{"id":"020aa53a-a3ea-4ab5-99e0-be62929d2458","arxiv_id":"2607.27728","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"A Koopman-Chebyshev framework identifies the governing PDE among four candidate linear equations from noiseless observations, using a data-projected discrepancy instead of eigenvalue comparisons.","lead":"The paper proposes a numerical method that recovers which linear PDE (from a small, fixed candidate set) generated observed data, by comparing data-driven and equation-driven Koopman operators in a Chebyshev spectral domain. The new piece handles observations on uniform or irregular grids, not just Chebyshev nodes, via least-squares coefficient recovery.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Validation may be circular: reference trajectories appear to be generated by the same truncated Chebyshev spectral ODE used to build K*, so the identification experiments may not test the method on genuine PDE observations.","rationale":"The reader's weakest assumption already flags that the data-generating numerical solution must be consistent with the Chebyshev spectral truncation used to build K*, and that coefficient values are assumed known. My concern sharpens the first point: the manuscript does not state how reference trajectories are computed, which boundary conditions are imposed, or what initial condition is used. The candidate PDEs are not well-posed on a bounded domain without boundary conditions, and the equation-driven K* is derived from bare Chebyshev differentiation matrices that do not encode boundary conditions. If the data are generated by integrating the same truncated ODE ȧ=N a, then K^ matches K* by construction and the near-machine-precision discrepancies in Table I are expected, not evidence of successful PDE identification. This is load-bearing because the empirical validation is the main support for the central claim. The proposed test—inspecting the code and, if needed, re-running with a genuine PDE solver and explicit boundary conditions—would settle whether the experiments are circular. Until this is clarified, a conditional verdict is appropriate. I do not move to REJECT because the framework may still be sound and the test may pass; but the current manuscript is insufficiently specified to accept the empirical claim at face value.","tokens_in":17128,"tokens_out":16077,"duration_ms":179285,"concrete_test":"Inspect the Code Ocean capsule/scripts: determine how the reference trajectories in Section IV-A are generated. If they are obtained by integrating the truncated spectral ODE ȧ=N a (or any Chebyshev spectral discretization using the same N), then re-generate data by solving the PDEs (37)–(40) with a standard high-resolution finite-difference/spectral method on [-1,1]^2 with explicit boundary conditions (e.g., Dirichlet or periodic), using the same initial condition, and re-run the full identification pipeline. If the diagonal entries in Table I no longer remain orders of magnitude below the off-diagonal entries (or the identified PDE changes), the current validation is circular and the central claim is not supported.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central claim—that the data-projected discrepancy d(K*,K^) in Eq. (24) identifies the true PDE—requires the observed coefficient sequence {â_k} to evolve under the same truncated operator exp(ΔtN) used to build K*. Section IV-A gives the four candidates (37)–(40) but never specifies the initial condition, boundary conditions, or numerical solver used to generate the 'reference trajectories.' The candidate PDEs are not well-posed on [-1,1]^2 without boundary conditions, and N built from the Chebyshev differentiation matrices in (13) does not include boundary terms. If the reference trajectories are produced by integrating the 64-dimensional spectral ODE ȧ=N a (18) with a fine internal step, then K^ is estimated from data that exactly satisfy a_{k+1}=exp(ΔtN)a_k; K^ then coincides with K* by construction, and the near-machine-precision diagonal entries in Table I (≈1e-12) are an artifact. In that case the experiments do not test identification from genuine PDE observations, and the claimed generality to arbitrary sampling is unsubstantiated.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a numerical framework, 'numerical spectrum linking', for identifying the governing PDE of a spatiotemporal process from discrete observations. It constructs an observation-driven Koopman matrix K̂ from Chebyshev spectral coefficients recovered from arbitrary sampling grids via a unified observation model, and an equation-driven Koopman matrix K* = exp(ΔtN) from candidate PDEs. The central theoretical observation is that for the candidate class (linear, constant-coefficient, no zero-order term), the Chebyshev coefficient-space generator N is strictly upper triangular and nilpotent, so K* has a single degenerate unit eigenvalue; the paper therefore introduces a data-projected discrepancy d(K*,K̂) (Eq. 24) as the identification criterion. Experiments on four advection/diffusion PDEs under direct Chebyshev, uniform, and irregular sampling, plus an observation-density study, are reported.","tokens_in":17503,"tokens_out":8177,"duration_ms":84721,"significance":"The mathematical statements in the paper are correct: the Chebyshev differentiation matrices are strictly upper triangular for the chosen ordering, N is nilpotent, and the rank transition of the observation matrix Φ at N = M is a clean, falsifiable prediction. The unified observation model and the data-projected discrepancy are sensible design choices, and the paper makes code available. However, the empirical validation is not yet convincing: the data-generation procedure is underspecified, and if the reference trajectories are produced by integrating the same spectral ODE used to build K*, the identification results are close to tautological. The paper also assumes known coefficients and noiseless data. If the experimental concerns are addressed, the framework could be a useful contribution to spectral Koopman-based PDE identification; at present the central claim is not fully supported.","major_comments":[{"comment":"The reference-trajectory generation is unspecified. The text says only that 'a finer internal integration step of 1.0e-5 is employed', but no initial condition, boundary conditions, or numerical solver is stated. If, as the natural reading suggests, the trajectories are generated by integrating the spectral ODE a_dot = N a (Eq. 18) with the same truncated Chebyshev operator N used to build K* = exp(ΔtN) in Eq. (20), then the data exactly satisfy a_{k+1}=K*a_k and K̂ obtained from Eq. (17) coincides with K* by construction. The near-machine-precision diagonal entries in Table I would then be consistency checks, not evidence of PDE identification. Please specify the full data-generation protocol and re-run the experiments with data produced by an independent PDE solver (e.g., finite differences or a pseudospectral method with explicit boundary conditions), so the recovered coefficients are","section":"IV-A and III-B"},{"comment":"The candidate coefficients a_x, a_y, ν are never assigned numerical values, and the paper does not state whether the K* matrices are built using the same coefficient values used in the data-generating dynamics. Since d(K*,K̂) in Eq. (24) depends on these values, the method as described assumes the coefficients are known a priori. This should be stated explicitly, and the authors should clarify that the framework identifies operator structure, not coefficients. If coefficient estimation is intended, the present formulation does not yet address it; the conclusion's mention of 'unknown coefficients' as future work (Section V) confirms that this is a limitation.","section":"IV-A, Eqs. (37)-(40)"},{"comment":"The empirical support is limited to noiseless data with known candidate coefficients, and no comparison is made to existing PDE-identification methods such as PDE-FIND. The abstract's claim that the framework 'accurately identifies the governing PDE from observations' is therefore only supported under idealized conditions. A noise-robustness analysis and at least one comparison with a baseline method would be necessary to substantiate the practical relevance argued in the introduction. This is especially important because the identification margin in Table II is small (≈3×10^-3), and its behavior under realistic perturbations is unknown.","section":"IV and V"}],"minor_comments":[{"comment":"The sentence 'This condition on N is therefore not an independent assumption but a direct consequence of the shape of Φ' is unclear and appears erroneous; please revise or remove.","section":"III-F, after Eq. (35)"},{"comment":"N is used both for the number of snapshots (Section III-A) and the number of observations (Section IV-A), while M denotes both the truncation multi-index (Section II-B) and the state dimension (Section III-C). This overloading is confusing; suggest N_t, N_obs and d or M_tot.","section":"Notation throughout"},{"comment":"The denominator uses K* A0; a brief justification that this normalization does not bias the criterion toward candidates with small ||K* A0|| would improve readability. (Not a correctness issue in the current experiments.)","section":"Eq. (24)"},{"comment":"The Code Ocean capsule is mentioned only in the footnote; for reproducibility, the main text should describe which scripts generate each table/figure and what software versions were used.","section":"Reproducibility"}],"recommendation":"major_revision","confidential_remarks":"The paper's central mathematical observation is solid, but the experimental validation appears to be potentially circular. The authors should be asked to provide a complete description of the data-generation procedure and to repeat experiments with independently generated PDE solutions. If the issues are resolved, the paper could be suitable for publication; in current form I cannot recommend acceptance."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Honest take: this is a solid incremental paper, and the math is mostly right. The new pieces — the unified observation model (33), least-squares coefficient recovery (35), and the data-projected discrepancy (24) — are sensible and extend the authors' ICASSP version [30] from direct Chebyshev sampling to arbitrary grids. The nilpotency remark is correct and worth stating: because Chebyshev differentiation is strictly upper triangular, the generator N is nilpotent for every candidate with no zero-order term, so K* has a degenerate unit eigenvalue and eigenvalue-based comparison is useless. That observation alone justifies the paper.\n\nThe main soft spot is the validation. The reference trajectories appear to be generated by integrating the same truncated Chebyshev spectral ODE ȧ = N a with a fine internal step, then sampling the Chebyshev expansion at the observation locations. Section IV-A says only that the spatial state is represented with the truncated Chebyshev expansion and that a finer internal step is used for reference trajectories; it never states the initial condition, boundary conditions, or whether the simulation solves the PDE or the spectral ODE (18). If the latter, the experiments are self-consistent but circular: the data exactly satisfy a_{k+1} = exp(Δt N) a_k, so K^ is approximately K* for the true model by construction. What is actually tested is the coefficient-recovery pipeline under resampling, which the observation-density study does fairly well (rank transition at N=M=64, seed-to-seed variance at low N). What is not tested is identification from genuine PDE observations with truncation error, boundary effects, or a different numerical solver.\n\nThe second limitation is that candidate coefficients (a_x, a_y, ν) are assumed known a priori; the method only selects among a finite library of fixed-coefficient linear PDEs. No noise, no coefficient estimation, no comparison to PDE-FIND or other discovery methods. The authors do list noise and unknowns as future work, but the abstract's claim of 'identifying governing PDEs from observations' overstates what the experiments demonstrate.\n\nSo: worth reading for the nilpotency observation and the clean formulation of the resampling problem; the code is available (CodeOcean) and the math claims check out. But the central empirical claim needs to be scoped down or re-tested on data from a genuine PDE solver. I'd send it to a serious referee, but I'd expect a major revision clarifying the data-generation procedure and softening the claims.","headline":"A solid incremental paper whose central empirical claim is undercut by self-consistency in the data generation.","tokens_in":17849,"tokens_out":6107,"would_cite":true,"duration_ms":56064,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["65M70","93B30","65M32"],"pacs":[],"model":"deepseek-v4-flash","headline":"A data-projected discrepancy between Koopman matrices built from observations and from candidate PDEs identifies the correct governing equation, even though all candidates share a degenerate eigenvalue spectrum.","keywords":["Koopman operator","Chebyshev spectral method","PDE identification","data-projected discrepancy","resampling","coefficient recovery","observation model","nilpotent generator"],"falsifier":"Run the four-candidate identification on data generated by the advection-diffusion equation with the candidate diffusivity ν perturbed ±10% around the true value. If the minimum data-projected discrepancy no longer selects the true structural form at the true parameter value, or the margin turns negative, then the method's reliance on a priori exact coefficients is exposed.","tokens_in":17048,"feed_emoji":"🧮","tokens_out":6419,"duration_ms":60350,"temperature":0.7,"pith_summary":"The paper claims that comparing where two Koopman operators send observed data—not their spectra—reveals which candidate PDE generated the data. It builds both operators in a Chebyshev spectral basis: one from recovered coefficients of the observed field, one from each candidate equation's spatial differential operator. A structural nilpotency result shows the equation-driven operator always has the same degenerate unit eigenvalue for every candidate, so eigenvalue comparisons cannot discriminate. Instead, the data-projected discrepancy, a normalized measure of how differently the two operators act on the recovered coefficient snapshots, is shown to be minimized by the true PDE across four canonical linear candidates and three sampling geometries. If correct, this gives a spectral-domain route from raw spatiotemporal measurements to the governing linear constant-coefficient PDE.","feed_headline":"Data-projected discrepancy identifies the true PDE","feed_subtitle":"When eigenvalues are degenerate for every candidate, comparing operator action on observations still picks the true equation.","key_machinery":"The load-bearing objects are the coefficient-space generator N of a candidate PDE and the observation model that maps arbitrary sampling grids into the Chebyshev basis. Chebyshev differentiation matrices are strictly upper triangular, so any linear constant-coefficient spatial differential operator without a zero-order term becomes a strictly upper triangular, nilpotent matrix N in coefficient space; consequently K* = exp(Δt N) carries no spectral information. The data-projected discrepancy d(K*,K̂) = ||(K*-K̂)Â0||_F/||K*Â0||_F uses the recovered coefficient snapshot matrix Â0 to compare operator action on data, remaining discriminative despite degenerate spectra. The observation matrix H =","core_discovery":"PDE identification is performed by comparing operators in the Chebyshev coefficient domain rather than by their spectra. An observation-driven Koopman matrix K̂ is fit to recovered Chebyshev coefficients, while each candidate PDE supplies an equation-driven matrix K* = exp(Δt N), with N the coefficient-space generator. The authors show N is strictly upper triangular for the candidate advection/diffusion operators, hence nilpotent, so every K* has the same degenerate unit eigenvalue regardless of dynamics. Spectral comparison is therefore uninformative. Instead, the data-projected discrepancy d(K*,K̂) = ||(K*-K̂)Â0||_F/||K*Â0||_F, with Â0 the recovered-coefficient snapshot matrix, compares ho","pith_inferences":["The nilpotency argument generalizes beyond the four tested examples to any linear constant-coefficient PDE without a zero-order term, so the data-projected discrepancy criterion is likely applicable to a wider candidate library of the same structural type.","The requirement that candidate coefficients be known a priori is a practical bottleneck; extending the framework to jointly estimate coefficients (e.g., by minimizing the discrepancy over parameter values) is a natural next step that the paper lists as future work.","The sharp transition at N=M suggests a design rule for sensor placement: measurements should be positioned to make the Chebyshev evaluation matrix Φ well-conditioned and full rank, since the number of observations alone does not guarantee identifiability.","For nonlinear or parameter-varying PDEs, the linear-operator comparison would need a different basis or a nonlinear extension; the data-projected discrepancy itself is basis-dependent and would not transfer unchanged."],"forward_implications":["Under direct Chebyshev, uniform, and irregular sampling, a minimum data-projected discrepancy selects the true PDE among the four candidates, with the correct entry several orders of magnitude below competing entries.","The identification margin stays positive for both uniform and irregular sampling once the number of observations reaches the spectral state dimension M=64; additional observations beyond this rank ceiling give no further benefit.","The nilpotency result means that eigenvalue-based model selection (e.g., matching Koopman eigenvalues) cannot work for this class of candidate PDEs, and any method relying on spectra alone will fail to discriminate.","The framework provides a practical observation-density guideline: collect at least as many independent observations as spectral modes to guarantee full column rank of the recovery matrix and thereby reliable identification.","The unified observation model reduces to the direct Chebyshev case when H=I, so the method is a strict generalization of earlier ideal-setting formulations."],"fun_headline_variants":["Eigenvalues blind, operators solve: PDE ID from data","Operator action beats spectra for PDE discovery","Degenerate eigenvalues? Compare operator actions instead","Chebyshev domain exposes true PDE via operator mismatch","When spectra lie, operator comparison finds the PDE"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The procedure assumes that the candidate PDEs are linear, constant-coefficient spatial differential operators with no zero-order term, and that the coefficient values in each candidate (a_x, a_y, ν) are known in advance rather than estimated from the data.","fun_headline_variants_meta":{"raw":{"variants":["Eigenvalues blind, operators solve: PDE ID from data","Operator action beats spectra for PDE discovery","Degenerate eigenvalues? Compare operator actions instead","Chebyshev domain exposes true PDE via operator mismatch","When spectra lie, operator comparison finds the PDE"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000328,"raw_usage":{"total_tokens":1657,"prompt_tokens":722,"completion_tokens":935,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":466,"completion_tokens_details":{"reasoning_tokens":863}},"tokens_in":466,"tokens_out":935,"duration_ms":9200,"temperature":1.0,"reasoning_tokens":863,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T02:28:34.273256+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the four-candidate identification on data generated by the advection-diffusion equation with the candidate diffusivity ν perturbed ±10% around the true value. If the minimum data-projected discrepancy no longer selects the true structural form at the true parameter value, or the margin turns negative, then the method's reliance on a priori exact coefficients is exposed.","supporting_citations":[],"review_version":1}