REVIEW 3 major objections 6 minor 51 references
ResKoopNet: Learning Koopman Representations for Complex Dynamics with Spectral Residuals
T0 review · 3 major / 6 minor · reviewed 2026-08-10 · deepseek-v4-flash
Pith's one-line read 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…
desk verdict Novel residual-based dictionary learning for Koopman spectra, but the theoretical guarantees are unsupported and the closed-form update is just EDMD. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
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.
What would settle it
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.
Extended reading notes
Core claim
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.
Load-bearing premise
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.
Editorial extensions
If this is right
- 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.
Reading between the lines
- 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.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
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.
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 (3)
- [Section 3.2, Eqs. (3.5)-(3.6) and Appendix A.3] 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.
- [Appendix A.4, Assumption A.4(b) and the SGD convergence argument] 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.
- [Appendix A.4, paragraph on spectral convergence and data-dependent dictionaries] 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.
minor comments (6)
- [Abstract and Section 1] 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 3.2, Eq. (3.6) and Algorithm 1] 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.
- [Appendix A.1] 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 4.3] 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 4.2 and Appendix A.8.2] 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.
- [Appendix A.4, Assumption A.4(a)] 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.
Circularity Check
The closed-form Koopman matrix K = G†A is, for fixed dictionary, the EDMD solution by construction; the paper's claim that it is derived from spectral-residual rather than prediction-error minimization is a renaming of a known result. The dictionary-learning loss and empirical benchmarks remain independent, so circularity is partial.
-
renaming known result
[Section 3.2, Eqs. (3.5)–(3.6) and the paragraph following Remark 3.2]
"minimizing J is also equivalent to the following minimization problem: min_˜K J = min_˜K 1/m ∥(ΨY − ΨX ˜K)V ∥²_F , (3.5) where each column of the matrix V is an (right) eigenvector vi of the matrix ˜K. Thus, with dictionary Ψ fixed, we can obtain a closed form for the optimal Koopman matrix ˜K as following ˜K = G†A, (3.6)... While they share a similar matrix expression, this expression is derived from minimizing the spectral residual in Eq. (3.3) rather than the prediction error that EDMD minimizes."
For fixed Ψ, V is the eigenvector matrix of K and is invertible. The map K ↦ KV is a bijection, so min_K ∥(ΨY − ΨX K)V∥_F is the same least-squares problem as min_K ∥ΨY − ΨX K∥_F; both have solution K = Ψ_X† Ψ_Y = G†A. Thus Eq. (3.6) is exactly the EDMD Koopman matrix by construction, and the asserted distinction between 'derived from minimizing the spectral residual' and 'prediction error that EDMD minimizes' is not realized for the closed-form K. The claimed theoretical foundation for the Koopman matrix therefore reduces to renaming a known result; only the data-dependent dictionary loss J(θ) provides independent content.
full rationale
The paper's main new algorithmic content is the dictionary-learning loop that minimizes the empirical spectral residual J(θ) over neural-network parameters, and the experimental comparisons use external ground truth (unit-circle spectrum, airfoil pressure field, visual-stimulus trial labels), so the overall method is not self-validating. However, one load-bearing step in the derivation chain is circular in the sense of renaming a known result: the closed-form optimal Koopman matrix in Eq. (3.6) coincides with EDMD's K = G†A because the eigenvector matrix V is invertible, making the projected residual minimization equivalent to EDMD's prediction-error minimization for a fixed dictionary. The paper's assertion that the shared expression arises from a fundamentally different theoretical foundation is therefore not supported by its own equations. The residual is also used both as the training loss and as the mode-ordering/evaluation criterion, but since the headline claims are checked against ground-truth physical and biological patterns, this is a metric-alignment caveat rather than a circular prediction. The self-citation [46] is not load-bearing. Appendix A.4's admission that J(θ) is non-convex despite Assumption A.4(b) using strong convexity is a gap in the claimed theoretical guarantees, but it is a correctness risk, not a circularity. Overall, the central contribution retains independent empirical content, so the paper is only partially circular.
Assumptions & free parameters
free parameters (5)
- Number of dictionary observables NK =
25-350 depending on experiment
- Neural network architecture (hidden layers, neurons) =
e.g., 3x300, 3x200
- Regularization parameter sigma and loss threshold epsilon =
not specified numerically
- SVD truncation ranks =
150 (turbulence), 24/300 (neural)
- Hankel-DMD time delay (baseline) =
150, 50, or 5 depending on experiment
assumptions (5)
- standard math The Koopman operator K on L2(Omega, mu) has a spectral decomposition into point and continuous spectra (Section 2).
- domain assumption The optimal dictionary functions spanning the Koopman invariant subspace lie in Barron space (Assumption A.4(a)).
- ad hoc to paper The loss J(theta) is strongly convex and Lipschitz continuous in theta (Assumption A.4(b)).
- ad hoc to paper K = G†A minimizes the spectral-residual loss J even though the eigenvector matrix V depends on K.
- standard math Empirical Gram matrices converge to L2 inner products as m goes to infinity (Eq. 3.2).
Cite this review
Pith. "Pith review of ResKoopNet: Learning Koopman Representations for Complex Dynamics with Spectral Residuals." pith.science (2026). https://pith.science/paper/JZHXKSAF
@misc{pith2026250100701,
author = {Pith},
title = {Pith review of: ResKoopNet: Learning Koopman Representations for Complex Dynamics with Spectral Residuals},
year = {2026},
howpublished = {\url{https://pith.science/paper/JZHXKSAF}},
note = {Machine review of arXiv:2501.00701}
}
read the original abstract
Analyzing the long-term behavior of high-dimensional nonlinear dynamical systems remains a significant challenge. While the Koopman operator framework provides a powerful global linearization tool, current methods for approximating its spectral components often face theoretical limitations and depend on predefined dictionaries. Residual Dynamic Mode Decomposition (ResDMD) advanced the field by introducing the \emph{spectral residual} to assess Koopman operator approximation accuracy; however, its approach of only filtering precomputed spectra prevents the discovery of the operator's complete spectral information, a limitation known as the `spectral inclusion' problem. We introduce ResKoopNet (Residual-based Koopman-learning Network), a novel method that directly addresses this by explicitly minimizing the \emph{spectral residual} to compute Koopman eigenpairs. This enables the identification of a more precise and complete Koopman operator spectrum. Using neural networks, our approach provides theoretical guarantees while maintaining computational adaptability. Experiments on a variety of physical and biological systems show that ResKoopNet achieves more accurate spectral approximations than existing methods, particularly for high-dimensional systems and those with continuous spectra, which demonstrates its effectiveness as a tool for analyzing complex dynamical systems.
Figures
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Reference graph
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Therefore, we cannot make the dictionary size consistent with the ResKoopNet example
For Kernel ResDMD, the dictionary size is theoretically determined to be the number of snapshots. Therefore, we cannot make the dictionary size consistent with the ResKoopNet example. Based on the above justifications, we believe our choices of dictionary sizes are reasonable ...
Reviewed August 10, 2026 · model on record in the stance chip above.
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