{"id":"2c5f215d-1ac3-464c-b7c7-9446c93d796b","arxiv_id":"2608.01018","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"high","formal_verification":"none","parameter_count":3,"one_line_summary":"A new RKHS construction gives the Zubov-Koopman operator a spectrum inside the unit disk, so the Zubov function can be estimated from data with a sectorially bounded error.","lead":"This paper redefines the Zubov-Koopman operator on a specially chosen function space so that its infinite-time action yields the domain of attraction's Zubov function. It provides a kernel-based learning algorithm with a provable error bound that shrinks as sample size grows, demonstrated on three nonlinear systems.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 4.1's error bound is unsupported: r(Ẑ)<1 is inferred from a noncommuting contraction, and the summability of ‖Ẑ^m‖ is not uniform in sample size.","rationale":"The reader's identified weak point is the global linearizability assumption, but on close reading Assumption 3.1 only requires a forward-invariant subset O, which may be a small neighborhood; local Sternberg linearization plausibly holds for all three examples (nonresonant hyperbolic equilibria). Thus that assumption is less obviously load-bearing. The more serious gap is in the proof of the statistical guarantee, which is the paper's main advertised contribution. The proof's step r(Ẑ)=r(Z^*T)≤r(Z^*) relies on a false inequality for noncommuting contractions; the sampling operator T=(S+λ/n)^{-1}S does not commute with Z^*. Moreover, the proof needs a uniform bound on Σ‖Ẑ^m‖ but only asserts finiteness for each fixed n. These are internal gaps, not scope limitations, and they affect the strongest claim in the abstract (sectorially bounded error scaling with sample size). A numerical test can settle whether the empirical spectral radius stays uniformly away from 1; if not, the theorem's proof fails. If the test passes, the concern becomes a request for a rigorous proof of the commuting/uniform bound rather than a rejection.","tokens_in":29653,"tokens_out":35365,"duration_ms":386480,"concrete_test":"Implement (8)-(9) for Example 2 with n=25²,50²,75²,100²,150² and for each n compute the spectral radius ρ(Ẑ) (equivalently, the eigenvalue of largest modulus of ZΩ or the conditioning of Φ+λI−Ω) and the partial sums S_M=Σ_{m=0}^M ‖Ẑ^m‖_{H̊→H̊} for M=10⁴. If ρ(Ẑ) approaches 1 or S_M grows faster than n^{p} with p < (s−d/2)/d, then Theorem 4.1's bound is not valid as stated; if ρ(Ẑ)≤ρ0<1 uniformly and S_M stays bounded, the gap is likely fixable.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central data-driven guarantee (Theorem 4.1) does not follow from the proof as written. The proof bounds |ζ̂(x)−ζ(x)| by (Σ_l ‖Z^l‖_{C̊→C̊})(Σ_m ‖Ẑ^m‖_{H̊→H̊}) ‖Ẑ−Z‖_{H̊→C̊}‖η‖. The second factor is asserted finite because r(Ẑ)<1, which is justified only by Ẑ=TZ with T=(S+λ/n)^{-1}S a contraction, and the claim r(Ẑ)≤r(Z). This is false in general: for noncommuting bounded operators, r(TZ) can exceed r(Z), and T does not commute with Z here (T is a spectral filter of the sampling operator S). Even if r(Ẑ)<1, the sum Σ_m ‖Ẑ^m‖ can be arbitrarily large when the spectral radius is close to 1; the proof provides no uniform-in-n bound, but such a bound is required to make the prefactor scale as d^{s−d/2}. Without it, the advertised sectorial error rate is not established, nor is the well-definedness of the Neumann series in (9) guaranteed. This is separate from Assumption 3.1; even for systems where linearizability holds, the empirical spectral-radius step fails.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":29999,"tokens_out":13079,"duration_ms":154669,"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":[{"comment":"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.","section":"Section 3.2, Assumption 3.1 and Theorem 3.2"},{"comment":"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":"Lemma 3.1"},{"comment":"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":"Section 4.3, Theorem 4.1"},{"comment":"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.","section":"Section 4.2, Corollary 4.1 and Proposition 4.3"}],"minor_comments":[{"comment":"Typo: 'infinite-dimensinoal' should be 'infinite-dimensional'.","section":"Introduction"},{"comment":"'The reminder of this paper' should be 'The remainder of this paper'.","section":"Organization and Notations"},{"comment":"The Gram matrix is defined with indices i,j=1,...,N after the sum uses n; please make the notation consistent.","section":"Definition 2.2"},{"comment":"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.","section":"Remark 2.1"},{"comment":"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":"Remark 4.2"},{"comment":"'outforms' should be 'outperforms'.","section":"Section 5.2"}],"recommendation":"major_revision","confidential_remarks":"The paper relies heavily on the author's own prior work for the linear–Sobolev space equivalence and related boundedness results; this is acceptable if those papers are available, but the referee may wish to confirm that the cited results actually contain the needed statements in the form used here. The central data-driven guarantee is not supported as written, so the manuscript needs substantive revision rather than minor polishing."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nWhat 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.\n\nThe 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.\n\nAssumption 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.\n\nOne 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.\n\nThe 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.\n\nWho 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.","headline":"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.","tokens_in":30454,"tokens_out":11002,"would_cite":false,"duration_ms":129378,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["47B32","93D30"],"pacs":[],"model":"deepseek-v4-flash","headline":"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}|","keywords":["domain of attraction","Zubov function","Zubov–Koopman operator","reproducing kernel Hilbert space","linear–radial kernel","spectral radius","kernel regression","data-driven estimation"],"falsifier":"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","tokens_in":29523,"feed_emoji":"🎯","tokens_out":16234,"duration_ms":155050,"temperature":0.7,"pith_summary":"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.","feed_headline":"One eigenvalue marks the whole basin of attraction","feed_subtitle":"A convex kernel method recovers the basin of attraction with an error bound that shrinks with sample size.","key_machinery":"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","core_discovery":"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","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Introduces the Zubov–Koopman operator and its data-driven learning; this paper redefines the operator on an RKHS to repair its missing spectral guarantees.","marker":"[22]"},{"why":"Establishes that Sobolev–Hilbert spaces are RKHSs with radial Sobolev kernels, the foundation for the linear–Sobolev construction and for scattered-data error bounds.","marker":"[23]"},{"why":"Supplies the smoothness and boundedness conditions under which Koopman operators act on Sobolev-type RKHSs, conditions the strong-continuity theorem extends to the weighted Zubov case.","marker":"[30]"},{"why":"Introduces the linear–radial product kernel and the linear–Sobolev space, the exact function space on which the new spectral theorem is proved.","marker":"[32]"},{"why":"Proves the Koopman semigroup is strongly continuous on the linear–Sobolev space, a step reused to show strong continuity of the Zubov–Koopman semigroup.","marker":"[43]"},{"why":"Provides the uniform kernel-regression error bound in terms of fill distance and smoothness that Theorem 4.1 converts into the sectorial Zubov-function error.","marker":"[44]"},{"why":"Gives the Zubov-PDE/viscosity characterization and exponential transform from maximal Lyapunov functions used to identify the invariant element as the Zubov function.","marker":"[21]"}],"fun_headline_variants":["Zubov-Koopman spectrum: one eigenvalue reveals the basin","A single eigenfunction pins down the domain of attraction","Kernel method zooms into the basin of attraction","Spectral trick shrinks error as data grows","RKHS operator captures the whole basin in one eigenfunction"],"cache_read_input_tokens":2816,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Zubov-Koopman spectrum: one eigenvalue reveals the basin","A single eigenfunction pins down the domain of attraction","Kernel method zooms into the basin of attraction","Spectral trick shrinks error as data grows","RKHS operator captures the whole basin in one eigenfunction"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000165,"raw_usage":{"total_tokens":1117,"prompt_tokens":801,"completion_tokens":316,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":545,"completion_tokens_details":{"reasoning_tokens":237}},"tokens_in":545,"tokens_out":316,"duration_ms":4212,"temperature":1.0,"reasoning_tokens":237,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T00:36:45.889646+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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","supporting_citations":[{"cited_title":"Learning regions of attraction in unknown dynamical systems via Zubov–Koopman lifting: Regularities and convergence,","cited_arxiv_id":null,"evidence_quote":"Introduces the Zubov–Koopman operator and its data-driven learning; this paper redefines the operator on an RKHS to repair its missing spectral guarantees."},{"cited_title":"𝐿 ∞-error bounds for approxi- mations of the Koopman operator by kernel extended dynamic mode decomposition,","cited_arxiv_id":null,"evidence_quote":"Supplies the smoothness and boundedness conditions under which Koopman operators act on Sobolev-type RKHSs, conditions the strong-continuity theorem extends to the weighted Zubov case."},{"cited_title":"Approximate interpolation with applications to selecting smoothing parameters,","cited_arxiv_id":null,"evidence_quote":"Provides the uniform kernel-regression error bound in terms of fill distance and smoothness that Theorem 4.1 converts into the sectorial Zubov-function error."},{"cited_title":"Physics-informed neural network Lyapunov functions: PDE characterization, learning, and verification,","cited_arxiv_id":null,"evidence_quote":"Gives the Zubov-PDE/viscosity characterization and exponential transform from maximal Lyapunov functions used to identify the invariant element as the Zubov function."}],"review_version":1}