Weak-form Extended Dynamic Mode Decomposition

Christopher W. Curtis; David M. Bortz

REVIEW 2 major objections 5 minor 25 references

WEDMD makes Koopman generator estimation exact on linear systems and noise-robust on nonlinear ones by replacing pointwise derivatives with temporal averaging.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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2026-08-01 01:00 UTC pith:II4H6HIP

load-bearing objection WEDMD is a sensible weak-form twist on EDMD with a clean linear exactness proof and a real SDE variance result, but the unproven full-rank assumption on G and the lack of benchmarks against related methods keep it from being a finished paper. the 2 major comments →

arxiv 2607.25950 v1 pith:II4H6HIP submitted 2026-07-28 math.DS

Weak-form Extended Dynamic Mode Decomposition

Christopher W. Curtis , David M. Bortz This is my paper

classification math.DS MSC 37M9965P9934A55

keywords weak-form EDMDKoopman generatorGalerkin projectiongeneralized eigenvalue problemnoise mitigationreduced-order forecastingstochastic differential equationsKoopman modes

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.

This paper introduces Weak-form Extended Dynamic Mode Decomposition (WEDMD), a method that approximates the Koopman generator rather than the Koopman operator. It integrates the eigenvalue equation against compactly supported test functions, avoiding numerical differentiation and converting noise into filtered averages. For linear deterministic systems, the resulting pencil is exact up to quadrature error, so the computed eigenpairs match the true Koopman generator. For nonlinear systems with finite monomial dictionaries, WEDMD produces enough accurate Koopman modes that selecting modes by reconstruction error yields reduced-order models with roughly 1% relative error, even under strong additive noise and, for fixed-point attractors, under Brownian motion.

Core claim

On the paper's own terms, WEDMD establishes that a weak Galerkin temporal form applied to the Koopman eigenvalue equation yields a generalized eigenproblem λGw = -Dw, where G and D are built from dictionary observables and test functions and their derivatives. For a linear system ẏ = Ay with canonical linear observables, the identity D = -ΛG holds analytically, so the computed spectrum is exactly the spectrum of e^{At} up to quadrature and noise. The same weak form acts as a low-pass filter on noise: variances of G and D depend only on filtered data correlations. For nonlinear systems, the method is a Galerkin approximation that makes the closure residual orthogonal to the test functions, an

What carries the argument

The central object is the pencil λGw = -Dw, built from a dictionary of observables Θ(y) and compactly supported bump test functions ψ_k(t) whose support width r is chosen by minimizing a per-mode reconstruction error E_c(r). Integration by parts replaces the derivative ẏ·∇φ with -φ dψ/dt, so no derivative estimation is needed; G and D are discrete Trapezoid-rule integrals Ψ_dΘ_d and Ψ̇_dΘ_d. For linear systems the identity D = -ΛG makes the pencil exact, and the Moore-Penrose pseudoinverse G^{-P} converts it to a standard eigenvalue problem. The support parameter r is tuned by changepoint detection and error minimization, which is what allows a small accurate subset of modes to be identified

Load-bearing premise

The method assumes the Gram matrix G has full column rank and that the finite monomial dictionary is nearly closed under the generator; the paper leaves the full-rank condition as a plausible but unproved statement, and for nonlinear dictionaries the weak form actually estimates a stochastic backward generator rather than the deterministic one.

What would settle it

Take a linear system ẏ = Ay with known diagonalizable A, sample clean data y_j = e^{A j δt} x, and run WEDMD with the canonical linear dictionary and sufficiently small δt. If any computed eigenvalue differs from an eigenvalue of A by more than the Trapezoid quadrature error, the exactness claim fails. Conversely, a negative control with a deliberately rank-deficient dictionary (e.g., duplicate or linearly dependent observables) should produce G singular and spurious eigenpairs; if the pseudoinverse still returns the true spectrum, the full-rank assumption is not actually load-bearing.

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

If this is right

On linear deterministic systems, WEDMD recovers the exact Koopman operator e^{At}; the only remaining errors are quadrature and noise, and the weak form suppresses noise by averaging.
The method avoids explicit differentiation, removing the variance-amplification problem of pointwise generator estimation; for additive Gaussian noise this keeps spectra and forecasts stable even at σ = 0.1.
For nonlinear attractors, a simple reconstruction-error filter picks out a small set of accurate Koopman modes, giving reduced-order models with errors near 1% on clean Duffing and Van der Pol data.
The averaging interpretation extends to linear stochastic systems: process noise enters only through zero-mean fluctuations for linear observables, so the weak-form operators depend on filtered correlations rather than instantaneous noisy derivatives.
For stochastic dynamics with nonlinear dictionaries, the weak form estimates the Kolmogorov backward generator of the stochastic process rather than the deterministic Koopman generator, delimiting when deterministic forecasting is possible.

Where Pith is reading between the lines

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

A natural testable extension is to use the same temporal weak form with adaptive or learned dictionaries, which could close the gap between the finite-dictionary Galerkin approximation and true Koopman invariance.
The Itô-correction distinction identified for SDEs suggests WEDMD could be deliberately repurposed as a data-driven estimator of the Kolmogorov backward generator from stochastic trajectories, rather than treating stochasticity only as noise to remove.
The reliance on full column rank of G suggests a practical diagnostic: monitor the condition number of G and adapt r, p, or the dictionary until it stabilizes, turning a heuristic hyperparameter choice into a data-driven criterion.
Since the method needs no sparsity promotion, it may combine naturally with dictionary pruning based on reconstruction error, yielding interpretable reduced-order Koopman models for high-dimensional systems.

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

WEDMD is a sensible weak-form twist on EDMD with a clean linear exactness proof and a real SDE variance result, but the unproven full-rank assumption on G and the lack of benchmarks against related methods keep it from being a finished paper.

read the letter

You should know this paper before the others in your stack. It does something genuinely useful: it takes the WSINDy weak-form idea and applies it to the continuous-time Koopman generator with a nonlinear observable dictionary. The linear deterministic exactness proof in Section 2.3 is clean — no fitted constants, just quadrature error and noise left as error sources. The SDE variance calculation for canonical observables is also real, and it explains the low-pass filtering effect without hand-waving. Those two analytic results are the core of the paper, and they hold up.

The numerical work is honest and reasonably convincing. For the Duffing and Van der Pol examples, the mode-filtering strategy gives good reduced-order forecasts under additive noise, and the authors openly report that WEDMD tracks the noisy trajectory rather than the underlying dynamics for the Van der Pol SDE case. That kind of limitation statement is refreshing and consistent with the Itô caveat in Section 2.3: for nonlinear dictionaries the weak form estimates the Kolmogorov backward generator, not the deterministic Koopman generator.

Now the soft spots, in proportion. The full-rank claim on G is the load-bearing assumption for the pseudoinverse reduction G^{-P}D w = -λ w, and it is only asserted as "it should be possible" in Section 2.1. If K+1 < N_d, G is rank-deficient by construction, and the stress-test note is right that the linear exactness claim and the computed spectra can then contain spurious modes. The paper never reports K for its examples, so we do not know whether the successes happen in the full-rank regime. This is a fixable issue — prove a rank bound under reasonable conditions or at least report and check K+1 ≥ N_d — but it is currently the weakest link. The residual-convergence argument in Section 2.2 is also explicitly an "infinite resolution" heuristic, not a theorem, though it is clearly labeled as such. And there are no benchmarks against WDMD, PRONE, or even standard EDMD, which is a missed opportunity given that the authors already distinguish their method from those two.

The self-selection of r by minimizing the reconstruction error on the same trajectories used to report forecasting skill is a mild circularity, but not damning; they filter modes by the same error metric, so it is more a tuning choice than a hidden fit.

Who gets value from this: anyone working on data-driven Koopman approximations or weak-form methods for noisy dynamical systems. It deserves a serious referee, but I would send it back with a request to address the rank condition, add comparison benchmarks, and check whether the conclusions survive in the K+1 < N_d regime. If those are addressed, the method stands as a practical contribution to the subfield.

Referee Report

2 major / 5 minor

Summary. The paper introduces WEDMD, a weak-form variant of EDMD for estimating the Koopman generator from time series. Observables are expanded in a monomial dictionary; the generator eigenvalue problem is integrated against compactly supported test functions to form the pencil λGw = -Dw, which is then reduced to a standard eigenvalue problem via the pseudoinverse of G. The authors prove that for linear deterministic systems with canonical observables the matrices satisfy D = -GΛ exactly up to quadrature, and they show for linear SDEs that the variances of G and D are governed by filtered covariance integrals, supporting a low-pass filtering interpretation. For nonlinear Duffing and Van der Pol examples, they demonstrate that filtering computed modes by reconstruction error yields good reduced-order forecasts under additive noise. The paper is clearly written and the numerical studies are thoughtfully designed, but the rank condition underpinning the pseudoinverse step is left as a conjecture without verification.

Significance. WEDMD is a promising methodological contribution: avoiding explicit derivative estimation is valuable, and the linear-SDE variance calculation is a concrete, useful theoretical result. The linear exactness argument is elegant and self-contained. The numerical evidence for mode filtering and forecasting is compelling in the additive-noise regime. However, the central reduction depends on full column rank of G, a condition that is neither proved nor empirically checked (K+1 is never reported), so the theoretical guarantees do not yet cover the practical algorithm. The residual-convergence argument in §2.2 is also heuristic. These gaps are fixable and do not invalidate the overall approach.

major comments (2)

[Section 2.1, Eq. λGw = -Dw and pseudoinverse reduction] The reduction to G^{-P}Dw = -λw is justified only if G has full column rank, but the paper only conjectures this ('it should be possible to choose r, s, and p to ensure G has full rank'). In the linear exactness argument of §2.3, D = -GΛ termwise turns the pencil into (λI - Λ)Gw = 0; if G is rank-deficient, any w ∈ null(G) produces a spurious eigenvalue, so 'we should get the exact Koopman operator e^{At}' fails even with noiseless data and zero quadrature error. The numerical sections never report K+1, so the reader cannot tell whether the experiments actually operated in the full-rank regime. The authors should either prove the rank condition under explicit assumptions on r, s, p and the trajectory, or replace the pseudoinverse step with a truncated-SVD/regularized formulation and prove a perturbation bound.
[Section 2.2, residual-convergence argument] The claim that taking r → 0+ while letting centers pass to a continuum limit 'essentially eliminate[s] the residual' is heuristic rather than a proof. The text itself concedes this is 'an infinite resolution data result.' The argument interchanges a limit in r with a continuum limit in K and assumes a dominated-convergence statement for the residual that is not established. Since the choice of r is a central hyperparameter and Algorithm 1 selects it by minimizing E_c(r), the theoretical convergence statement needs either a precise theorem with commuting limits or a clear demotion to a plausibility argument. As written, this is load-bearing for the claim that the weak form 'provides a systematic means of controlling approximation error.'

minor comments (5)

[Section 2.1] The sentence 'G^{-P} is the Moore-Penrose pseudoinverse of D' should read 'of G'; also the notation G^{-P} is not defined.
[Algorithm 1] p is initialized in step 1 but immediately overwritten in step 4; clarify the role of each step.
[Sections 3.1 and 3.2] Report K+1 (or r, s, p and N_T) so that the full-rank condition can be checked by the reader.
[Section 2.2] The displayed limit '=0 R(t,x)Λ ∫...' appears to have a missing equality/line break; please re-set the equation.
[Figures 3 and 8] Panels are referenced as (a)-(f) in the text, but the captions do not describe all panels; please expand the captions.

Circularity Check

1 steps flagged

No central circularity: the linear exactness proof and SDE filtering analysis are self-contained. One low-level data-leakage step: the support radius r and mode-filter thresholds are chosen by minimizing errors over the full trajectory, including the interval later reported as 'prediction'.

specific steps

fitted input called prediction [Section 2.4 and Algorithm 1; Section 3.1 (Damped Duffing)]
"Ec(r) measures the relative accuracy to which a given eigenvector and eigenvalue act as Koopman eigenfunctions and values over the length of the time series. We then determine r by minimizing Ec(r). ... we select those modes (wl, λl) such that E5,l < .1 ... To test this, we generate modes using data for 0≤t≤18 and then use our reduced mode model to predict dynamics from 18≤t≤20."

The hyperparameter r and the mode-selection threshold are chosen by minimizing reconstruction errors E_c(r)/E_l computed over the entire trajectory 0≤t≤t_f, which includes the forecast interval 18≤t≤20 (and analogously for the other experiments). The modes are then fit on 0≤t≤18 and the error on 18≤t≤20 is reported as a prediction. Because the selection criterion already used the test interval, the reported forecast error is partially in-sample; the 'prediction' is statistically forced by the same data rather than being an independent out-of-sample check. This affects the forecasting validation, not the analytic claims of the paper.

full rationale

The central derivation is not circular. In Section 2.3 the paper proves, for linear deterministic systems with the canonical dictionary, that Dkn = −λnGkn termwise from the exact flow y(t)=e^{At}x. The pencil λGw=−Dw then reduces by linear algebra to eigenpairs of Λ provided G has full column rank. That rank condition is asserted only as 'it should be possible to choose r, s, and p to ensure G has full rank' (Section 2.1), which is an omitted proof and a genuine correctness risk (the skeptic's rank-deficiency scenario is real), but it is not a circular reduction: no fitted parameter is inserted into the derivation, and D = −GΛ is obtained from the assumed dictionary, not assumed as the desired answer. The stochastic filtering analysis in Section 2.3 is similarly self-contained, and the paper explicitly flags the Itô-correction limitation for nonlinear dictionaries. Self-citations to WSINDy work ([17], [18]) are used only for the bump-function ansatz and the initial changepoint estimate of r; they are methodological and not load-bearing for the exactness or closure theorems. The one genuine circularity in the paper is in the forecasting validation: r and the mode-filtering thresholds are chosen by minimizing errors computed over the full trajectory, including the interval later reported as 'prediction', so the reported forecast quality is partly in-sample. This is a minor, localized issue and does not undermine the main analytic results; hence the low score.

Axiom & Free-Parameter Ledger

4 free parameters · 5 axioms · 0 invented entities

No new physical or mathematical entities are introduced; the bump test functions are standard constructions from the WSINDy literature. The central burden is the unproven full-rank/closure assumption and the Itô-correction caveat for nonlinear stochastic data.

free parameters (4)

test-function support radius r = not tabulated; selected by changepoint detection then minimizing E_c(r)
Controls low-pass filtering and Galerkin resolution; hyperparameter tuned on each dataset in Algorithm 1.
mode filtering threshold = E<0.1 clean Duffing/VdP, E<1 additive noise, E<2 Duffing SDE
Hand-chosen threshold for selecting modes with small reconstruction error; directly affects the reported forecast quality.
dictionary degree N_ord = 3, 4, 5, and 6 in different runs
Monomial dictionary truncation; higher N_ord improves mode accuracy but not uniformly.
test-function overlap and edge parameters s, τ_l = not specified in text
Algorithm requires s∈(0,1) and τ_l≪1 with p derived from them; no concrete values are reported for the examples.

axioms (5)

domain assumption Finite monomial dictionary Θ(x) well-approximates the relevant Koopman eigenfunctions (dictionary closure).
Section 2.2 says approximation quality depends on 'the degree to which the observables form a closed or nearly closed subspace'; no adaptive dictionary is used.
ad hoc to paper The data Gram matrix G has full column rank, allowing the pseudoinverse eigenvalue problem.
Section 2.1: 'it should be possible to choose r, s, and p to ensure G has full rank' - asserted, not proved.
domain assumption The ordinary chain rule applies even for stochastic data; for nonlinear dictionaries the Itô correction is treated as a shift to the Kolmogorov backward generator.
Section 2.3 explicitly says the Itô correction is a deterministic shift for nonlinear dictionaries and is not removed by the weak form.
domain assumption Quadrature error from the trapezoid rule and discretization is negligible relative to the approximation error.
Discretized G and D are formed via the trapezoid rule, requiring at least one interior data point per test-function support; no convergence rate is proved.
standard math Linear systems considered are diagonalizable, A = VΛV^{-1}.
Used in Section 2.3 to transform the pencil; generic but an assumption.

pith-pipeline@v1.3.0-alltime-deepseek · 11736 in / 14130 out tokens · 133371 ms · 2026-08-01T01:00:30.543345+00:00 · methodology

0 comments

read the original abstract

In this work, we develop a weak-form version of Extended Dynamic Mode Decomposition that we call WEDMD. We establish a number of analytic results about the method and show explicitly how the weak form is able to mitigate the impacts of noise in linear stochastic differential equations. In nonlinear systems, we likewise show how the method is able to generate excellent approximations to Koopman modes without making recourse to sparsity promoting approaches. This excellent approximation property allows for meaningful forecasting even in the presence of noisy data.

Figures

Figures reproduced from arXiv: 2607.25950 by Christopher W. Curtis, David M. Bortz.

**Figure 2.** Figure 2: As seen, the relative error from our model relative to the numerically [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗

**Figure 3.** Figure 3: For the Duffing equation, noisy data generated by additive noise (left [PITH_FULL_IMAGE:figures/full_fig_p016_3.png] view at source ↗

**Figure 4.** Figure 4: Comparison of Duffing dynamics generated by reduced model using [PITH_FULL_IMAGE:figures/full_fig_p017_4.png] view at source ↗

**Figure 5.** Figure 5: Comparison of Duffing dynamics generated by reduced model using [PITH_FULL_IMAGE:figures/full_fig_p017_5.png] view at source ↗

**Figure 6.** Figure 6: Mode error for the Van der Pol oscillator for [PITH_FULL_IMAGE:figures/full_fig_p018_6.png] view at source ↗

**Figure 7.** Figure 7: Relative error using a reduced mode model for [PITH_FULL_IMAGE:figures/full_fig_p019_7.png] view at source ↗

**Figure 8.** Figure 8: For the Van der Pol equation, noisy data generated by additive noise [PITH_FULL_IMAGE:figures/full_fig_p020_8.png] view at source ↗

**Figure 9.** Figure 9: Comparison of Van der Pol dynamics generated by reduced model [PITH_FULL_IMAGE:figures/full_fig_p021_9.png] view at source ↗

**Figure 10.** Figure 10: Comparison of Van der Pol dynamics generated by reduced model [PITH_FULL_IMAGE:figures/full_fig_p021_10.png] view at source ↗

discussion (0)

Reference graph

Works this paper leans on

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[1] [1]

D. J. Alford-Lago, C. W. Curtis, A. T. Ihler, and O. Issan , Deep learning enhanced dynamic mode decomposition , Chaos, 32 (2022), p. 033116

2022

[2] [2]

Azencot, N

O. Azencot, N. B. Erichson, V. Lin, and M. W. Mahoney , Forecasting sequential data using consistent koopman autoencoders , in Proceedings of the 37 th International Conference on Machine Learning (ICML), 2020, pp. 475--485

2020

[3] [3]

Bennett, R

W. Bennett, R. G. McClarren, E. Smith, and M. Derman , Weak-dmd: A galerkin approach to the problem of noise in the dynamic mode decomposition algorithm , arXiv preprint arXiv:2604.14350, (2026)

Pith/arXiv arXiv 2026

[4] [4]

Berkooz, P

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

2012

[5] [5]

D. M. Bortz, D. A. Messenger, and V. Dukic , Direct estimation of parameters in ode models using WENDy : Weak-form estimation of nonlinear dynamics , Bull. Math. Biol., 85 (2023), p. 110

2023

[6] [6]

S. L. Brunton, M. Budi s i\' c , E. Kaiser, and J. N. Kutz , Modern koopman theory for dynamical systems , SIAM Rev., 64 (2022), pp. 229--340

2022

[7] [7]

S. L. Brunton, J. L. Proctor, and J. N. Kutz , Discovering governing equations from data by sparse identification of nonlinear dynamical systems , Proc. Natl. Acad. Sci. USA, 113 (2016), pp. 3932--3937

2016

[8] [8]

M. J. Colbrook, A. Herwig, and J. N. Kutz , PRONE : Petrov--Galerkin operator learning unifies DMD , SINDy & Koopmanism , arXiv:2606.27982, (2026)

Pith/arXiv arXiv 2026

[9] [9]

C. W. Curtis, D. Jay Alford-Lago, E. Bollt, and A. Tuma , Machine learning enhanced hankel dynamic-mode decomposition , Chaos, 33 (2023), p. 083133

2023

[10] [10]

Kaheman, S

K. Kaheman, S. L. Brunton, and J. N. Kutz , Physics-informed machine learning for nonlinear partial differential equations and dynamical systems , Phys. Rev. E, 101 (2020), p. 033302

2020

[11] [11]

Kaheman, J

K. Kaheman, J. N. Kutz, and S. L. Brunton , SIND y- PI : A robust algorithm for parallel implicit sparse identification of nonlinear dynamics , Proc. A, 477 (2021), p. 20200232

2021

[12] [12]

Kaiser, J

E. Kaiser, J. N. Kutz, and S. L. Brunton , Data-driven discovery of koopman eigenfunctions for control , Mach. Learn. Sci. Technol., 2 (2021), p. 035023

2021

[13] [13]

S. Klus, F. Nüske, S. Peitz, J.-H. Niemann, C. Clementi, and C. Schütte , Data-driven approximation of the koopman generator: Model reduction, system identification, and control , Phys. D, 406 (2020), p. 132416

2020

[14] [14]

J. N. Kutz, S. L. Brunton, B. W. Brunton, and J. L. Proctor , Dynamic Mode Decomposition: Data-Driven Modeling of Complex Systems , SIAM, Philadelphia, 2016

2016

[15] [15]

Lasota and M

A. Lasota and M. C. Mackey , Chaos, Fractals, and Noise: Stochastic Aspects of Dynamics , vol. 97 of Applied Mathematical Sciences, Springer, New York, 2 ed., 1994

1994

[16] [16]

Lusch, J

B. Lusch, J. N. Kutz, and S. L. Brunton , Deep learning for universal linear embeddings of nonlinear dynamics , Nat. Commun., 9 (2018), p. 4950

2018

[17] [17]

D. A. Messenger and D. M. Bortz , Weak SINDy for partial differential equations , J. Comput. Phys., 449 (2021), p. 110525

2021

[18] [18]

Simul., 19 (2021), pp

height 2pt depth -1.6pt width 23pt, Weak SINDy : Galerkin-based data-driven model selection , Multiscale Model. Simul., 19 (2021), pp. 1474--1497

2021

[19] [19]

Mezi \'c , Koopman operator, geometry, and learning of dynamical systems , Notices Amer

I. Mezi \'c , Koopman operator, geometry, and learning of dynamical systems , Notices Amer. Math. Soc., 68 (2021), p. 1

2021

[20] [20]

P. J. Schmid , Dynamic mode decomposition of numerical and experimental data , J. Fluid Mech., 656 (2009), pp. 5--28

2009

[21] [21]

Towne, O

A. Towne, O. T. Schmidt, and T. Colonius , Spectral proper orthogonal decomposition and its relationship to dynamic mode decomposition and resolvent analysis , J. Fluid Mech., 847 (2018), pp. 821--867

2018

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