arXiv:2509.20596v2

Data-Driven State Observers for Measure-Preserving Systems

Wentao Tang

eess.SYmath.DS37A0537C0537C4047A1047B2093B2893B53

Paper metadata

Original submission: September 24, 2025
Latest revision: August 12, 2026 · version 2
Claimed MSC: 37A05, 37C05, 37C40, 47A10, 47B20, 93B28, 93B53
arXiv categories: eess.SY, math.DS
Paper license: CC BY-NC-SA 4.0 ↗
Comments: 30 pages, 11 figures, submitted to IEEE Transactions on Cybernetics on August 11, 2026. This work has been submitted to the IEEE for possible publication. Copyright may be transferred without notice, after which this version may no longer be accessible

Abstract

The use of data-driven control strategies on systems with not fully measurable states induces the problem of learning-based state observation. Motivated by this need, the present work proposes a data-driven approach for the synthesis of state observers for discrete-time nonlinear systems with measure-preserving dynamics. To this end, Kazantzis--Kravaris/Luenburger (KKL) observers are shown to be well-defined, where the observer design boils down to determining a nonlinear injective mapping of states and its pseudo-inverse. For its learning-based construction, the KKL observer is related to the Koopman operator, well-defined on the square-integrable function space and restrictable to a Sobolev-type reproducing kernel Hilbert space (RKHS). Hence, observer synthesis algorithms, based on kernel interpolation/regression routines for the desired injective mapping in the observer and its pseudo-inverse, are proposed in various settings of the available dataset -- (i) many orbits, (ii) single long orbit, and (iii) snapshots. Theoretical error analyses are provided, and numerical studies on a chaotic Lorenz system are demonstrated.

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Audit summary

Audited against arXiv v2

Not a correctness certificate. A “Correct” result may include yellow typos or minor formal corrections that do not affect substantive soundness. It means this audit found no unresolved substantive error under the stated criteria; it does not replace expert scrutiny or formal verification.

Current report

Detailed mathematical audit

Generated August 18, 2026

01Statements4 reported findingsContains wrong statements

The claimed injectivity of every deep KKL map under uniform backward distinguishability is false, as is the resulting observer representation with a left inverse. An explicit smooth measure-preserving interval system satisfies all stated assumptions but makes the one-component KKL map noninjective. The three learning guarantees are not verified because their proofs do not establish the printed uniform conclusions.

Theorem 1Incorrect

Uniform backward distinguishability does not force the deep KKL map to be injective

Page 10 · Theorem 1 · arXiv:2509.20596v2

Take $\mathbb X=[-1,1]$ , $f(x)=-x$ , $h(x)=e^x$ , observer order $m=1$ , and $\beta=1/2$ . The map $f$ is invertible with continuous inverse, $h$ is bounded, and Assumption 4 holds because $h(f^{-1}(x))=e^{-x}$ is injective. Yet the series in Definition 5 gives $\zeta_0(\beta,x)=\frac{e^{-x}+\beta e^x}{1+\beta}.$ If $x_0=(\log 2)/2$ and $x_\pm=x_0\pm1/4$ , then $x_-\ne x_+$ while $\zeta_0(1/2,x_-)=\zeta_0(1/2,x_+)$ . Thus every hypothesis of Theorem 1 holds and its conclusion fails. Repair classification: No repair supplied; stronger assumptions controlling cancellation among the delayed observations are necessary.

Theorem 2Incorrect

The asserted observer with a left pseudo-inverse need not exist

Pages 10–11 · Theorem 2 and equations (13)–(14) · arXiv:2509.20596v2

The same counterexample satisfies Assumption 1, with $|\det Df|=1$ ; Assumption 2, using normalized Lebesgue measure; Assumption 3 for every finite Sobolev order; and Assumption 4. For $\beta=1/2$ , however, $\zeta$ is not injective, so no left pseudo-inverse $\zeta^\dagger$ exists on $\zeta(\mathbb X)$ . Consequently the displayed state reconstruction in (13) is false under the stated assumptions. The Koopman-series identity by itself does not restore injectivity.

Theorems 3 and 4Not able to verify

The uniform KRR error bounds are not established

Pages 14–17 · Theorems 3 and 4 · arXiv:2509.20596v2

These theorems add hypotheses under which a Lipschitz extension of $\zeta^\dagger$ is assumed to exist, so the counterexample to Theorem 1 alone does not disprove their restricted implications. Their proofs nevertheless leave two nontrivial obligations unresolved: the cited KRR theorem applies to the marginal distribution of the actual noisy inputs $\widetilde\zeta(x)$ , whereas condition (iv) is imposed on the integral operator for $\nu_z=\zeta_\sharp\nu$ ; and the derived sup-norm estimate is stated only on $\widetilde\zeta(\mathbb X)$ , while (20) evaluates $\widehat\zeta^\dagger$ at $\zeta(x)$ . Lipschitz continuity is assumed for $\zeta^\dagger$ , not for the learned map, so proximity of these two input sets does not close the latter step. No independent proof of (20) was supplied.

Theorem 5Not able to verify

The snapshot guarantee is not established

Pages 18–20 · Lemmas 3–4 and Theorem 5 · arXiv:2509.20596v2

The spectral construction used to justify Algorithm 3 contains false intermediate claims. In Lemma 3, the printed radius is an infimum over a set containing $\lambda$ and is therefore zero; moreover, an $L^2$ spectral projection need not preserve $H^s$ . Lemma 4 assumes that every fine arc supplies a small-residual vector and calls a Gram matrix unitary, neither of which follows. For the identity Koopman operator, arcs away from $1$ have no spectral mass and every normalized vector has residual $|1-\lambda|$ , directly contradicting the asserted mesh-scale construction. The theorem also inherits the unresolved KRR transfer in Theorem 4, so its conclusion remains unverified rather than proved false.

02Proofs4 reported findingsContains incorrect or incomplete proofs

The injectivity proof is contradicted by an explicit example and reverses the conclusion of its own empty-set assertion. The learning proofs misapply the cited KRR estimate to a different input distribution and do not transfer a bound from the approximate-input set to the exact-input set. The snapshot proof additionally relies on false spectral lemmas.

Proof of Theorem 1Incorrect as written

Finite derivative equalities at $\beta$ are incorrectly transported to $\beta=0$

Page 10 · proof of Theorem 1 · arXiv:2509.20596v2

Backward distinguishability identifies derivatives of the holomorphic difference at $\beta=0$ . The proof instead assumes that the first $m$ derivatives vanish at an arbitrary fixed $\beta$ and then says that evaluation at $\beta=0$ gives the same information. A finite jet at one point does not determine a holomorphic function's finite jet at another point. The sets $\mathbb K_\ell$ therefore are not shown to be empty; the final paragraph also literally changes $\mathbb K_m=\varnothing$ into the existence of a distinct pair. The explicit interval counterexample in Part 1 shows that this is not a repairable omission under the printed hypotheses.

Proofs of Theorems 3 and 4Incomplete as written

The cited learning estimate is applied to the wrong input law and on the wrong evaluation set

Pages 14–17 · proofs of Theorems 3 and 4 · arXiv:2509.20596v2

Smale–Zhou's estimate is formulated for independent pairs drawn from one fixed joint law and its integral operator is defined using that law's input marginal. Here the KRR inputs are $\widetilde\zeta(x)$ , but condition (iv) uses the different marginal $\zeta_\sharp\nu$ . No comparison of the two integral operators or their ranges is proved. Even granting an estimate on $\widetilde\zeta(\mathbb X)$ , the last triangle argument controls $\widehat\zeta^\dagger(z)$ for a nearby $z\in\widetilde\zeta(\mathbb X)$ , not the printed quantity $\widehat\zeta^\dagger(\zeta(x))$ ; no modulus of continuity for the learned function is provided. Repair classification: No repair supplied.

Smale–Zhou, learning-theory sampling framework ↗

Lemmas 3 and 4Incorrect as written

The approximate-eigenfunction spanning argument is false

Pages 18–20 · Lemmas 3 and 4 · arXiv:2509.20596v2

Because $\lambda\in\mathbb J$ , the radius $\inf_{\lambda'\in\mathbb J}|\lambda'-\lambda|$ in Lemma 3 equals zero, which would assert an $H^s$ eigenfunction at every spectral point; continuous spectrum need not have eigenfunctions. Replacing infimum by supremum does not repair the proof, since $E'(\mathbb J)g$ need not lie in $H^s$ . Lemma 4 then assigns mesh-scale residuals to every equally spaced grid point even if its arc has zero spectral projection. The identity operator gives a direct obstruction: away from $\lambda=1$ , the residual is exactly $|1-\lambda|$ for every unit vector. The subsequent assertion that the Gram matrix is unitary is also false. Repair classification: No repair supplied; a new finite-dimensional spectral approximation theorem with explicit support and regularity hypotheses is required.

Equation (19)Typo

The displayed KRR coefficient formula contains an extra vector factor

Page 14 · equation (19) · arXiv:2509.20596v2

The printed matrix $G_\kappa c_k+\alpha I$ is dimensionally undefined. Differentiating the objective displayed immediately above gives $\widehat c_k=(G_\kappa+(\alpha/2)I)^{-1}x_k$ . The intended correction is mechanically determined and does not affect the substantive audit findings.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper: arXiv:2509.20596v2
Authors listed: Wentao Tang
Audit date: August 18, 2026

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You are a mathematical auditor. Evaluate only the correctness of the paper's central statements, the correctness and completeness of its proofs, and evidence-backed non-novelty. Do not make editorial comments or evaluate importance, exposition, style, or suitability for publication. Keep the report neutral, precise, and professional. PART 1 — CORRECTNESS OF THE MAIN STATEMENTS Identify the paper's central statements and assess each one separately using exactly one of these outcomes: - Correct: the statement has a correct, verifiable proof; or, despite a defect in the printed proof, you can give a fully verified repair or independent proof. - Incorrect: you can give strong formal evidence that the statement as written is false, preferably a counterexample or a complete contradiction. - Not able to verify: the supplied proof does not establish the statement, and you can neither supply a fully verified proof nor disprove it. 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For every defect, identify its smallest exact location, explain its downstream dependency, and classify any proposed repair as one of: - Verified repair: supply enough reasoning to verify that the repair works throughout every affected use. - Plausible repair only: describe the possible repair but do not treat it as established. - No repair supplied. Assign the overall proofs status by the following priority: 1. Contains incorrect or incomplete proofs, if at least one proof is Incorrect as written or Incomplete as written. 2. Contains unverified proofs, if there is no proved incorrectness or incompleteness but at least one material proof is Not able to verify. 3. Correct, if all audited proofs are Correct and complete, apart from any typographical or minor formal findings allowed by Part 4. PART 3 — NOVELTY OF THE MAIN STATEMENTS This is a one-sided, evidence-only check. Evaluate only central results. 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If the list is empty, the section may say only “No non-novelty findings.” PART 4 — TECHNICAL AND EVIDENTIARY RULES 1. Criticism must never be overstated. Call a statement incorrect, a proof incorrect or incomplete, or a result non-new only when strong formal evidence supports that exact judgment. Otherwise use Not able to verify where that outcome exists, or remain silent in the novelty section. 2. Review a specified paper version. Give exact page, theorem, equation, or proof locations; restate relevant quantifiers and hypotheses; and check domains, endpoints, parameter ranges, constants, and dependencies explicitly. Apply a defect only to the regimes and downstream results that actually depend on it. 3. Immediate-consequence closure: when deciding whether a conclusion is supported, treat the argument as including every immediate formal consequence of facts established earlier in the paper and available at that point. An immediate formal consequence is a local, mechanically checkable inference using only definitions already in force, elementary logic, direct substitution or renaming, weakening of an inequality or set inclusion, or a monotonicity or symmetry property explicit in a definition or already proved. 4. Such an inference must introduce no new lemma, estimate, construction, case analysis, limiting or compactness argument, external result, or new issue of uniformity or parameter dependence. Check the direction of monotonicity and the dependence of every threshold or constant explicitly. Do not exchange pointwise and uniform conclusions, change exceptional sets, interchange quantifiers or limits, or extend an endpoint under this rule unless the required fact was already established. 5. If the claimed conclusion follows after immediate-consequence closure, it is fully supported. 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P(delta)

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t>0

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Data-Driven State Observers for Measure-Preserving Systems

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

Uniform backward distinguishability does not force the deep KKL map to be injective

The asserted observer with a left pseudo-inverse need not exist

The uniform KRR error bounds are not established

The snapshot guarantee is not established

Finite derivative equalities at β\betaβ are incorrectly transported to β=0\beta=0β=0

The cited learning estimate is applied to the wrong input law and on the wrong evaluation set

The approximate-eigenfunction spanning argument is false

The displayed KRR coefficient formula contains an extra vector factor

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Finite derivative equalities at $\beta$ are incorrectly transported to $\beta=0$