda, M. Putinar, and I. Mezi´ c. Data-driven spectr al analysis of the Koopman operator. Applied and Computational Harmonic Analysis , 48(2):599–629, 2020

work page 2020
[24]

Krishnan, S

M. Krishnan, S. G¨ u˘ gercin, and P. A. Tarazaga. A wavele t-based dynamic mode decomposition for modeling mechanical systems from partia l observations. Mechanical Systems and Signal Processing , 187:109919, 2023

work page 2023
[25]

J.N. Kutz. Data-Driven Modeling & Scientiﬁc Computation: Methods for Co mplex Systems & Big Data . OUP Oxford, 2013

work page 2013
[26]

Q. Li, F. Dietrich, E. M. Bollt, and I. G. Kevrekidis. Ext ended dynamic mode de- composition with dictionary learning: A data-driven adapt ive spectral decomposition of the Koopman operator. Chaos: An Interdisciplinary Journal of Nonlinear Science , 27(10), 2017

work page 2017
[27]

S. Mallat. A Wavelet Tour of Signal Processing, Third Edition: The Sparse Way. Academic Press, Inc., USA, 3rd edition, 2008

work page 2008
[28]

Mallat and W.L

S. Mallat and W.L. Hwang. Singularity detection and pro cessing with wavelets. IEEE Transactions on Information Theory , 38(2):617–643, 1992

work page 1992
[29]

Mauroy, Y

A. Mauroy, Y. Susuki, and I. Mezi´ c. Introduction to the Koopman operator in dynam- ical systems and control theory. In The Koopman operator in systems and control: concepts, methodologies, and applications , pages 3–33. Springer, 2020

work page 2020
[30]

Mauroy, Y Susuki, and I

A. Mauroy, Y Susuki, and I. Mezi´ c. Koopman operator in systems and control . Springer, 2020

work page 2020
[31]

Mayo and A.C

A.J. Mayo and A.C. Antoulas. A framework for the solutio n of the generalized real- ization problem. Linear Algebra and its Applications , 425(2):634–662, 2007. Special Issue in honor of Paul Fuhrmann

work page 2007
[32]

I. Mezi´ c. Spectrum of the Koopman operator, spectral expansions in functional spaces, and state-space geometry. Journal of Nonlinear Science , 30(5), 2020

work page 2020
[33]

R. Mohr, M. Fonoberova, and I. Mezi´ c. Koopman reduced- order modeling with conﬁ- dence bounds. SIAM Journal on Applied Dynamical Systems , 24(3):2070–2101, 2025

work page 2070
[34]

T. W. Muld, G. Efraimsson, and D. S. Henningson. Flow str uctures around a high- speed train extracted using proper orthogonal decompositi on and dynamic mode de- composition. Computers & Fluids , 57:87–97, 2012

work page 2012
[35]

Nakao and I

H. Nakao and I. Mezi´ c. Spectral analysis of the Koopman operator for partial diﬀer- ential equations. Chaos: An Interdisciplinary Journal of Nonlinear Science , 30(11), 2020. Preprint. 2026-05-15 C.Tilki, S.G¨ u˘ gercin: Wavelet-Based Observables for Koopman Analysis 33

work page 2020
[36]

Nakatsukasa, O

Y. Nakatsukasa, O. S` ete, and L. N. Trefethen. The AAA al gorithm for rational approximation. SIAM Journal on Scientiﬁc Computing , 40(3):A1494–A1522, 2018

work page 2018
[37]

E Otto and C

S. E Otto and C. W Rowley. Linearly recurrent autoencode r networks for learning dynamics. SIAM Journal on Applied Dynamical Systems , 18(1):558–593, 2019

work page 2019
[38]

S. E. Otto and C. W. Rowley. Koopman operators for estima tion and control of dynamical systems. Annual Review of Control, Robotics, and Autonomous Systems , 4(Volume 4, 2021):59–87, 2021

work page 2021
[39]

Peherstorfer and K

B. Peherstorfer and K. E. Willcox. Data-driven operato r inference for nonintrusive projection-based model reduction. Computer Methods in Applied Mechanics and En- gineering, 306:196–215, 2016

work page 2016
[40]

D. B. Percival and A. T. Walden. Wavelet Methods for Time Series Analysis . Cam- bridge Series in Statistical and Probabilistic Mathematic s. Cambridge University Press, 2000

work page 2000
[41]

E. Qian, B. Kramer, B. Peherstorfer, and K. Willcox. Lif t & learn: Physics-informed machine learning for large-scale nonlinear dynamical syst ems. Physica D: Nonlinear Phenomena, 406:132401, 2020

work page 2020
[42]

Rajendra and V

P. Rajendra and V. Brahmajirao. Modeling of dynamical s ystems through deep learning. Biophysical Reviews, 12:1311 – 1320, 2020

work page 2020
[43]

T.J. Rivlin. An Introduction to the Approximation of Functions . Blaisdell book in numerical analysis and computer science. Dover Publicatio ns, 1981

work page 1981
[44]

C. W. Rowley, I. Mezi´ c, S. Bagheri, P. Schlatter, and D. S. Henningson. Spectral analysis of nonlinear ﬂows. Journal of Fluid Mechanics , 641:115 – 127, 2009

work page 2009
[45]

P. J. Schmid. Dynamic mode decomposition of numerical a nd experimental data. Journal of Fluid Mechanics , 656:5–28, 2010

work page 2010
[46]

P. J. Schmid. Dynamic mode decomposition and its varian ts. Annual Review of Fluid Mechanics, 54(1):225–254, 2022

work page 2022
[47]

P. J. Schmid, L. Li, M. P. Juniper, and O. Pust. Applicati ons of the dynamic mode decomposition. Theoretical and Computational Fluid Dynamics , 25:249–259, 2011

work page 2011
[48]

Seena and H

A. Seena and H. J. Sung. Dynamic mode decomposition of tu rbulent cavity ﬂows for self-sustained oscillations. International Journal of Heat and Fluid Flow , 32(6):1098– 1110, 2011

work page 2011
[49]

Susuki, A

Y. Susuki, A. Mauroy, and Z. Drmaˇ c. Koopman resolvents of nonlinear discrete-time systems: Formulation and identiﬁcation. 2024 European Control Conference (ECC) , pages 627–632, 2024

work page 2024
[50]

Susuki, A

Y. Susuki, A. Mauroy, and I. Mezi´ c. Koopman resolvent: A Laplace-domain analysis of nonlinear autonomous dynamical systems. SIAM Journal on Applied Dynamical Systems, 20(4):2013–2036, 2021

work page 2013
[51]

Tantet, V

A. Tantet, V. Lucarini, and H. Dijkstra. Resonances in a chaotic attractor crisis of the Lorenz ﬂow. Journal of Statistical Physics , 170:584–616, 02 2018. Preprint. 2026-05-15 C.Tilki, S.G¨ u˘ gercin: Wavelet-Based Observables for Koopman Analysis 34

work page 2018
[52]

T. Tao. An Introduction to Measure Theory . Graduate Studies in Mathematics. American Mathematical Society, 2021

work page 2021
[53]

G. Teschl. Ordinary Diﬀerential Equations and Dynamical Systems . Graduate studies in mathematics. American Mathematical Society, 2012

work page 2012
[54]

M. O. Williams, I. G. Kevrekidis, and C. W. Rowley. A data -driven approxima- tion of the Koopman operator: Extending dynamic mode decomp osition. Journal of Nonlinear Science, 25(6):1307–1346, 2015

work page 2015
[55]

K. Yosida. Functional Analysis . Grundlehren der mathematischen Wissenschaften. Springer Berlin Heidelberg, 1966. Preprint. 2026-05-15

work page 1966