Data-Driven Domain of Attraction Estimation via Convergent Koopman-Zubov Approximation

Wentao Tang

Data-Driven Domain of Attraction Estimation via Convergent Koopman-Zubov Approximation

T0 review · 4 major / 6 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The paper proves that the Zubov–Koopman operator on the constant-augmented linear–Sobolev RKHS has spectral radius below 1 and a unique invariant element equal to the Zubov function, with data-driven estimates converging at rate $d^{s-d/2}|

desk verdict A promising kernel-based Zubov–Koopman construction with a good algorithm and decent numerics, but Theorem 4.1's spectral-radius step is invalid as written and Assumption 3.1 is far too strong to cover the examples. read the letter →

arxiv 2608.01018 v2 pith:OOJ4SNAG submitted 2026-08-02 eess.SY cs.SY

Wentao Tang This is my paper · ORCID

classification eess.SYcs.SY MSC 47B3293D30

keywords domainofattractionZubovfunctionZubov–KoopmanoperatorreproducingkernelHilbertspacelinear–radialspectralradiusregressiondata-drivenestimation

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper tries to establish that the Zubov–Koopman operator — the weighted composition operator behind Zubov's equation for the domain of attraction (DOA) — can be given a function space on which it has the spectral structure needed for computation. On the constant-augmented linear–Sobolev reproducing kernel Hilbert space, the operator has spectral radius strictly below $1$ and a unique invariant element that is exactly the Zubov function, so the formal limit $\zeta = \lim_{t\to\infty}Z_t\mathbf{1}$ becomes rigorous. The paper further proves that a regularized kernel regression from snapshot data estimates this Zubov function with a sectorial error bound $|\hat\zeta(x)-\zeta(x)|\lesssim d^{s-d/2}|x|$, where $d$ is the fill distance of the samples and $s$ the smoothness index. A sympathetic reader would care because this converts DOA estimation from a hard nonlinear PDE search or nonconvex neural-network training into a convex kernel problem with quantitative guarantees.

What carries the argument

The load-bearing object is the linear–radial kernel $\mathring\kappa(x,x')=(x^\top x')\rho(|x-x'|)$, where $\rho$ is a radial Sobolev kernel; its RKHS is the linear–Sobolev space $\mathring H^s(\mathbb X)$, the span of products $x_k g_k(x)$ with $g_k\in H^s(\mathbb X)$. The kernel is origin-preserving: the canonical feature satisfies $\|\mathring\phi(0)\|=0$ and $\|\mathring\phi(x)\|\propto |x|$. This local-linear structure is what lets the proof reduce the nonlinear dynamics, via the $\mathring C^s$-conjugacy, to the linear flow $e^{tJ}$, whose Koopman spectrum on monomials is explicitly controllable and strictly inside the unit disk when $J$ is Hurwitz. The same origin-preserving geometry

What would settle it

Run the proposed kernel estimator on an asymptotically stable system with a finite DOA but no global smooth linearization on any forward-invariant set — for example, a stable equilibrium whose linearization has a resonance that blocks a $\mathring C^s$ conjugacy, or a system where the linearizing map exists only on a small neighborhood. The central claim predicts $|\hat\zeta(x)-\zeta(x)|\lesssim d^{s-d/2}|x|$ with the stated fill-distance rate. If the measured sup-error does not decay at that rate, or if the learned finite-rank operator has spectral radius $\ge 1$, the theorem's hypothesis is

Watch

Extended reading notes

Core claim

The central claim is that the Zubov–Koopman semigroup $Z_{\Delta t}g=\exp(-\int_0^{\Delta t}\omega\circ S_f^\tau\,d\tau)\,(g\circ S_f^{\Delta t})$, defined on the linear–Sobolev RKHS $\mathring H^s(\mathbb X)$ and its constant augmentation $\mathbb G_\oplus=\mathbf 1\oplus\mathring H^s(\mathbb X)$, has, for every fixed $\Delta t>0$, spectral radius $r(Z_{\Delta t})<1$ on $\mathring H^s(\mathbb X)$. Consequently the operator on $\mathbb G_\oplus$ has a unique invariant element $\zeta=\mathbf 1-(I-Z_{\Delta t})^{-1}\eta$, where $\eta=1-e^{-\omega_{\Delta t}}$, and this element is the Zubov function whose nonzero level sets delineate the DOA. The data-driven version replaces $Z_{\Delta t}$ by a

Load-bearing premise

The load-bearing premise is that the nonlinear flow can be straightened into its linearization on a whole forward-invariant set inside the basin (a $\mathring C^s$-conjugacy $\psi$ with $\psi(S_f^t x)=e^{tJ}\psi(x)$); this is much stronger than local linearizability, and if it fails the proof that the Zubov–Koopman operator has spectral radius below $1$ has no basis.

Editorial extensions

If this is right

Computing a DOA estimate becomes a finite-dimensional convex optimization whose solution has the closed form $Z=(\Phi+\lambda I)^{-1}$, so the method scales to modest data sizes without nonconvex training.
The spectral-radius theorem justifies the infinite Neumann series $\hat\zeta=\mathbf 1-(I-\hat Z)^{-1}\eta$, so the learned operator can be iterated to convergence with a guarantee that the iteration is well posed.
The sectorial error bound gives a spatial accuracy profile: near the origin the estimate is tight, while uncertainty grows linearly in $|x|$; with a lattice sample the uniform error decays like $n^{-(s/d-1/2)}$.
For smooth enough systems ($s\ge d$), the rate is at least $\tilde O(n^{-1/2})$, meaning a few thousand snapshots can already separate on-DOA from off-DOA states, as the numerical examples show.

Reading between the lines

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

Editorial inference: Assumption 3.1 is the fragile point; a natural stress test is a stable equilibrium with a resonance that prevents a global smooth linearization — the paper's theory is silent there, even though the method may still work.
Editorial inference: the error envelope $C d^{s-d/2}|x|$ could be used to build a certified inner approximation of the DOA by thresholding $\hat\zeta$ away from the boundary, turning pointwise misclassification into a quantified guarantee.
Editorial inference: replacing the linear factor $x^\top x'$ with a kernel tailored to a different invariant object (a limit cycle, an invariant torus, or a manifold) is the natural route to carry the same spectral argument beyond equilibrium basins, a direction the conclusion explicitly leaves open.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Wentao Tang's paper gives a concrete way to estimate the domain of attraction by putting the Zubov–Koopman operator on the augmented linear–Sobolev RKHS. The construction is sensible, the kernel learning problem is convex and explicit (the Z matrix is (Φ+λI)^{-1}), and the numerical results on three examples show misclassification rates in the low single digits with sub-second training times. The code is released. That is real value.

What is new: the combination of the linear–radial kernel with the Zubov operator, the spectral radius theorem on that space, and the claim of a sectorial error bound. The writing is mostly clear, and the author engages the prior literature, including the EDMD Zubov work and PINN alternatives.

The soft spots are serious, though. The central guarantee in Theorem 4.1 does not follow from the proof. The step asserting r(hat Z) ≤ r(Z) because hat Z = T Z with T = (S+λ/n)^{-1}S a contraction is not valid without commutativity, and T does not commute with Z. A contraction multiplying a stable operator on the left can produce a spectral radius larger than the original. And even if r(hat Z)<1 for each dataset, the factor Σ_m ||hat Z^m|| in the error bound could grow with n; the proof needs a uniform bound and doesn't have one. Without that, the advertised d^{s-d/2}|x| rate is unsupported. This is a load-bearing gap, not a cosmetic one.

Assumption 3.1 is also a problem. A global C̊^s conjugacy to the linearized flow on a forward-invariant subset of the DOA is far stronger than local Hartman–Grobman, and the paper does not verify it for any of the three examples. For the reversed van der Pol and the two-machine system it is hard to see how it can hold. So the theoretical guarantees do not cover the numerics as presented.

One smaller point: the method is not fully data-driven. Computing η(x) = 1 - exp(-∫_0^{Δt} ω(S_τ f(x)) dτ) requires the model and an integration along the trajectory, not just snapshot data.

The proof of Lemma 3.1 also glosses over derivatives of the exponential weight. That part may be fixable because the integral converges, but as written it is handwavy.

Who should read this: anyone working on Koopman-based stability or DOA estimation. The idea is worth testing, and the algorithm is easy to implement. But I would not cite Theorem 4.1 as a theorem in its current form. The paper deserves a serious referee: the flaws are identifiable and possibly repairable with a proper perturbation argument or a different norm in which the empirical operator is contractive. I'd send it to review, but the author should expect a hard requirement to fix the spectral-radius step and either verify or weaken Assumption 3.1.

Referee Report

4 major / 6 minor

Summary. The paper proposes a data-driven method for estimating the domain of attraction (DOA) of a nonlinear system via a Zubov–Koopman operator. The operator is defined on an RKHS formed as the product of a linear kernel and a Sobolev kernel (the linear–Sobolev space), augmented by the constant function space. The main theoretical claims are: (i) under a global smooth linearizability assumption (Assumption 3.1) and a sufficient-cost condition (Assumption 3.2), the Zubov–Koopman operator restricted to the linear–Sobolev space has spectral radius strictly less than 1 (Theorem 3.2); (ii) on the augmented space, it has a unique invariant element which is the Zubov function (Theorem 3.3); and (iii) the regularized kernel-based estimate satisfies the sectorial error bound |ζ̂(x)−ζ(x)| ≲ d^{s−d/2}|x|, where d is the fill distance (Theorem 4.1). Three two-dimensional numerical examples illustrate the method.

Significance. If the theorems were fully established, the paper would provide a useful operator-theoretic justification for a convex, kernel-based DOA estimation procedure with explicit error rates, complementing SOS and neural-network approaches. The linear–radial RKHS construction is a natural way to make the Zubov–Koopman operator contractive, and the numerical experiments demonstrate reasonable empirical performance with modest computational cost. The sectorial error bound, if valid, would be a distinctive contribution. However, several load-bearing proof steps are incomplete or incorrect, and the main assumption is strong and unverified in the examples. For these reasons the contribution is currently conditional rather than established.

major comments (4)

[Section 3.2, Assumption 3.1 and Theorem 3.2] Assumption 3.1 postulates a global C̊^s homeomorphism conjugating the nonlinear flow to its linearization, which is far stronger than local Hartman–Grobman or Sternberg linearization. No verification is provided for the numerical examples in Section 5. In addition, the proof of Theorem 3.2 asserts that 'any state in O must reach O_1 within a uniformly bounded amount of time,' so that O is contained in some X_m. This is not implied by the stated assumptions: O is a forward-invariant subset of the DOA and may be noncompact, and the conjugating coordinate does not make the absorption time uniform. In Example 5.1, for instance, points arbitrarily close to the boundary of the unit disk take arbitrarily long to enter any fixed inner ball. The decomposition into X_m and X\O_1 therefore does not follow as written.
[Lemma 3.1] The final step of Lemma 3.1 concludes r(Z_t)<1 from the facts that monomials are eigenfunctions of the linearized Koopman operator and are dense in H̊^s(O). Dense eigenfunctions with eigenvalues inside the unit disk do not, by themselves, imply a spectral radius strictly below 1; a uniform spectral gap or a direct norm estimate is required. In this specific setting such an estimate may be recoverable because e^{tJ} is a strict contraction, but the written basis argument is not valid. Since Theorem 3.2 relies on Lemma 3.1, this is a load-bearing gap.
[Section 4.3, Theorem 4.1] The proof of r(Ẑ)<1 is incorrect. It states that because Ẑ^* = Z^*T with T=(S+λ/n)^{-1}S contractive, one has r(Ẑ)=r(Ẑ^*)≤r(Z^*)=r(Z). For noncommuting bounded operators this implication is false; r(TZ) can exceed r(Z) even when T is a self-adjoint contraction. Moreover, even accepting r(Ẑ)<1, the proof needs the summability of Σ_m‖Ẑ^m‖ to be uniform in the sample size n. The argument provides no such uniform bound, and a spectral radius strictly below 1 does not by itself control the magnitude of the sum. Consequently the displayed error bound and the well-definedness of the Neumann series in equation (9) are not established.
[Section 4.2, Corollary 4.1 and Proposition 4.3] The proof of Corollary 4.1 reduces the linear–radial kernel regression to independent scalar regressions by writing ĝ_k = ĥ_k/e_k and claiming that each ĥ_k is obtained by a separate scalar optimization. The regression objective is coupled across components through the factors x_i,k in the evaluation functional, so this reduction is not valid. In addition, the equivalence of norms for ĝ_k and ĥ_k requires a lower bound on |x| away from zero; the stated assumption only gives ε_x = min_i |x_i| > 0, which may tend to zero as n grows, making any constant dependent on ε_x incompatible with the advertised d^{s−d/2} rate. Proposition 4.3 inherits this issue.

minor comments (6)

[Introduction] Typo: 'infinite-dimensinoal' should be 'infinite-dimensional'.
[Organization and Notations] 'The reminder of this paper' should be 'The remainder of this paper'.
[Definition 2.2] The Gram matrix is defined with indices i,j=1,...,N after the sum uses n; please make the notation consistent.
[Remark 2.1] The formula for the native space of the Wendland kernel appears to read s=k+1/2, which is inconsistent with the standard Wendland smoothness s=k+(d+1)/2. Please clarify the displayed formula.
[Remark 4.2] The exponent notation for the rate in terms of n is ambiguous: the displayed expression should be n^{-(s/d−1/2)} when d ~ n^{-1/d}; please re-typeset for clarity.
[Section 5.2] 'outforms' should be 'outperforms'.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: core proofs are self-contained; the two self-citations are independent mathematical lemmas.

full rationale

The central spectral-radius result (Theorem 3.2) is proved from the dynamical-systems assumptions: under the C̊^s-homeomorphism of Assumption 3.1 the nonlinear flow is conjugated to the linearized flow, and the Hurwitz property of the Jacobian plus the cost-growth Assumption 3.2 yield r(Z)<1. This is a direct proof, not a reduction of the output to the input. The unique invariant element (Theorem 3.3) follows from the Neumann series (I−Z)^{-1} justified by r(Z)<1, and equals the Zubov function by Lemma 2.6; this is a mathematical identity, not a circular definition. The data-driven error bound (Theorem 4.1) uses the external scattered-data approximation bound of Wendland–Rieger (Lemma 4.1) and the reproducing property of the linear–radial kernel (the feature norm is proportional to |x|), which is a standard learning-theoretic guarantee rather than a fitted parameter renamed as a prediction. The two load-bearing self-citations—Corollary 2.1 (linear–Sobolev space equivalence to the linear–radial RKHS, cited to [32]) and Lemma 2.3 (strong continuity of the Koopman semigroup, cited to [43])—are independent mathematical results with stated assumptions that do not include the target results, so per the review rules they are real evidence and do not raise the circularity score. The manuscript itself does not assert any limitation or missing support, and the proof gap identified in Theorem 4.1 (the step r(Ẑ) ≤ r(Z) for noncommuting T and Z, and the non-uniform summability of ||Ẑ^m||) is a correctness concern, not a circular equivalence. The strong Assumption 3.1 is likewise a hypothesis rather than a circular input. Therefore, the derivation chain is not circular; the score of 2 reflects only the presence of self-citations for foundational lemmas, which do not constitute circularity.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central theoretical result rests on standard RKHS/scattered-data theory, on regularity assumptions about the dynamics and cost, and critically on a global linearizability assumption (Assumption 3.1) introduced specifically to enable the spectral proof. The numerical experiments tune cost and kernel hyperparameters that are not part of the theoretical claim.

free parameters (3)

cost coefficient α = Examples: 3.0 then 5.0 (Example 1), 2.0 (Example 2), 1/2 (Example 3)
The cost function ω(x)=α|x|^2 is a user-chosen input; the numerical classification accuracy depends on tuning α, though the theoretical claims hold for any admissible ω.
kernel scale σ = 1.5, then 2.25 (Example 1); 4.5 (Example 2); unstated for Example 3
σ in the Wendland kernel is a hyperparameter tuned to improve misclassification rates. The theory's error bound does not explicitly track σ.
regularization λ = not reported in numerical sections
λ appears in the ridge regression (6) and in the error bounds, but the paper does not state the values used in the experiments.

assumptions (5)

domain assumption f ∈ [C̊^s(X)]^d, s ≥ 1, and the origin is exponentially stable
Used in Theorem 3.1 for strong continuity and in Theorem 3.2 for the exponential stability requirement.
ad hoc to paper Assumption 3.1: existence of a C̊^s homeomorphism ψ on a forward-invariant O ⊂ A such that ψ(S_t f(x)) = e^{tJ}ψ(x)
This global linearizability assumption is introduced specifically to prove the spectral radius theorem; it is not a standard result and is not verified in the numerical examples.
domain assumption Assumption 3.2: cost ω is sufficiently large outside a forward-invariant set O_1 ⊂ O
Ensures exponential decay of the cost term outside O_1, used in the proof of Theorem 3.2.
domain assumption X is a bounded Lipschitz domain satisfying an interior cone condition (in Theorem 4.1)
Required for the scattered-data approximation bounds from Wendland-Rieger (Lemma 4.1).
standard math Standard RKHS and scattered-data approximation results (product kernel property, Wendland kernel equivalence, Sobolev embedding)
Background results used for the learning error analysis and the characterization of the RKHS.

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Cite this review

Pith. "Pith review of Data-Driven Domain of Attraction Estimation via Convergent Koopman-Zubov Approximation." pith.science (2026). https://pith.science/paper/OOJ4SNAG

@misc{pith2026260801018,
  author       = {Pith},
  title        = {Pith review of: Data-Driven Domain of Attraction Estimation via Convergent Koopman-Zubov Approximation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OOJ4SNAG}},
  note         = {Machine review of arXiv:2608.01018}
}

read the original abstract

The computation of a domain of attraction (DOA) around an equilibrium point is a key issue in nonlinear stability analysis, which boils down to the difficult problem of searching for a Zubov function. With an operator-theoretical viewpoint of nonlinear systems, the concept of Zubov--Koopman operator has been introduced. However, due to the lack of convergence guarantee on the infinite-times action of Zubov--Koopman operator, the Zubov function estimate is unamenable to a theoretical bound under data-based learning errors. In this paper, considering a reproducing kernel Hilbert space (RKHS) with a linear--radial product kernel, the operator is proved to have a spectrum inside the unit circle. Hence, by augmenting this RKHS with constant-valued functions, the Zubov function that characterizes the DOA is obtained as the unique invariant element under the operator's action. This new RKHS formulation allows an efficient kernel-based estimation, which has an at most sectorially bounded error that scales down with the sample size. The proposed approach is tested with numerical examples, showing high accuracy of on-DOA/off-DOA classification of states, with two order-of-magnitude faster computation than neural networks.