{"id":"2d16106b-8bb5-4f9d-99e8-ed863c65f85a","arxiv_id":"2509.09891","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"The authors define Koopman and Perron-Frobenius operators for decoupled McKean-Vlasov SDEs and prove EDMD converges to the projected operators.","lead":"Mean-field SDEs describe huge groups of interacting particles; this paper computes their spectral properties from simulation data. The key trick is to freeze the law of the process so the associated transfer operators stay linear, then approximate them with EDMD.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Convergence to K_T is proven, but K_T is a time-inhomogeneous, µ0-dependent operator with no semigroup; eigenvalues near 1 are not shown to indicate metastability of the mean-field system.","rationale":"The reader's weakest assumption precisely names the gap I find most load-bearing. Theorem 4.7 is a convergence statement for a fixed finite-time operator K_T associated with a time-inhomogeneous decoupled process. Even if every assumption holds and the EDMD matrix converges, the paper's central advertised application—identifying slowly evolving patterns and metastable sets from eigenvalues near one—requires a spectral theory that connects K_T to the long-time behavior of the nonlinear McKean–Vlasov equation. No such theorem is supplied; §3.5 merely asserts the connection by analogy with autonomous Markov processes. The µ0-dependence of K_T makes this especially concrete: changing the initial law of the MV equation changes the operator being approximated, so any 'timescale' read off from the spectrum is not an intrinsic property of the model. The proposed numerical test directly probes this dependence and the semigroup failure. I do not see a more fundamental flaw in the convergence argument itself; the secondary mismatch between Assumption 3.3 (Lipschitz basis) and the indicator dictionaries used in some examples is a repairable gap, not the central issue. Hence the reader's CONDITIONAL verdict remains appropriate: core convergence is plausible, but the claimed dynamical interpretation needs either a stationarity/slow-variation assumption and proof, or a numerical validation showing µ0-independence of the relevant eigenvalues.","tokens_in":26649,"tokens_out":8088,"duration_ms":101233,"concrete_test":"For the Kuramoto-on-circle example, fix lag T=0.5 and the monomial basis. Estimate \\hat K_T from EDMD with (i) ξ~Unif[0,2π] as in the paper and (ii) ξ drawn from the known invariant density ρ∝e^{-cos(2x)}. If the second eigenvalue differs materially, K_T depends on µ0 and the eigen-spectrum cannot define a stationary metastable timescale. Additionally, compare the spectrum of the empirical K_T squared with a directly estimated \\hat K_{2T}; if they differ beyond sampling error, the semigroup property required for a timescale interpretation fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The core theorem (Thm 4.7) establishes convergence of the EDMD matrix to (C_N+E(h))G_N^{-1}, i.e., to the Galerkin projection of the finite-time Koopman operator K_T for the decoupled process (3). This is a legitimate approximation statement. The load-bearing gap is the spectral interpretation in §3.5. For the decoupled SDE, b and σ depend on t through µ_t, so the family {K_T}_{T>0} is not a semigroup: K_{s+t} ≠ K_s K_t in general. Moreover, K_T itself depends on the initial law µ0 of the reference McKean–Vlasov equation, through the transition kernel p(µ0,0,T,x,y) in (8) and through µ_t in (3). Therefore the eigenvalues of K_T are properties of a finite-time, initial-condition-dependent operator. The metastability/timescale vocabulary borrowed from autonomous transfer operator theory (e.g., [27,50]) does not apply without an additional argument: no theorem in the paper connects the spectrum of a single K_T to slow timescales or metastable sets of the nonlinear McKean–Vlasov dynamics. The numerical experiments (e.g., §5.2) start from uniform initial data rather than an invariant measure, so the reported second eigenvalue could reflect transient relaxation rather than a genuine slow metastable transition. This is not an internal inconsistency in the convergence proof; it is an unproven extrapolation in the paper's advertised claim.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops transfer-operator methodology for McKean–Vlasov SDEs. Since the McKean–Vlasov equation is nonlinear, the authors work with the decoupled SDE (3), in which the law µ_t is treated as an externally prescribed time-dependent coefficient. They define Koopman and Perron–Frobenius operators for this decoupled process, form Galerkin projections onto a finite dictionary with respect to the initial law µ0, and estimate the projected matrices by Monte Carlo EDMD. The central theoretical result (Theorem 4.7) asserts almost-sure convergence of the data-driven EDMD matrix to (C_N + E(h))G_N^{-1} as the number of samples M tends to infinity, and then to the exact projected Koopman matrix C_N G_N^{-1} as the time step h tends to zero. The paper also presents numerical experiments for the Cormier model, the Kuramoto model on the circle, and a Kuramoto model on the sphere, interpreting eigenvalues close to one as evidence of metastability.","tokens_in":27020,"tokens_out":10298,"duration_ms":105869,"significance":"If the results are taken as stated, the paper provides a rigorous convergence guarantee for EDMD in a mean-field SDE setting, extending a well-established data-driven framework to a class of nonlinear, measure-dependent dynamics. The proof strategy is largely sound: the Gram and structure matrices are Monte Carlo estimates of well-defined expectations, and no parameter is fit to reproduce a target spectrum. The numerical examples illustrate the method on nontrivial benchmark models. However, the advertised spectral interpretation in terms of metastability and slow timescales is not established for the time-inhomogeneous decoupled operator, and the definition of the transfer operators contains a measure-theoretic inconsistency. These issues concern the main interpretive claim of the paper, so the contribution, while promising, is not yet fully supported.","major_comments":[{"comment":"The transition density p is introduced in §2.2 as a spatial probability density (see Eq. (4), where q(µ0,0,t,z)=∫ p(µ0,0,t,x,z) µ0(dx)). But Eq. (8) defines K_T f(x)=∫ p(µ0,0,T,x,y) f(y) µ0(dy), and the proof of Lemma 3.1 uses ∫ p(µ0,0,T,x,y) µ0(dy)=1. These identities are incompatible with p being a density with respect to Lebesgue measure unless µ0 is Lebesgue measure. Thus the definitions of K_T and P_T as written are not well-defined for a general initial law µ0. The Monte Carlo target C_N in Eq. (13) uses the correct expectation E[ψ_i(X_T^{ξ,µ})ψ_j(ξ)], so the convergence theorem can be salvaged, but the operator definitions and the duality statement must be rewritten consistently (for example, by using Lebesgue measure as the reference and deriving the appropriate duality, or by explicitly defining p as a density with respect to µ0 and adjusting Eq. (4)).","section":"§3.1, Eq. (8); §3.2, Lemma 3.1"},{"comment":"The spectral/metastability interpretation is an unproven extrapolation. For the decoupled SDE (3), the coefficients depend on t through µ_t, so the family {K_T}_{T>0} is not a semigroup, and K_T itself depends on the initial law µ0. The standard results cited (e.g., [27,50]) concern autonomous Markov semigroups. No theorem in this paper connects the eigenvalues of a single finite-time K_T to slow timescales or metastable sets of the nonlinear McKean–Vlasov system. The experiments in §5.2 start from uniform initial data rather than an invariant measure, so the reported second eigenvalue could reflect transient relaxation rather than a genuine metastable transition. The central advertised claim—detection of metastable sets—therefore requires either a proof under additional assumptions or a substantial caveat limiting the claim to the auxiliary decoupled process.","section":"§3.5 and §5"}],"minor_comments":[{"comment":"In Eq. (2), the notation 'X_t = ξ' should read 'X_0 = ξ'.","section":"§2.1, Eq. (2)"},{"comment":"The notation 'µ=Law(X_t)' in Eq. (3) is ambiguous and appears circular. The superscript µ is used as the fixed law flow of the McKean–Vlasov SDE (2), not the law of the decoupled process. This should be clarified.","section":"§2.2, Eq. (3)"},{"comment":"The statement that 'both Koopman and Perron–Frobenius operators can be extended or restricted to be defined on L2(X,µ0)' is asserted without justification. For a non-stationary process and arbitrary f∈L2(µ0), the expectation E[f(X_T^x)] need not be finite or well-defined µ0-a.e. For the bounded dictionary used here this is harmless, but the extension claim should be formulated more carefully or omitted.","section":"§3.4"},{"comment":"The L2(Ω) convergence of \\widehat G_N^{-1} to G_N^{-1} is asserted in the proof of Theorem 4.6 but not proved in Lemma 4.3. It follows from continuity of inversion on the set of invertible matrices, but this step should be stated explicitly.","section":"§4.2, Lemma 4.3"},{"comment":"The sentence 'Since all except the first two eigenvalues are small, there is only one form of metastability' is imprecise: the number of metastable sets is not determined solely by the number of eigenvalues close to one without additional assumptions about the operator and the system.","section":"§5.2"},{"comment":"There are several typos: 'Mckean' in the first paragraph of Section 4; 'X_t = ξ' in Eq. (2); and in Assumption 3.3(a) the bound involving γ_N appears to contain a typo (it should likely be |(ψ_1(x),...,ψ_N(x))|^2 < γ_N or a matching definition).","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The paper is within the journal's scope and the convergence theorem is a useful contribution. The main risk is the unproven spectral interpretation: as written, the paper claims metastability detection for mean-field systems, but the object whose spectrum is computed is a time-inhomogeneous, initial-law-dependent operator without a semigroup structure. This is not a fatal flaw in the convergence proof, and the paper can likely be made publishable by either proving a metastability statement under appropriate assumptions or reframing the contribution as the approximation of auxiliary transfer operators without the metastability claim. The definitional inconsistency in Eq. (8) should also be fixed before publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Read this for a careful convergence theorem for EDMD applied to decoupled McKean-Vlasov SDEs, not for a proven theory of metastability in mean-field systems.\n\nThe genuinely new piece is the combination: freezing the law via the decoupling device, defining Koopman and Perron-Frobenius operators for the resulting time-inhomogeneous SDE, and proving L2 and almost-sure convergence of the data-driven Galerkin matrices to the projected Koopman operator. The law-estimation step is handled honestly, including the remark that IPS data only gives L2 convergence because the particle trajectories are dependent. The proofs are mostly adaptations of the authors' earlier work [49], but the adaptation is careful and the Wasserstein error bounds on the estimated law are cleanly integrated. That part deserves credit.\n\nThe soft spot is exactly where the reader's stress-test lands, and it is load-bearing. Section 3.5 asserts that eigenvalues close to one indicate slow timescales and metastable sets, but the operator whose spectrum is computed is K_T for a decoupled SDE with coefficients depending on t through mu_t. This is not a semigroup, and K_T itself depends on the initial law mu0. No theorem connects the spectrum of a single finite-time operator to metastability of the nonlinear McKean-Vlasov equation. The numerical experiments start from uniform initial data, so the reported second eigenvalue could reflect transient relaxation rather than a genuine slow transition. This is not an internal inconsistency in the convergence proof; it is an unproven extrapolation in the advertised claim. The paper could be repaired by restricting to the stationary case, adding a perturbation result for slowly varying mu_t, or toning down the metastability language until a connection is established.\n\nMinor but real: the numerics have no error bars, no convergence checks, and no code or data deposit. That makes the empirical claims harder to verify, though the examples are simple enough that the qualitative conclusions are plausible.\n\nWho this is for: researchers working on data-driven methods for mean-field SDEs, and EDMD practitioners who want a rigorous statement for a specific linearization. The convergence result is useful and likely correct. It deserves a serious referee, but the referee should insist on addressing the spectral interpretation gap and improving the numerical appendix.\n\nRecommendation: send to peer review, with the caveat that the spectral interpretation must be either proven, restricted, or softened.","headline":"Solid EDMD convergence theory for a decoupled McKean-Vlasov SDE, but the paper's advertised metastability claims rest on an unproven leap from a time-inhomogeneous finite-time operator to the nonlinear mean-field system.","tokens_in":27493,"tokens_out":1839,"would_cite":false,"duration_ms":24585,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["37M25","60H10","65C30"],"pacs":[],"model":"deepseek-v4-flash","headline":"EDMD applied to the decoupled McKean–Vlasov SDE converges almost surely to the exact projected Koopman operator.","keywords":["transfer operators","Koopman operator","Perron–Frobenius operator","McKean–Vlasov SDE","mean-field dynamics","EDMD","Galerkin projection","metastability"],"falsifier":"Take a mean-field SDE with a time-periodic law $\\mu_t$ (e.g., a forced Kuramoto model whose center of mass oscillates), simulate the decoupled scheme for $M=10^6$, $h=0.001$, and compute the second eigenvalue $\\lambda_2$ of the EDMD matrix with a fine dictionary. If $\\lambda_2 \\approx 1$ while direct particle simulations show all particles mixing between the candidate sets identified by the corresponding eigenfunction at the same time scale, then the claim that eigenvalues near one detect metastable sets would be falsified.","tokens_in":26536,"feed_emoji":"🧮","tokens_out":7348,"duration_ms":69722,"temperature":0.7,"texified_at":"2026-08-05T20:29:48.773110+00:00","pith_summary":"The paper shows that transfer operator theory—the Koopman and Perron–Frobenius operators that expose global dynamical behaviour—can be extended to mean-field (McKean–Vlasov) stochastic differential equations, the limiting dynamics of large interacting particle systems. The key move is to work with a 'decoupled' McKean–Vlasov SDE in which the law of the process is a fixed time-dependent parameter; this restores the linearity and Markov property that transfer operators need. The authors prove that the data-driven EDMD estimator of the Galerkin-projected Koopman operator converges almost surely to the exact projected operator as the number of trajectories goes to infinity and the time step goes to zero. This provides the theoretical foundation for using spectral methods to detect metastable sets, slow timescales, and invariant distributions in mean-field systems, with numerical demonstrations on the Cormier and Kuramoto models.","texify_model":"deepseek-v4-flash","texify_usage":{"total_tokens":4892,"prompt_tokens":894,"completion_tokens":3998,"prompt_tokens_details":{"cached_tokens":0},"prompt_cache_hit_tokens":0,"prompt_cache_miss_tokens":894,"completion_tokens_details":{"reasoning_tokens":3162}},"feed_headline":"EDMD almost surely converges for McKean–Vlasov SDEs","feed_subtitle":"The decoupled formulation restores linearity, unlocking data-driven spectral analysis of mean-field systems.","key_machinery":"The decoupled McKean–Vlasov SDE (equation 3) is the central object: it is an ordinary Itô diffusion whose coefficients depend on the fixed law $\\mu_t$ of the original mean-field limit. This restores the Markov property and makes the Koopman operator $K_T f(x) = E[f(X^{x,\\mu}_T)]$ linear and contractive. The finite-dimensional approximation is built on the Galerkin identity $K_N^T = C_N G_N^{-1}$, where $C_N$ and $G_N$ are the structure and Gram matrices of the dictionary basis; EDMD replaces these exact moments by Monte Carlo averages over independent trajectories of the decoupled scheme (equations 16–17). The almost-sure convergence is carried by a strong law of large numbers for i.i.d. samples plus Lips","core_discovery":"The central claim is that a data-driven approximation of transfer operators for mean-field SDEs is legitimate. By decoupling the law from the dynamics—replacing the McKean–Vlasov SDE with the standard SDE $dX_t = b(t,X_t,\\mu_t)dt + \\sigma(t,X_t,\\mu_t)dW_t$ where $\\mu_t$ is the original law—the Koopman and Perron–Frobenius operators become well-defined linear operators between function spaces. The paper proves (Theorem 4.7) that the EDMD matrix $\\hat{K}_{N,M} = \\hat{C}_{N,M}\\hat{G}_{N,M}^{-1}$ converges almost surely to $C_N G_N^{-1}$ as the sample size $M \\to \\infty$ and, subsequently, as the discretization step $h \\to 0$ along with the error in the decoupling measure. This means the eigenvalues and eigenvectors of the computed m","pith_inferences":["The convergence theorem suggests a practical recipe: estimate the law µ via a short particle simulation, fix it, then run many independent decoupled trajectories; the cost separates into law estimation and EDMD estimation, which may allow adaptive refinement of each.","The spectral interpretation is the least secure part: since µ_t varies in time, the Koopman family does not form a semigroup, so a single matrix's spectrum may not correspond to physically meaningful timescales for time-dependent mean-field dynamics. A direct test would be to compare EDMD eigenvalues with transition rates computed by long-time particle simulations for a non-stationary µ_t.","The framework naturally extends to generator-based EDMD (gEDMD) on the decoupled SDE, which would eliminate the arbitrary lag time T and might yield sharper spectra.","Because the almost-sure result depends only on the vanishing of sup_k W2(µ_{t_k}, \\hat µ_{t_k}), more efficient sampling of the law (multilevel, quasi-Monte Carlo) could be plugged in without changing the main proof structure."],"forward_implications":["EDMD can be applied to McKean–Vlasov SDEs with rigorous convergence guarantees, so spectral quantities (eigenvalues, eigenfunctions) reported from data are provably close to the projected Koopman operator.","Metastable sets and transition timescales of the decoupled process—and, heuristically, of the mean-field system—can be read off from the dominant eigenvalues of the estimated matrix.","The Perron–Frobenius operator is approximated by the transpose-based estimator \\hat P_{N,M}^T = \\hat C_{N,M}^T \\hat G_{N,M}^{-1}, so invariant densities and eigenfunctions are accessible with the same data.","If particle-system data are used instead of decoupled data, only L2 convergence holds; the decoupled scheme is the one that yields almost-sure convergence and clean eigenvalue estimates.","The method identifies the known metastable hemispheres in the sphere Kuramoto model and the two metastable intervals in the circle Kuramoto model, matching analytical predictions."],"fun_headline_variants":["EDMD converges for McKean–Vlasov SDEs","Data-driven spectral tools for mean-field SDEs","Transfer operators learned from data for McKean–Vlasov","Mean-field SDEs: EDMD almost surely converges","Spectral analysis of mean-field systems without PDEs"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The spectral interpretation—that eigenvalues close to one indicate slow timescales and metastable sets—assumes the decoupled process behaves like an autonomous Markov process; but when the law $\\mu_t$ is time-dependent, the family of finite-time Koopman operators is not a semigroup, and no theorem in the paper connects the spectrum of a single operator to the metastability of the nonlinear McKean–Vlasov equation.","fun_headline_variants_meta":{"raw":{"variants":["EDMD converges for McKean–Vlasov SDEs","Data-driven spectral tools for mean-field SDEs","Transfer operators learned from data for McKean–Vlasov","Mean-field SDEs: EDMD almost surely converges","Spectral analysis of mean-field systems without PDEs"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000259,"raw_usage":{"total_tokens":1422,"prompt_tokens":745,"completion_tokens":677,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":489,"completion_tokens_details":{"reasoning_tokens":593}},"tokens_in":489,"tokens_out":677,"duration_ms":7403,"temperature":1.0,"reasoning_tokens":593,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-04T18:31:03.499706+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a mean-field SDE with a time-periodic law $\\mu_t$ (e.g., a forced Kuramoto model whose center of mass oscillates), simulate the decoupled scheme for $M=10^6$, $h=0.001$, and compute the second eigenvalue $\\lambda_2$ of the EDMD matrix with a fine dictionary. If $\\lambda_2 \\approx 1$ while direct particle simulations show all particles mixing between the candidate sets identified by the corresponding eigenfunction at the same time scale, then the claim that eigenvalues near one detect metastable sets would be falsified.","supporting_citations":[],"review_version":1}