{"id":"d0c5e5a1-2056-41ae-9238-682dcebf00b4","arxiv_id":"2501.00701","paper_version":4,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"ResKoopNet learns Koopman eigenpairs by minimizing the spectral residual over neural-network dictionaries, reporting fuller spectra and better latent-state separation than existing Koopman methods.","lead":"ResKoopNet uses a neural network to learn a Koopman dictionary by directly minimizing the spectral residual, a measure of how accurately each eigenvalue-eigenfunction pair describes the dynamics. The authors report more complete spectra for a pendulum, turbulent airfoil flow, and mouse visual cortex recordings than EDMD, ResDMD, and Hankel-DMD.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The claimed theoretical guarantees rest on a false convexity assumption and ignore finite-sample overfitting of the learned dictionary; minimizing the training spectral residual does not imply convergence to the true Koopman spectrum.","rationale":"The paper's abstract promises theoretical guarantees. The only theoretical support is Appendix A.4, which assumes strong convexity of the neural-network loss (Assumption A.4(b)). This is acknowledged false in the same appendix. Moreover, the empirical spectral residual in Eq. (3.3) is minimized on the training snapshots; no generalization analysis is given, so a sufficiently expressive network can make the residual artificially small. ResDMD's convergence theorems assume a fixed dictionary and therefore do not cover data-dependent dictionaries. The reader correctly flags the flawed derivation of Eq. (3.6) as a weakness, but I judge the convergence/overfitting gap to be more load-bearing: even if the closed-form were repaired, the claimed guarantees would still not follow. The method may still work as a heuristic, and the experiments are suggestive, so CONDITIONAL remains appropriate.","tokens_in":20326,"tokens_out":12723,"duration_ms":122564,"concrete_test":"Split the pendulum dataset into disjoint training and held-out snapshot sets; train ResKoopNet on the training set, then evaluate Eq. (3.3) on the held-out set using the learned dictionary and eigenpairs. If the held-out spectral residual exceeds the training residual by a substantial margin (or does not decrease as training progresses), the learned spectrum is not generalizing and the paper's theoretical-guarantee claim fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The most load-bearing concern is that the advertised 'theoretical guarantees' (Abstract) are not supported by the paper's convergence analysis. In Appendix A.4, Assumption A.4(b) asserts J(θ) is 'strongly convex and Lipschitz continuous in θ', but the very next paragraph admits 'In practice, J(θ) is non-convex due to the neural network'. Strong convexity is used to invoke an O(1/n) SGD rate and to conclude J(θ_n) → 0; without it, the conclusion does not follow. Even setting convexity aside, J(θ) is the empirical spectral residual (Eq. 3.3) computed on the same snapshots used to train the dictionary Ψ(x;θ). No sample-splitting, regularization, or generalization bound is provided, so the network can drive the training residual to zero by overfitting without approximating the true Koopman operator. ResDMD's rigorous convergence results (Colbrook & Townsend, 2024) are for a fixed dictionary; they do not transfer to a data-dependent dictionary. Thus the central claim that minimizing the spectral residual yields 'a more precise and complete Koopman operator spectrum' with 'theoretical guarantees' is unsubstantiated. A secondary flaw is the derivation of Eq. (3.6): the differentiation in Appendix A.3 treats V as independent of K, although V is the eigenvector matrix of K; a fixed-V interpretation yields G†A, but the paper does not state this, so the claim that K=G†A minimizes J is not rigorously established.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces ResKoopNet, a method that trains a neural-network parameterized dictionary by minimizing the empirical spectral residual (Eq. 3.3) over computed Koopman eigenpairs, with the aim of addressing the spectral inclusion problem and recovering both discrete and continuous spectra. The authors propose an alternating scheme: update the dictionary parameters θ by gradient descent on the residual loss J(θ), and at each step set the Koopman matrix to K(θ) = (G(θ)+σI)^{-1}A(θ), which they claim minimizes J. The method is tested on a pendulum, a turbulent airfoil flow, and calcium-imaging data from mouse visual cortex, where it is compared with EDMD, EDMD-DL, Hankel-DMD, and ResDMD. The paper also includes a convergence discussion based on Barron-space approximation theory and an SGD rate argument in Appendix A.4.","tokens_in":20698,"tokens_out":5438,"duration_ms":57403,"significance":"If the theoretical claims were established, ResKoopNet would be a meaningful contribution: it would offer a principled way to learn dictionaries directly for spectral accuracy, potentially reducing the number of observables needed for continuous spectra and scaling to high-dimensional systems. The empirical results, particularly the pendulum spectrum with NK=300 and the recovery of the pressure field in the turbulence example, are suggestive and the work addresses a real limitation of ResDMD, namely that filtering precomputed spectra cannot discover missing spectral components. However, the advertised theoretical guarantees are not currently supported by the analysis in the manuscript. The derivation of the closed-form Koopman matrix has a load-bearing gap, and the convergence argument relies on assumptions that the paper itself acknowledges are not satisfied. The empirical evaluation is therefore the main strength of the paper, but the central theoretical claims need substantial revision before publication.","major_comments":[{"comment":"The derivation of the closed-form optimal Koopman matrix K = G†A differentiates the loss J = (1/m)||(ΨY − ΨX K)V||_F^2 with respect to K while treating the eigenvector matrix V as a constant. However, V is defined as the eigenvector matrix of K, so V depends on K. The calculation in Appendix A.3 omits the terms arising from ∂V/∂K. Consequently, the paper does not establish that K = G†A minimizes J over all K. At best, the calculation shows that K = G†A minimizes the residual for a fixed V, which is not the objective being optimized. This is a load-bearing issue because the method's theoretical foundation rests on the claim that Eq. (3.6) is the optimal Koopman matrix for the spectral-residual loss. The authors should either provide a rigorous derivation that accounts for the V dependence, or explicitly state that Eq. (3.6) is a heuristic choice and analyze the actual objective being minimized by the algorithm.","section":"Section 3.2, Eqs. (3.5)-(3.6) and Appendix A.3"},{"comment":"The convergence analysis assumes that J(θ) is strongly convex and Lipschitz continuous in θ, and uses this assumption to invoke an O(1/n) SGD rate and to conclude that J(θ_n) tends to zero. In the very next paragraph, the paper acknowledges that 'In practice, J(θ) is non-convex due to the neural network.' Strong convexity is therefore not satisfied, and the stated SGD convergence result does not apply. The claim that J(θ_n) → 0 as n and m tend to infinity is thus unsupported. The authors need to replace this argument with a non-convex convergence analysis, or significantly weaken the theoretical claims to state what is actually proven.","section":"Appendix A.4, Assumption A.4(b) and the SGD convergence argument"},{"comment":"The paper invokes ResDMD's convergence results, e.g., Colbrook and Townsend [8, Theorem B.1, Lemma B.1], to argue that J(θ) → 0 implies convergence of the computed eigenpairs and pseudospectrum to the true Koopman spectrum. Those results are for a fixed dictionary, whereas ResKoopNet uses a dictionary Ψ(x; θ) that is trained on the same snapshots used to define the empirical residual. No sample-splitting, regularization, or generalization bound is provided to show that minimizing the training residual controls the true spectral residual. The sentence claiming that uniform convergence follows from density of B_NK and Dini's theorem is not justified for data-dependent dictionaries. A theorem with explicit assumptions relating the number of snapshots m, the dictionary size NK, and the network capacity to the spectral error is needed to substantiate the abstract's claim of theoretical guarantees.","section":"Appendix A.4, paragraph on spectral convergence and data-dependent dictionaries"}],"minor_comments":[{"comment":"The abstract states that the approach 'provides theoretical guarantees while maintaining computational adaptability,' but no formal theorem for ResKoopNet is stated in the main text; the convergence discussion in Appendix A.4 is conditional on assumptions that are acknowledged to be violated. The wording should be tempered to match the actual results.","section":"Abstract and Section 1"},{"comment":"Eq. (3.6) presents the optimal Koopman matrix as K = G†A, but Algorithm 1 and Remark 3.2 use the regularized form K = (G+σI)^{-1}A. The relationship between these two expressions and the role of σ in the theory should be clarified.","section":"Section 3.2, Eq. (3.6) and Algorithm 1"},{"comment":"The statement 'the source code will be available at this link' does not include an actual URL. For a reproducibility-focused journal, a working link should be provided.","section":"Appendix A.1"},{"comment":"The paper says 'we trained dictionaries on all snapshots from each mouse to avoid overfitting,' but this means the evaluation of clustering is performed on the same data used for training. This should be stated explicitly as an in-sample evaluation, and the possibility of overfitting to trial identities should be discussed.","section":"Section 4.3"},{"comment":"The Hankel-DMD comparison in the turbulence example is described as having small spectral residuals, yet the paper argues that the residual metric 'does not fully extend to the Hankel-DMD setting.' If the residual is not a valid comparison for Hankel-DMD, the quantitative comparison in Figure 9 and the associated discussion should be framed more carefully.","section":"Section 4.2 and Appendix A.8.2"},{"comment":"Assumption A.4(a) assumes that there is a finite-dimensional invariant subspace B_NK spanned by optimal dictionary functions. For systems with continuous spectra, such a finite-dimensional exactly invariant subspace generally does not exist; this assumption should be stated and discussed, since it is not satisfied by the pendulum example with continuous spectrum.","section":"Appendix A.4, Assumption A.4(a)"}],"recommendation":"major_revision","confidential_remarks":"The core difficulty is that the theoretical framework is presented as rigorous but the two pillars — the closed-form optimality of K = G†A and the SGD convergence under strong convexity — are not sound as written. The empirical results are interesting and likely salvageable, but the authors should be asked to either provide a correct theory or clearly reposition the paper as a heuristic method with strong empirical validation. I also note that the self-citation [46] is not load-bearing for the main claims, so I do not see a citation-integrity concern."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"ResKoopNet's core idea is worth taking seriously: minimizing the spectral residual as a dictionary-learning objective is a clean, novel departure from EDMD-DL's prediction loss and VAMP's variational score. That alone makes the paper worth a look. The pendulum and turbulence experiments are suggestive, and the comparison to ResDMD under the same residual metric is fair. But the advertised 'theoretical guarantees' do not survive contact with the appendix, and the closed-form 'optimal' Koopman matrix in Eq. (3.6) is just EDMD.\n\nThe load-bearing problem is in Section 3.2 / Appendix A.3. The derivation differentiates J with respect to K while holding the eigenvector matrix V fixed, even though V is the eigenvector matrix of K. No proof is given that G†A actually minimizes J; in fact, the fixed-V minimizer is exactly the EDMD matrix. So the paper hasn't shown that minimizing the spectral residual yields a new Koopman matrix—only that, for a given dictionary, EDMD's K is the least-squares solution of a projected residual. The convergence analysis in Appendix A.4 is also broken: Assumption A.4(b) asserts J(θ) is strongly convex, and the next paragraph admits it is non-convex. The O(1/n) SGD rate and the conclusion J(θ_n)→0 don't follow. On top of that, the residual is evaluated on the same snapshots used to train the dictionary, with no sample splitting or generalization bound, so driving J to zero could just be overfitting.\n\nThere are smaller issues: no code is provided, despite the appendix saying it 'will be available at this link'; the neural dynamics experiment relies on a clustering metric (DBI) that, while reasonable, is presented without variance estimates; and the comparisons to Hankel-DMD, while fair in spirit, are not apples-to-apples in dictionary size.\n\nWhat's genuinely new is using the residual as a training loss rather than a post-hoc filter. If the authors fix the derivation (or explicitly state that K=G†A is a heuristic fixed-point choice), drop the overclaim about guarantees, and release code with quantitative spectral error metrics, this could be a solid contribution.\n\nWho is this for? Koopman practitioners who want to explore dictionary learning for continuous spectra. A serious referee should see it, because the core idea is novel and the experiments are thought-provoking. But it should go back for major revision, not be accepted as is.","headline":"Novel residual-based dictionary learning for Koopman spectra, but the theoretical guarantees are unsupported and the closed-form update is just EDMD.","tokens_in":21217,"tokens_out":4310,"would_cite":false,"duration_ms":36657,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["37M10","65P99","68T07"],"pacs":[],"model":"deepseek-v4-flash","headline":"ResKoopNet claims that minimizing the spectral residual over neural-network-learned observables recovers a more complete Koopman spectrum — discrete eigenvalues and continuous bands — than the filtering approach of ResDMD, using far fewer…","keywords":["Koopman operator","spectral residual","spectral inclusion problem","neural network dictionary learning","continuous spectrum","pseudospectrum","dynamic mode decomposition","high-dimensional dynamical systems"],"falsifier":"Test the optimality claim on a tiny system with a known Koopman spectrum: compute the spectrum from ResKoopNet's fixed-$V$ closed form $K = G^\\dagger A$ and compare it with the spectrum of a $K$ obtained by minimizing the same loss jointly over $K$ and its eigenvector matrix $V$, with no fixed-$V$ shortcut; if the two spectra differ materially, the closed form does not minimize the residual and the theoretical claim collapses. A complementary check is a random-seed sweep of the pendulum experiment at the claimed $N_K = 300$: the paper's own appendix shows the method is unstable with one or two hidden layers, so repeated runs that fail to place eigenvalues on the unit circle would falsify the efficiency claim.","tokens_in":20098,"feed_emoji":"🌀","tokens_out":11560,"duration_ms":104179,"temperature":0.7,"pith_summary":"The paper sets out to cure a known failure mode of data-driven Koopman spectral analysis: the 'spectral inclusion' problem, where a computed spectrum is confined to a subset of the true spectrum and can simply miss parts of it. Its proposed method, ResKoopNet, treats the spectral residual — a data-estimated measure of how far a candidate eigenvalue–eigenfunction pair is from satisfying the Koopman eigenvalue equation — as the objective to minimize, rather than as a post-hoc filter for rejecting polluted eigenvalues as ResDMD does. A neural network learns the dictionary of observables while the Koopman matrix is updated in closed form at each step, so the optimization searches for genuinely accurate eigenpairs instead of refining a precomputed set. The payoff, if the claims hold, is practical: the full pendulum spectrum is recovered with roughly 300 observables where ResDMD needs hundreds more, the dominant pressure field of a turbulent airfoil flow emerges as the lowest-residual Koopman mode, and eigenfunctions from mouse visual-cortex recordings separate neatly by stimulus identity where three comparison methods fail.","feed_headline":"Spectral-residual loss captures Koopman spectra other methods miss","feed_subtitle":"Neural-network observables trained on spectral residual recover fuller spectra from less data.","key_machinery":"The load-bearing object is the empirical spectral residual $\\text{cres}(\\lambda, \\phi)^2 = (1/m)\\, v^*[\\Psi_Y^* \\Psi_Y - \\lambda(\\Psi_X^* \\Psi_Y)^* - \\bar\\lambda \\Psi_X^* \\Psi_Y + |\\lambda|^2 \\Psi_X^* \\Psi_X]\\, v$, which estimates, from snapshots alone, how far the pair $(\\lambda, \\phi = \\Psi v)$ deviates from the Koopman eigenvalue equation $K\\phi = \\lambda\\phi$. The paper's move is to sum this residual over all eigenpairs and minimize the sum as a loss — equivalently $\\|(\\Psi_Y - \\Psi_X K)V\\|_F^2$ — while a feedforward network re-shapes the dictionary $\\Psi(x; \\theta)$. The Koopman matrix is updated by the closed form $K = G^\\dagger A$ at each step, and the pseudospectrum, which captures continuous spectral regions, is assembled by scanning candidate complex frequencies and recording where the minimal residual falls below a threshold.","core_discovery":"On its own terms, ResKoopNet establishes that the Koopman spectrum can be learned directly by minimizing the total spectral residual $J = (1/m)\\|(\\Psi_Y - \\Psi_X K)V\\|_F^2$ over all computed eigenpairs, where $V$ is the eigenvector matrix of the Koopman matrix $K$ and $\\Psi$ is a dictionary of observables parameterized by a neural network. For a fixed dictionary the paper derives a closed-form optimal Koopman matrix, $K = G^\\dagger A$ with $G = (1/m)\\Psi_X^* \\Psi_X$ and $A = (1/m)\\Psi_X^* \\Psi_Y$, and alternates this update with gradient descent on the network parameters until $J(\\theta)$ falls below a threshold. Because a vanishing spectral residual implies $\\|K\\phi - \\lambda\\phi\\|$ is small, the computed eigenpairs inherit the convergence guarantees of the residual-based spectral theory: as data and dictionary size grow, eigenpairs and pseudospectra converge to the operator's true discrete and continuous spectra. The central discovery is therefore that spectral accuracy, not predictive accuracy, should be the training objective, and that this objective can be optimized through a dictionary learned from data rather than fixed in advance.","pith_inferences":["The paper's guarantee chain, read closely, rests on the ResDMD residual-to-spectrum theorem plus a Barron-space approximation argument; the closed-form $K = G^\\dagger A$ update is what a skeptical reader would test first, since the derivation holds $V$ fixed even though $V$ is defined by $K$.","Because the objective is purely spectral, nothing in the loss rewards prediction of future states; a natural extension is adding a small prediction-error term to test whether spectral accuracy and forecasting accuracy are compatible at fixed dictionary size, and whether spectral training improves long-horizon forecasts on chaotic systems.","The continuous-spectrum claim is demonstrated through grid-based pseudospectra; an independent check would compare ResKoopNet's $\\varepsilon$-pseudospectrum with an analytically known continuous spectrum — for instance a rotation system — to verify that the recovered band has the right location and width."],"forward_implications":["If the central claim holds, the spectral inclusion problem is resolved: optimizing eigenpairs directly against the spectral residual recovers discrete eigenvalues and continuous spectral bands that ResDMD's filter-only approach can miss, including spectra that are otherwise trivial.","Koopman-mode extraction improves on real high-dimensional data: the lowest-residual mode in the turbulent airfoil example reconstructs the dominant pressure field, a global spatial structure that kernel-based ResDMD and Hankel-DMD do not recover from the same dataset.","Substantially smaller dictionaries suffice for accurate spectra: roughly 300–350 learned observables approximate the pendulum's full unit-circle spectrum, compared with roughly 460–964 fixed basis functions for ResDMD.","Koopman eigenfunctions learned this way separate latent dynamical states: on five mouse visual-cortex recordings, eigenfunctions cluster cleanly by the six video stimuli (low Davies-Bouldin index), while Hankel-DMD, EDMD with RBF bases, and kernel ResDMD show no comparable separation."],"supporting_citations":[{"why":"Defines the spectral residual and its convergence theory, supplies the pseudospectrum framework, and provides the ResDMD baselines (pendulum, turbulence) that ResKoopNet claims to improve; the spectral-accuracy arguments inherit this paper's Theorem B.1.","marker":"[8]"},{"why":"Establishes EDMD and its data matrices $\\Psi_X, \\Psi_Y$; ResKoopNet's closed form $K = G^\\dagger A$ is explicitly contrasted with EDMD's prediction-error minimization that this reference codifies.","marker":"[44]"},{"why":"Proves EDMD convergence and frames spectral pollution, the failure mode that motivates the spectral-residual approach.","marker":"[19]"},{"why":"Provides Hankel-DMD, the time-delay-embedding benchmark used in all three experiments, together with the spectral theory of measure-preserving systems underlying the pendulum's continuous spectrum.","marker":"[1]"},{"why":"Introduces EDMD with dictionary learning, the alternating optimization paradigm ResKoopNet adapts (dictionary versus Koopman matrix), and informs the RBF basis choice in the neural-dynamics experiment.","marker":"[22]"},{"why":"Supplies the Kernel-ResDMD implementation used as a high-dimensional baseline in the turbulence and neural-dynamics experiments.","marker":"[9]"},{"why":"Gives the L2 direct approximation theorem for neural networks in Barron space on which Appendix A.4's dictionary-approximation argument rests.","marker":"[42]"},{"why":"Supplies the SGD convergence rate ($O(1/n)$ under convexity assumptions) used in Appendix A.4 to argue that the loss $J(\\theta)$ decreases with iterations.","marker":"[4]"}],"fun_headline_variants":["Minimize spectral residual to recover full Koopman spectra","Neural Koopman learning via direct spectral residual loss","Spectral residual training loss reveals hidden Koopman modes","Optimize spectral residual to compute complete Koopman eigenpairs","Train on spectral residual, not prediction, for Koopman spectra"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the closed-form Koopman matrix $K = G^\\dagger A$ truly minimizes the spectral-residual loss, even though the derivation differentiates the loss while holding the eigenvector matrix $V$ fixed even though $V$ is by definition the eigenvector matrix of $K$; if that optimality claim fails, the theoretical guarantees reduce to those of EDMD with a residual-weighted dictionary.","fun_headline_variants_meta":{"raw":{"variants":["Minimize spectral residual to recover full Koopman spectra","Neural Koopman learning via direct spectral residual loss","Spectral residual training loss reveals hidden Koopman modes","Optimize spectral residual to compute complete Koopman eigenpairs","Train on spectral residual, not prediction, for Koopman spectra"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001014,"raw_usage":{"total_tokens":4316,"prompt_tokens":1016,"completion_tokens":3300,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":632,"completion_tokens_details":{"reasoning_tokens":3215}},"tokens_in":632,"tokens_out":3300,"duration_ms":21851,"temperature":1.0,"reasoning_tokens":3215,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T22:44:29.032859+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Test the optimality claim on a tiny system with a known Koopman spectrum: compute the spectrum from ResKoopNet's fixed-$V$ closed form $K = G^\\dagger A$ and compare it with the spectrum of a $K$ obtained by minimizing the same loss jointly over $K$ and its eigenvector matrix $V$, with no fixed-$V$ shortcut; if the two spectra differ materially, the closed form does not minimize the residual and the theoretical claim collapses. A complementary check is a random-seed sweep of the pendulum experiment at the claimed $N_K = 300$: the paper's own appendix shows the method is unstable with one or two hidden layers, so repeated runs that fail to place eigenvalues on the unit circle would falsify the efficiency claim.","supporting_citations":[{"cited_title":"Data-driven approximation of the koopman operator: Extending dynamic mode decomposition","cited_arxiv_id":null,"evidence_quote":"Establishes EDMD and its data matrices $\\Psi_X, \\Psi_Y$; ResKoopNet's closed form $K = G^\\dagger A$ is explicitly contrasted with EDMD's prediction-error minimization that this reference codifies."},{"cited_title":"On convergence of extended dynamic mode decomposition to the koopman operator","cited_arxiv_id":null,"evidence_quote":"Proves EDMD convergence and frames spectral pollution, the failure mode that motivates the spectral-residual approach."},{"cited_title":"Ergodic theory, dynamic mode decomposition, and computation of spectral properties of the koopman operator","cited_arxiv_id":null,"evidence_quote":"Provides Hankel-DMD, the time-delay-embedding benchmark used in all three experiments, together with the spectral theory of measure-preserving systems underlying the pendulum's continuous spectrum."},{"cited_title":"Extended dynamic mode decomposition with dictionary learning: A data-driven adaptive spectral decomposition of the koopman operator","cited_arxiv_id":null,"evidence_quote":"Introduces EDMD with dictionary learning, the alternating optimization paradigm ResKoopNet adapts (dictionary versus Koopman matrix), and informs the RBF basis choice in the neural-dynamics experiment."},{"cited_title":"Residual dynamic mode decomposition: robust and verified koopmanism","cited_arxiv_id":null,"evidence_quote":"Supplies the Kernel-ResDMD implementation used as a high-dimensional baseline in the turbulence and neural-dynamics experiments."}],"review_version":1}