{"id":"f745235f-c8d4-453c-9ce2-6e97aaa3304b","arxiv_id":"2607.25950","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"WEDMD uses compactly supported test functions and a generalized eigenvalue problem to approximate Koopman generator eigenpairs directly from noisy time-series data, avoiding derivative estimation.","lead":"This paper develops WEDMD, a way to estimate the Koopman operator of a nonlinear system from noisy time-series data by integrating against smooth bump functions instead of estimating derivatives. The method matters because it offers a noise-robust route to spectral analysis and forecasting for data-driven dynamical systems.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The linear exactness and pseudoinverse reduction rely on unproven full column rank of G; when K+1 < N_d, G is rank-deficient and spurious eigenpairs can arise.","rationale":"The reader identified G full column rank as the weakest assumption, and my analysis agrees. The paper's own language ('it should be possible') signals that this is a conjecture rather than a proven condition, and the algorithm proceeds through the pseudoinverse regardless. This is the most load-bearing gap because it affects even the clean, linear, deterministic case where the paper claims exactness; it is not merely a technicality about noise or dictionary closure. The concrete test I propose would settle whether rank deficiency actually produces spurious eigenvalues in the linear setting, and would clarify whether the nonlinear experiments operate in the full-rank regime. I do not propose changing the verdict to REJECT because the method has clear empirical support and the gap is addressable by either proving the rank condition under explicit parameter choices or by adding a rank check and regularization strategy. The reader's CONDITIONAL verdict is appropriate, so I recommend UNCHANGED.","tokens_in":12103,"tokens_out":8528,"duration_ms":86465,"concrete_test":"Run WEDMD on a known linear system (e.g., A = [[−0.1, 1], [−1, −0.1]]) with clean data. Use the canonical dictionary (N_d = 2) and a monomial dictionary of order 3 (N_d = 9). For each, choose test-function parameters so that K+1 is first smaller than N_d (e.g., K+1 = 3) and then larger (e.g., K+1 = 15). Compute rank(G) and the WEDMD eigenvalues. If spurious eigenvalues appear only when K+1 < N_d, and all eigenvalues match the exact e^{At} eigenvalues when K+1 ≥ N_d, then the full-rank assumption is confirmed as the deciding factor. If spurious eigenvalues appear even in the full-rank case, the problem is deeper.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central reduction in Section 2.1 replaces the pencil λGw = −Dw with the standard eigenvalue problem G^{-P}Dw = −λw, which is justified only if G has full column rank. The paper only conjectures this: \"it should be possible to choose r, s, and p to ensure G has full rank.\" No proof, rank-check, or parameter-selection rule is given. This is load-bearing for both parts of the central claim. First, the linear exactness proof in Section 2.3 establishes D = −GΛ termwise, so the pencil becomes (λI − Λ)Gw = 0. If G is rank-deficient, the null space of G introduces spurious eigenvalues that are not eigenvalues of Λ, so the claim \"we should get the exact Koopman operator e^{At}\" fails even with zero quadrature error and noiseless data. Second, in the nonlinear experiments the dictionary has N_d = (N_ord+1)^2 = 36 for N_ord = 5, while the number of test functions K+1 is determined implicitly by r, s, p; if K+1 < N_d, G is rank-deficient by construction. In that regime the pseudoinverse does not reduce to a standard eigenproblem: components of w in null(G) are unconstrained, and the computed spectrum can contain spurious modes. The paper never reports K for its examples, so it is unknown whether the reported successes occur in the full-rank regime. The contribution is otherwise promising, but this unproven rank condition is the weakest load-bearing step in the argument.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces WEDMD, a weak-form variant of EDMD for estimating the Koopman generator from time series. Observables are expanded in a monomial dictionary; the generator eigenvalue problem is integrated against compactly supported test functions to form the pencil λGw = -Dw, which is then reduced to a standard eigenvalue problem via the pseudoinverse of G. The authors prove that for linear deterministic systems with canonical observables the matrices satisfy D = -GΛ exactly up to quadrature, and they show for linear SDEs that the variances of G and D are governed by filtered covariance integrals, supporting a low-pass filtering interpretation. For nonlinear Duffing and Van der Pol examples, they demonstrate that filtering computed modes by reconstruction error yields good reduced-order forecasts under additive noise. The paper is clearly written and the numerical studies are thoughtfully designed, but the rank condition underpinning the pseudoinverse step is left as a conjecture without verification.","tokens_in":12481,"tokens_out":5941,"duration_ms":57059,"significance":"WEDMD is a promising methodological contribution: avoiding explicit derivative estimation is valuable, and the linear-SDE variance calculation is a concrete, useful theoretical result. The linear exactness argument is elegant and self-contained. The numerical evidence for mode filtering and forecasting is compelling in the additive-noise regime. However, the central reduction depends on full column rank of G, a condition that is neither proved nor empirically checked (K+1 is never reported), so the theoretical guarantees do not yet cover the practical algorithm. The residual-convergence argument in §2.2 is also heuristic. These gaps are fixable and do not invalidate the overall approach.","major_comments":[{"comment":"The reduction to G^{-P}Dw = -λw is justified only if G has full column rank, but the paper only conjectures this ('it should be possible to choose r, s, and p to ensure G has full rank'). In the linear exactness argument of §2.3, D = -GΛ termwise turns the pencil into (λI - Λ)Gw = 0; if G is rank-deficient, any w ∈ null(G) produces a spurious eigenvalue, so 'we should get the exact Koopman operator e^{At}' fails even with noiseless data and zero quadrature error. The numerical sections never report K+1, so the reader cannot tell whether the experiments actually operated in the full-rank regime. The authors should either prove the rank condition under explicit assumptions on r, s, p and the trajectory, or replace the pseudoinverse step with a truncated-SVD/regularized formulation and prove a perturbation bound.","section":"Section 2.1, Eq. λGw = -Dw and pseudoinverse reduction"},{"comment":"The claim that taking r → 0+ while letting centers pass to a continuum limit 'essentially eliminate[s] the residual' is heuristic rather than a proof. The text itself concedes this is 'an infinite resolution data result.' The argument interchanges a limit in r with a continuum limit in K and assumes a dominated-convergence statement for the residual that is not established. Since the choice of r is a central hyperparameter and Algorithm 1 selects it by minimizing E_c(r), the theoretical convergence statement needs either a precise theorem with commuting limits or a clear demotion to a plausibility argument. As written, this is load-bearing for the claim that the weak form 'provides a systematic means of controlling approximation error.'","section":"Section 2.2, residual-convergence argument"}],"minor_comments":[{"comment":"The sentence 'G^{-P} is the Moore-Penrose pseudoinverse of D' should read 'of G'; also the notation G^{-P} is not defined.","section":"Section 2.1"},{"comment":"p is initialized in step 1 but immediately overwritten in step 4; clarify the role of each step.","section":"Algorithm 1"},{"comment":"Report K+1 (or r, s, p and N_T) so that the full-rank condition can be checked by the reader.","section":"Sections 3.1 and 3.2"},{"comment":"The displayed limit '=0 R(t,x)Λ ∫...' appears to have a missing equality/line break; please re-set the equation.","section":"Section 2.2"},{"comment":"Panels are referenced as (a)-(f) in the text, but the captions do not describe all panels; please expand the captions.","section":"Figures 3 and 8"}],"recommendation":"major_revision","confidential_remarks":"The paper would be strengthened by including code/data availability, though this is not essential. The main technical risk is the lack of any rank verification; if the authors add a rank theorem or, failing that, a truncated-SVD formulation with a perturbation bound, and report K in the experiments, the contribution would be suitable for publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"WEDMD is a sensible weak-form twist on EDMD with a clean linear exactness proof and a real SDE variance result, but the unproven full-rank assumption on G and the lack of benchmarks against related methods keep it from being a finished paper.\n\nYou should know this paper before the others in your stack. It does something genuinely useful: it takes the WSINDy weak-form idea and applies it to the continuous-time Koopman generator with a nonlinear observable dictionary. The linear deterministic exactness proof in Section 2.3 is clean — no fitted constants, just quadrature error and noise left as error sources. The SDE variance calculation for canonical observables is also real, and it explains the low-pass filtering effect without hand-waving. Those two analytic results are the core of the paper, and they hold up.\n\nThe numerical work is honest and reasonably convincing. For the Duffing and Van der Pol examples, the mode-filtering strategy gives good reduced-order forecasts under additive noise, and the authors openly report that WEDMD tracks the noisy trajectory rather than the underlying dynamics for the Van der Pol SDE case. That kind of limitation statement is refreshing and consistent with the Itô caveat in Section 2.3: for nonlinear dictionaries the weak form estimates the Kolmogorov backward generator, not the deterministic Koopman generator.\n\nNow the soft spots, in proportion. The full-rank claim on G is the load-bearing assumption for the pseudoinverse reduction G^{-P}D w = -λ w, and it is only asserted as \"it should be possible\" in Section 2.1. If K+1 < N_d, G is rank-deficient by construction, and the stress-test note is right that the linear exactness claim and the computed spectra can then contain spurious modes. The paper never reports K for its examples, so we do not know whether the successes happen in the full-rank regime. This is a fixable issue — prove a rank bound under reasonable conditions or at least report and check K+1 ≥ N_d — but it is currently the weakest link. The residual-convergence argument in Section 2.2 is also explicitly an \"infinite resolution\" heuristic, not a theorem, though it is clearly labeled as such. And there are no benchmarks against WDMD, PRONE, or even standard EDMD, which is a missed opportunity given that the authors already distinguish their method from those two.\n\nThe self-selection of r by minimizing the reconstruction error on the same trajectories used to report forecasting skill is a mild circularity, but not damning; they filter modes by the same error metric, so it is more a tuning choice than a hidden fit.\n\nWho gets value from this: anyone working on data-driven Koopman approximations or weak-form methods for noisy dynamical systems. It deserves a serious referee, but I would send it back with a request to address the rank condition, add comparison benchmarks, and check whether the conclusions survive in the K+1 < N_d regime. If those are addressed, the method stands as a practical contribution to the subfield.","headline":"WEDMD is a sensible weak-form twist on EDMD with a clean linear exactness proof and a real SDE variance result, but the unproven full-rank assumption on G and the lack of benchmarks against related methods keep it from being a finished paper.","tokens_in":752,"tokens_out":2673,"would_cite":true,"duration_ms":33525,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["37M99","65P99","34A55"],"pacs":[],"model":"deepseek-v4-flash","headline":"WEDMD makes Koopman generator estimation exact on linear systems and noise-robust on nonlinear ones by replacing pointwise derivatives with temporal averaging.","keywords":["weak-form EDMD","Koopman generator","Galerkin projection","generalized eigenvalue problem","noise mitigation","reduced-order forecasting","stochastic differential equations","Koopman modes"],"falsifier":"Take a linear system ẏ = Ay with known diagonalizable A, sample clean data y_j = e^{A j δt} x, and run WEDMD with the canonical linear dictionary and sufficiently small δt. If any computed eigenvalue differs from an eigenvalue of A by more than the Trapezoid quadrature error, the exactness claim fails. Conversely, a negative control with a deliberately rank-deficient dictionary (e.g., duplicate or linearly dependent observables) should produce G singular and spurious eigenpairs; if the pseudoinverse still returns the true spectrum, the full-rank assumption is not actually load-bearing.","tokens_in":11988,"feed_emoji":"📈","tokens_out":4550,"duration_ms":39285,"temperature":0.7,"pith_summary":"This paper introduces Weak-form Extended Dynamic Mode Decomposition (WEDMD), a method that approximates the Koopman generator rather than the Koopman operator. It integrates the eigenvalue equation against compactly supported test functions, avoiding numerical differentiation and converting noise into filtered averages. For linear deterministic systems, the resulting pencil is exact up to quadrature error, so the computed eigenpairs match the true Koopman generator. For nonlinear systems with finite monomial dictionaries, WEDMD produces enough accurate Koopman modes that selecting modes by reconstruction error yields reduced-order models with roughly 1% relative error, even under strong additive noise and, for fixed-point attractors, under Brownian motion.","feed_headline":"Weak-form EDMD recovers exact Koopman spectra on linear systems","feed_subtitle":"Temporal averaging filters noise; filtering accurate modes forecasts nonlinear dynamics to ~1% error.","key_machinery":"The central object is the pencil λGw = -Dw, built from a dictionary of observables Θ(y) and compactly supported bump test functions ψ_k(t) whose support width r is chosen by minimizing a per-mode reconstruction error E_c(r). Integration by parts replaces the derivative ẏ·∇φ with -φ dψ/dt, so no derivative estimation is needed; G and D are discrete Trapezoid-rule integrals Ψ_dΘ_d and Ψ̇_dΘ_d. For linear systems the identity D = -ΛG makes the pencil exact, and the Moore-Penrose pseudoinverse G^{-P} converts it to a standard eigenvalue problem. The support parameter r is tuned by changepoint detection and error minimization, which is what allows a small accurate subset of modes to be identified","core_discovery":"On the paper's own terms, WEDMD establishes that a weak Galerkin temporal form applied to the Koopman eigenvalue equation yields a generalized eigenproblem λGw = -Dw, where G and D are built from dictionary observables and test functions and their derivatives. For a linear system ẏ = Ay with canonical linear observables, the identity D = -ΛG holds analytically, so the computed spectrum is exactly the spectrum of e^{At} up to quadrature and noise. The same weak form acts as a low-pass filter on noise: variances of G and D depend only on filtered data correlations. For nonlinear systems, the method is a Galerkin approximation that makes the closure residual orthogonal to the test functions, an","pith_inferences":["A natural testable extension is to use the same temporal weak form with adaptive or learned dictionaries, which could close the gap between the finite-dictionary Galerkin approximation and true Koopman invariance.","The Itô-correction distinction identified for SDEs suggests WEDMD could be deliberately repurposed as a data-driven estimator of the Kolmogorov backward generator from stochastic trajectories, rather than treating stochasticity only as noise to remove.","The reliance on full column rank of G suggests a practical diagnostic: monitor the condition number of G and adapt r, p, or the dictionary until it stabilizes, turning a heuristic hyperparameter choice into a data-driven criterion.","Since the method needs no sparsity promotion, it may combine naturally with dictionary pruning based on reconstruction error, yielding interpretable reduced-order Koopman models for high-dimensional systems."],"forward_implications":["On linear deterministic systems, WEDMD recovers the exact Koopman operator e^{At}; the only remaining errors are quadrature and noise, and the weak form suppresses noise by averaging.","The method avoids explicit differentiation, removing the variance-amplification problem of pointwise generator estimation; for additive Gaussian noise this keeps spectra and forecasts stable even at σ = 0.1.","For nonlinear attractors, a simple reconstruction-error filter picks out a small set of accurate Koopman modes, giving reduced-order models with errors near 1% on clean Duffing and Van der Pol data.","The averaging interpretation extends to linear stochastic systems: process noise enters only through zero-mean fluctuations for linear observables, so the weak-form operators depend on filtered correlations rather than instantaneous noisy derivatives.","For stochastic dynamics with nonlinear dictionaries, the weak form estimates the Kolmogorov backward generator of the stochastic process rather than the deterministic Koopman generator, delimiting when deterministic forecasting is possible."],"fun_headline_variants":["Weak-form EDMD exact on linear, robust to noise","WEDMD: weak form tames noise, nails Koopman spectra","Weak-form EDMD filters noise, exact linear spectra","WEDMD: noise-robust weak form gives exact Koopman spectra","Weak-form EDMD: exact spectra from noisy data"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The method assumes the Gram matrix G has full column rank and that the finite monomial dictionary is nearly closed under the generator; the paper leaves the full-rank condition as a plausible but unproved statement, and for nonlinear dictionaries the weak form actually estimates a stochastic backward generator rather than the deterministic one.","fun_headline_variants_meta":{"raw":{"variants":["Weak-form EDMD exact on linear, robust to noise","WEDMD: weak form tames noise, nails Koopman spectra","Weak-form EDMD filters noise, exact linear spectra","WEDMD: noise-robust weak form gives exact Koopman spectra","Weak-form EDMD: exact spectra from noisy data"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000628,"raw_usage":{"total_tokens":2667,"prompt_tokens":599,"completion_tokens":2068,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":343,"completion_tokens_details":{"reasoning_tokens":1980}},"tokens_in":343,"tokens_out":2068,"duration_ms":12149,"temperature":1.0,"reasoning_tokens":1980,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T01:00:30.543345+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a linear system ẏ = Ay with known diagonalizable A, sample clean data y_j = e^{A j δt} x, and run WEDMD with the canonical linear dictionary and sufficiently small δt. If any computed eigenvalue differs from an eigenvalue of A by more than the Trapezoid quadrature error, the exactness claim fails. Conversely, a negative control with a deliberately rank-deficient dictionary (e.g., duplicate or linearly dependent observables) should produce G singular and spurious eigenpairs; if the pseudoinverse still returns the true spectrum, the full-rank assumption is not actually load-bearing.","supporting_citations":[],"review_version":1}