{"id":"f5d5de19-885e-4352-91f1-4bf7fafaaa3a","arxiv_id":"2502.05204","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"Matching invariant measures in time-delay coordinates can uniquely identify dynamical systems from noisy, slowly-sampled data, with proof and numerical demonstrations.","lead":"This paper identifies dynamical systems by matching their long-run statistics (invariant measures) instead of matching trajectories point by point, which is more robust to noise, chaos, and slow sampling. The authors prove that matching invariant measures in time-delay coordinates makes the identified system unique up to a coordinate change, and they demonstrate the approach on chaotic and high-dimensional systems.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 5.2's uniqueness guarantee depends on a matching-orbit condition not supplied by the invariant measures, so the abstract's claim of identifiability 'from the invariant measure alone' overstates the proven result.","rationale":"The reader's weakest_assumption correctly identifies the matching-orbit condition in Theorem 5.2 as the critical point on which the paper's strongest advertised claim depends. My review of the proof of Theorem 5.2 confirms that condition 1 is not a technical convenience but is used essentially: it is what turns the per-observable conjugacies Θ_i from Theorem 5.1 into the identity at the point x*, which then, via injectivity of the vector observable Y, yields agreement of the two systems along an entire orbit, allowing Lemma A.8 to upgrade conjugacy to equality on supp(µ). The delay-coordinate invariant measures alone, even for m distinct observables, do not provide such a point; they only imply conjugacies. Therefore the abstract's phrase 'guarantee unique system identifiability from the invariant measure alone' is stronger than what Theorem 5.2 proves. The mathematical content of Theorems 5.1 and 5.2 appears coherent and I found no internal inconsistency in the derivations as stated; the issue is the mismatch between the advertised claim and the theorem's hypothesis. The reader's CONDITIONAL verdict remains appropriate: the theoretical results are likely defensible once the statement is sharpened, and the numerical support should be strengthened. No change to the verdict is warranted from this stress-test pass.","tokens_in":31293,"tokens_out":11888,"duration_ms":120283,"concrete_test":"Train two independent neural-network reconstructions of the Lorenz-63 time-τ map by minimizing the paper's J2 delay-coordinate invariant-measure loss to numerical zero, with identical architecture and data but different random initializations. Then check (i) whether the two optimized maps agree on supp(µ) and (ii) whether they share any orbit segment of length m−1. If both reach J2 ≈ 0 but differ on supp(µ) and share no such segment, then the delay-coordinate invariant measures alone do not enforce uniqueness, confirming that condition 1 of Theorem 5.2 is load-bearing and the abstract's 'from the invariant measure alone' claim is unsupported. If every optimizer achieving J2 ≈ 0 collapses to the same map, the concern is mitigated.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper advertises that time-delay invariant measures can guarantee unique system identifiability from the invariant measure alone. The strongest supporting statement, Theorem 5.2, requires condition 1: existence of x* in B_{µ,T} ∩ supp(µ) with T^k(x*) = S^k(x*) for 1 ≤ k ≤ m−1. In the proof (Appendix A.6.2), this condition is used to show each conjugacy Θ_{y_i} from Theorem 5.1 fixes x*, which then—together with injectivity of Y—forces S^k(x*) = T^k(x*) for all k, enabling Lemma A.8 to conclude equality on supp(µ). Without condition 1, the m delay-coordinate measure equalities yield only m distinct topological conjugacies S = Θ_i^{-1} T Θ_i, not equality. The invariant measures themselves encode no information about such a common finite orbit segment; it must be assumed a priori. Thus the uniqueness conclusion is not obtained 'from the invariant measure alone.' Section 5.2 admits 'provided that a suitable initial condition also holds,' but the abstract and Section 5 introduction omit this caveat. This is a real gap between the advertised central claim and the proven theorem, though the theorem as stated appears mathematically sound.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes an Eulerian, invariant-measure-based approach to dynamical system identification. The forward model is the stationary solution of a Fokker-Planck equation, approximated either by an upwind finite-volume discretization (Section 3) or by a data-adaptive Galerkin projection of the Perron-Frobenius operator with Monte Carlo integration (Section 4). The paper's main theoretical contribution is in Section 5, where delay-coordinate invariant measures are shown to provide identifiability: Theorem 5.1 proves topological conjugacy on the support from equality of one delay-coordinate invariant measure, and Theorem 5.2 claims uniqueness on the support from finitely many delay-coordinate invariant measures and a suitable initial condition. Numerical experiments cover the Van der Pol oscillator, the Lorenz-63 system, a Hall-effect thruster signal, and a 30-dimensional Lorenz-96 system.","tokens_in":31558,"tokens_out":7439,"duration_ms":73643,"significance":"If the technical gaps identified below are repaired, the paper makes a valuable contribution: the delay-coordinate invariant measure construction is a genuinely new theoretical tool for invariant-measure-based system identification, and the m+1 dimensional delay-coordinate trick in Theorem 5.1 is elegant. The Galerkin formulation with variance-reducing data-adaptive meshes is also practically motivated and potentially scalable. The proofs are largely self-contained and the paper demonstrates the method on a diverse set of problems. However, the advertised claim of unique identifiability 'from the invariant measure alone' is stronger than Theorem 5.2 actually proves, and the claimed operator-norm convergence in Theorem 4.1 is not established by the proof. The numerical experiments are currently single-run and lack reproducibility details.","major_comments":[{"comment":"The abstract and the introduction claim that delay-coordinate invariant measures guarantee unique identifiability 'from the invariant measure alone,' but Theorem 5.2's condition 1 requires the existence of x* in B_{μ,T} ∩ supp(μ) such that T^k(x*) = S^k(x*) for 1 ≤ k ≤ m−1. This matching-orbit condition presumes information about how T and S are aligned at a finite orbit segment and is not encoded in any of the invariant measures listed in condition 2. In the proof in Appendix A.6.2, this condition is what forces each conjugacy Θ_{y_i} to fix x*, and through it the m delay-coordinate measure equalities yield equality of the maps; without condition 1, those equalities yield only m distinct topological conjugacies. Section 5.2 accurately states that uniqueness holds 'provided that a suitable initial condition also holds,' but the abstract and the Section 5 opening paragraph do not carry this caveat. Please either soften the advertised claim or show that the orbit-alignment condition can be obtained from data rather than assumed a priori.","section":"Abstract and Section 5.1, Theorem 5.2"},{"comment":"Theorem 4.1 states operator-norm convergence, ∥P^{(n,ε)} − P∥_{L^1→L^1} → 0, but the proof establishes at most strong convergence. The final displayed chain in the proof of Theorem 4.1 fixes an arbitrary f with ∥f∥=1 and shows ∥P^{(n,ε)}f − Pf∥_{L^1} → 0 for each f; it never takes a supremum over the unit ball. Pointwise strong convergence of uniformly bounded Markov operators does not imply convergence in operator norm, and the projections Q^{(n)} used in the proof typically do not converge to the identity in operator norm. As written, the claimed norm convergence is not proven. Please either supply a proof of the operator-norm statement under additional assumptions on T and μ, or restate Theorem 4.1 as strong convergence and adjust the subsequent claims accordingly.","section":"Section 4.1, Theorem 4.1, and Appendix A.3"},{"comment":"The Lorenz-96 experiment is the main numerical evidence for the scalability of the unstructured-mesh method, but Eq. (27) is not fully specified. The paper does not state how M^{(ε)}(v_θ) is estimated during optimization: how many Monte-Carlo sample pairs (x, Φ^{Δt}_{v_θ}(x)) are used per objective evaluation, whether the same fixed observed samples are reused, how the flow map is integrated, or how gradients with respect to θ are computed through the soft partition-of-unity and the k-means cells. The construction of M* from the observed trajectory is also unspecified. In addition, Table 1 and Figures 5, 8, and 11 report single runs without seeds, error bars, or a code release, so the 'comprehensive numerical tests' are not reproducible and the claimed robustness to noise and slow sampling is not quantitatively established. Please provide full experimental details and at least repeated-seed statistics for the main comparisons.","section":"Section 4.3.2, Eq. (27)"}],"minor_comments":[{"comment":"There are several typos, including 'observerved' in Section 1, 'correspoinding' in Section 3.2, 'identificaiton' in Section 6, and 'Lebegue's' in Appendix A.3. The notation T|^k_{supp(μ)} in Appendix A.6.2 is also awkward and should be rewritten for clarity.","section":"Throughout"},{"comment":"The Hall-effect thruster experiment reports no quantitative error between the simulated and observed invariant measures, and the post-training rescaling of v_θ and D is not described. Please add the rescaling procedure and a quantitative comparison, even if only the Wasserstein distance used in Table 1.","section":"Section 3.3.2"},{"comment":"The loss J2 in the Lorenz-63 comparison is the sum of a state-coordinate term and a delay-coordinate term, so the experiment does not isolate the contribution of the delay-coordinate matching to the successful reconstruction. Consider also reporting results with the delay term only, or with a sweep over the relative weight of the two terms.","section":"Section 5.3, Figure 11"},{"comment":"The iterated limit 'lim_{n→∞} lim_{ε→0}' in Theorem 4.1 should be distinguished from simultaneous refinement of the discretization and regularization parameters, since the proof only treats the iterated order. Please state explicitly which mode of convergence is intended and used in later sections.","section":"Theorem 4.1 and Appendix A.3"},{"comment":"The variance-reduction analysis assumes i.i.d. samples from μ, while the numerical experiments use trajectory data. This distinction should be stated explicitly, and the paper should justify transferring the i.i.d.-based optimal partition principle to correlated trajectory samples.","section":"Section 4.2"}],"recommendation":"major_revision","confidential_remarks":"The manuscript fits the journal's scope and I see no citation or novelty concerns. The main issues are the overclaim in the abstract relative to Theorem 5.2 and the unproven operator-norm convergence in Theorem 4.1. If the authors can repair the proof or restate the theorem, and provide the missing experimental details, I would support publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The core of this paper is worth a serious look. The delay-coordinate identifiability results in Section 5 are new as far as I know: Theorem 5.1 (equality of one delay-coordinate invariant measure implies topological conjugacy on the support) and Theorem 5.2 (several such measures plus an initial condition force equality) are not in the cited literature, and the proofs in the appendix are coherent. Theorem 4.1's convergence guarantee for the regularized Galerkin projection of the Perron–Frobenius operator is also a genuine contribution. The Lorenz-63 experiment in Figure 11 gives a nice concrete demonstration that delay-coordinate matching (loss J2) succeeds where state-coordinate matching (J1) fails.\n\nThat said, the paper oversells what is proven. The abstract says the approach can \"guarantee unique system identifiability from the invariant measure alone,\" but Theorem 5.2 requires condition 1: there exists x* in the basin and on the support such that T^k(x*) = S^k(x*) for k=1,...,m-1. The proof uses that condition to align the conjugacies from each observable and to put the orbits on a common track. The invariant measures themselves encode no information about such a common finite orbit segment. The author does acknowledge this inside Section 5.2 (\"provided that a suitable initial condition also holds\"), but the abstract and the Section 5 introduction drop it. That's a real gap between the advertised claim and the theorem as stated. The theorem itself seems mathematically sound; the framing is what needs correcting.\n\nThe numerical support is promising but under-reported: no code, no seeds, no error bars, and the Lorenz-96 objective in (27) leaves open how M^(ε)(v_θ) is estimated during optimization—the mesh cells come from k-means on observed data, but the transition matrix for the parameterized flow needs its own sample base. Also, the reported success metrics in several experiments are the same objectives being minimized, so they partially validate in-sample. Those are fixable issues, not fatal flaws.\n\nThis paper deserves peer review. A good referee will want the abstract toned down, the matching-orbit condition made prominent, and the numerical protocols tightened. The theoretical results justify the referee time. I'd bring it to a reading group, though I'd pair it with a careful read of the proof of Theorem 5.2.","headline":"Genuinely new identifiability theorems, but the abstract oversells Theorem 5.2 by omitting a matching-orbit condition that the invariant measures alone don't supply.","tokens_in":32105,"tokens_out":1300,"would_cite":true,"duration_ms":14422,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["37M10","37C40","65P99"],"pacs":[],"model":"deepseek-v4-flash","headline":"Matching the long-run statistics of a dynamical system in time-delay coordinates identifies it up to topological conjugacy, and with extra observables, uniquely on the attractor.","keywords":["invariant measures","system identification","Fokker-Planck equation","Perron-Frobenius operator","time-delay embedding","Takens embedding","data-adaptive mesh","PDE-constrained optimization"],"falsifier":"Compute delay-coordinate invariant measures for two non-conjugate diffeomorphisms that share a state-coordinate invariant measure (for instance, torus rotations with different rotation parameters or a modified cat map composed with a non-conjugate twist), using a generic observable and embedding dimension above twice the box-counting dimension; the theorems imply the measures must differ, so equality would refute the identifiability claim.","tokens_in":31027,"feed_emoji":"🌀","tokens_out":8043,"duration_ms":75359,"temperature":0.7,"pith_summary":"This paper argues that dynamical systems can be identified from their long-run statistics—their invariant measures—instead of from pointwise trajectories, and that this switch repairs three chronic failures of trajectory-based identification: sensitivity to noise, chaos, and slow sampling. The forward model is the stationary solution of a Fokker–Planck equation, computed as the dominant eigenvector of a finite-volume or Galerkin Markov matrix, so the fitting problem becomes a PDE-constrained optimization. The paper's central theoretical contribution is that the inverse problem becomes well-posed when the invariant measure is lifted to time-delay coordinates: equality of delay-coordinate invariant measures forces the two systems to be topologically conjugate on the support, and finitely many such measures from distinct observables, together with a single matching orbit, force them to coincide there. A companion computational contribution makes the approach practical in high dimensions by approximating the Perron–Frobenius operator on a data-adaptive unstructured mesh with Monte–Carlo integration. If the claims hold, modelers can recover attractor dynamics from noisy, sparsely observed data without resolving individual trajectories.","feed_headline":"Delay coordinates make invariant measures identify the dynamics","feed_subtitle":"Matching long-run statistics in delay coordinates recovers a system up to a conjugacy, or exactly given a shared orbit.","key_machinery":"The time-delay map $Ψ^{{(m)}}$_{(y,T)}(x) = (y(x), y(T(x)), …, y($T^{{m−1}}$(x))) and its pushforward measure ̂$μ^{{(m)}}$_{(y,T)} = $Ψ^{{(m)}}$_{(y,T)#} μ. The argument turns on the fact that an (m+1)-dimensional delay vector determines both $Ψ^{{(m)}}$(x) and $Ψ^{{(m)}}$(T(x)), so equality of the (m+1)-dimensional delay-coordinate measures lets one read off the dynamics' graph and build the conjugacy Θ = ($Ψ^{{(m)}}$_{(y,T)})^{-1} ∘ $Ψ^{{(m)}}$_{(y,S)}. On the computational side, the forward model is the unique fixed point of a column-stochastic Markov matrix M_ε = (1−ε)M + εU—the teleportation-regularized upwind finite-volume discretization of the Fokker–Planck equation—and the high-dimensional extension replaces the uniform mesh with a data-adaptive partition and a regularized Galerkin projection of the Perron–Frobenius operator approximated by Monte–Carlo integration.","core_discovery":"The discovery is that the non-uniqueness that plagues invariant-measure-based system identification disappears under a Takens-style time-delay coordinate change. There are two theorems. Theorem 5.1: if two diffeomorphisms have the same invariant measure in (m+1)-dimensional delay coordinates, with m exceeding twice the box-counting dimension of the support, then they are topologically conjugate on the support of that measure, for almost every $C^{1}$ observable. Theorem 5.2: using m delay-coordinate invariant measures from m distinct observables, plus an initial condition in the basin of the measure whose first m−1 iterates agree for the two systems, the systems coincide on the whole support. The key observation is that a single point in (m+1)-dimensional delay coordinates encodes both the current delay-coordinate state and its image under one step of the dynamics, so the invariant measure in those coordinates carries dynamical information that the state-coordinate invariant measure discards.","pith_inferences":["The orbit-matching condition in Theorem 5.2 is not obtainable from the invariant measures alone; making the uniqueness result fully data-driven would require identifying such an initial condition from the observed time series, for example by locating a distinguished orbit in the delay embedding, which is a testable extension.","The identifiability guarantee holds for almost every observable in the sense of prevalence, so a specific fixed sensor could in principle be exceptional; one could audit a given sensor by numerically checking injectivity of its delay-coordinate map on the attractor.","The natural synthesis of the paper's two halves—applying the Section 4 data-adaptive mesh and Markov-matrix matching to delay-coordinate invariant measures—is left as future work, and testing it on the 30-dimensional Lorenz-96 example would be a direct benchmark of the combined pipeline."],"forward_implications":["System identification from invariant statistics becomes well-posed in delay coordinates: the recovered dynamics are determined up to topological conjugacy from a single delay-coordinate invariant measure, and uniquely on the support when several observables and a matching orbit are available.","The Eulerian approach inherits robustness to noise, chaos, and slow or irregular sampling, because occupation measures converge to the invariant measure under mild assumptions, whereas trajectory-matching objectives degrade under these conditions.","The data-adaptive mesh and Galerkin approximation of the Perron–Frobenius operator extend invariant-measure matching to high-dimensional systems with low-dimensional attractors, making it feasible to compare full Markov matrices rather than only dominant eigenvectors.","Because delay-coordinate invariant measures can be formed from scalar time series, the identifiability guarantees apply in partial-observation settings where only one or a few functions of the state are measured."],"supporting_citations":[{"why":"Supplies the PDE-constrained optimization formulation, the Fokker–Planck forward model with teleportation regularization, and the adjoint-state gradient computation that Section 3 builds on.","marker":"[11]"},{"why":"Provides the fractal Takens embedding theorem and the Whitney embedding generalization that the delay-coordinate identifiability theorems extend, including the embedding dimension condition m > 2d and the periodic-point assumptions.","marker":"[13]"},{"why":"The origin of time-delay embedding, which motivates the delay-coordinate invariant measures as the coordinate change used in Theorems 5.1 and 5.2.","marker":"[19]"},{"why":"Ulam's Galerkin projection of the Perron–Frobenius operator, whose regularized, data-adaptive extension is developed and proven convergent in Section 4.","marker":"[17]"},{"why":"The SINDy baseline representing trajectory-based identification, against which the invariant-measure approach is compared under slow sampling in Section 3.3.","marker":"[8]"},{"why":"The data-driven model calibration approach that first uses invariant measures and occupation measures for system identification, including the Hall-effect thruster application used here.","marker":"[2]"},{"why":"Provides the prevalence notion of 'almost every observable' on which Theorems 5.1 and 5.2 rely for their genericity statements.","marker":"[28]"},{"why":"The finite volume method reference for the upwind Fokker–Planck discretization used in the forward model of Section 3.","marker":"[38]"}],"fun_headline_variants":["Delay coordinates make invariant measures uniquely identify dynamics","Time-delay embedding solves invariant-measure identifiability","Invariant measures plus delay coordinates pin down the system","Delay embedding converts statistics into dynamical fingerprints"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The uniqueness result depends on knowing an initial condition whose future visits agree for the two systems, and that knowledge is not contained in the invariant measures being matched.","fun_headline_variants_meta":{"raw":{"variants":["Delay coordinates make invariant measures uniquely identify dynamics","Time-delay embedding solves invariant-measure identifiability","Invariant measures plus delay coordinates pin down the system","Delay embedding converts statistics into dynamical fingerprints"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00033,"raw_usage":{"total_tokens":1872,"prompt_tokens":1014,"completion_tokens":858,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":630,"completion_tokens_details":{"reasoning_tokens":799}},"tokens_in":630,"tokens_out":858,"duration_ms":8449,"temperature":1.0,"reasoning_tokens":799,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-09T19:45:16.448366+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute delay-coordinate invariant measures for two non-conjugate diffeomorphisms that share a state-coordinate invariant measure (for instance, torus rotations with different rotation parameters or a modified cat map composed with a non-conjugate twist), using a generic observable and embedding dimension above twice the box-counting dimension; the theorems imply the measures must differ, so equality would refute the identifiability claim.","supporting_citations":[{"cited_title":"Optimal transport for parameter identification of chaotic dynamics via invariant measures","cited_arxiv_id":null,"evidence_quote":"Supplies the PDE-constrained optimization formulation, the Fokker–Planck forward model with teleportation regularization, and the adjoint-state gradient computation that Section 3 builds on."},{"cited_title":"Embedology","cited_arxiv_id":null,"evidence_quote":"Provides the fractal Takens embedding theorem and the Whitney embedding generalization that the delay-coordinate identifiability theorems extend, including the embedding dimension condition m > 2d and the periodic-point assumptions."},{"cited_title":"Detecting strange attractors in turbulence","cited_arxiv_id":null,"evidence_quote":"The origin of time-delay embedding, which motivates the delay-coordinate invariant measures as the coordinate change used in Theorems 5.1 and 5.2."},{"cited_title":"Finite approximation for the frobenius-perron operator","cited_arxiv_id":null,"evidence_quote":"Ulam's Galerkin projection of the Perron–Frobenius operator, whose regularized, data-adaptive extension is developed and proven convergent in Section 4."},{"cited_title":"A data-driven approach to model calibration for nonlinear dynamical systems","cited_arxiv_id":null,"evidence_quote":"The data-driven model calibration approach that first uses invariant measures and occupation measures for system identification, including the Hall-effect thruster application used here."},{"cited_title":"almost ev- ery","cited_arxiv_id":null,"evidence_quote":"Provides the prevalence notion of 'almost every observable' on which Theorems 5.1 and 5.2 rely for their genericity statements."},{"cited_title":"Finite volume methods for hyperbolic problems, volume 31","cited_arxiv_id":null,"evidence_quote":"The finite volume method reference for the upwind Fokker–Planck discretization used in the forward model of Section 3."}],"review_version":1}