{"id":"b75e1da0-cc07-47d7-af79-1ec5134d8949","arxiv_id":"2412.16588","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":3,"one_line_summary":"A new kernel collocation method solves the PDE for the nonlinear part of Koopman principal eigenfunctions, with claimed error estimates and numerical demonstrations.","lead":"The paper proposes a kernel-based method to compute the principal eigenfunctions of the Koopman operator without building the operator itself, splitting each eigenfunction into a linear part and a nonlinear PDE correction. If valid, the approach could give more rigorous, cheaper analysis of stability and long-term dynamics in nonlinear systems.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The stability estimate collapses: a nonzero homogeneous solution to Dh=0 exists for the linear saddle, falsifying Theorem 5's injectivity claim.","rationale":"I agree with the reader that the well-posedness of the PDE (11)-(23) is the weakest link. My pass strengthens this into a direct counterexample: for a linear saddle, the homogeneous equation has nonzero global solutions with the stated boundary conditions. This is not a boundary-effect or numerical artifact; it is an algebraic identity. The proof of Theorem 5 also does not address the fact that characteristics of a first-order hyperbolic equation do not all pass through the equilibrium, so boundary data at a single point cannot determine a unique solution on a neighborhood. The RKHS assumption of Theorem 6 is a further obstruction, but the failure of Theorem 5 alone is sufficient to invalidate the central claim of rigorous error estimates. I credit the authors for sharing code and for demonstrating accurate numerical reconstructions in several examples; those results suggest the method may be useful when the domain is chosen appropriately (e.g., within a basin or along stable manifolds). However, the paper as written claims the rigorous bounds as a main contribution, and the counterexample shows those bounds are not valid under the stated hypotheses. No formal verification or other independent support repairs this. Thus the rejection stands.","tokens_in":14609,"tokens_out":9001,"duration_ms":75426,"concrete_test":"Compute, by hand or symbolically, the action of D on h(x)=x1^3 x2^2 for the linear saddle f(x)=(−x1, x2) with λ=−1: verify ∇h·f − λh = 0 and that h(0)=0, ∇h(0)=0 on Ω=(−1,1)^2. If these identities hold, Theorem 5's injectivity assertion is false and any error estimate relying on the stability inequality (24) cannot hold on this domain.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 5 is the load-bearing assertion: it claims the first-order transport operator D h = ∇h · f − λ h is a bounded bijection between Sobolev spaces on a neighborhood Ω of a hyperbolic equilibrium, with h(xe)=0 and ∇h(xe)=0. The proof is invalid: it invokes elliptic regularity for a hyperbolic operator and infers global injectivity from hyperbolicity of the equilibrium. The inference is not merely unsupported but false. For the linear saddle f(x)=(−x1, x2) on Ω=(−1,1)^2, so G=0, choose λ=−1 and any fixed w, e.g., w=(1,0). The function h(x)=x1^3 x2^2 satisfies Dh = −x1(3x1^2 x2^2) + x2(2x1^3 x2) + x1^3 x2^2 = 0, together with h(0)=0 and ∇h(0)=0. Hence ker(D) contains a nonzero smooth function, contradicting Step 1 of Theorem 5. Consequently the stability bound (24) and the convergence estimate of Theorem 6, which both depend on a bounded inverse D^{-1}, are not established for arbitrary bounded domains Ω. The numerical examples may succeed on domains where the PDE is well-posed, but the paper's rigorous guarantees do not follow from the stated assumptions.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a kernel collocation method for approximating the principal Koopman eigenfunctions of a nonlinear system near a hyperbolic equilibrium. The eigenfunction is decomposed as φ_λ(x)=w^T x+h(x), where w is a left eigenvector of the linearization at the equilibrium and h solves the linear transport PDE (22) with boundary conditions h(x_e)=0 and ∇h(x_e)=0. The PDE is discretized by collocation and solved in a reproducing kernel Hilbert space via the representer formula (17)-(19). The paper claims rigorous error estimates: Theorem 5 asserts that the operator Dh=∇h·f−λh is a bounded bijection between Sobolev spaces on a bounded neighborhood Ω, yielding the stability bound (24), and Theorem 6 derives a fill-distance convergence rate from that bound. Numerical experiments in Sections 5.1–5.4 demonstrate recovery of known analytical eigenfunctions and level sets for the Duffing and gradient systems.","tokens_in":14971,"tokens_out":6393,"duration_ms":57091,"significance":"If Theorems 5 and 6 were valid, the paper would make a useful contribution: it would compute Koopman eigenfunctions without constructing the Koopman operator, with convergence guarantees and an explicit mechanism to avoid spectral pollution. The numerical demonstrations in Sections 5.1 and 5.2 show that the collocation approach can recover the nonlinear part h when the true eigenfunction is known. However, the advertised theoretical guarantees are the paper's central claim, and they are not established: the main well-posedness theorem is false as stated, and the convergence theorem depends on it. The numerical experiments are suggestive but do not compensate for the failure of the theoretical framework.","major_comments":[{"comment":"The proof of Theorem 5 is invalid and its statement is false. Step 1 infers injectivity of D from hyperbolicity of the equilibrium, but hyperbolicity does not imply injectivity of a first-order transport operator on a bounded domain with only point constraints at the equilibrium. For the linear saddle f(x)=(-x1,x2), λ=-1, w=(1,0), and Ω=(-1,1)^2, the function h(x)=x1^3 x2^2 satisfies Dh=0, h(0)=0, and ∇h(0)=0, so ker(D) contains a nonzero smooth function. Step 2 invokes elliptic regularity for a first-order hyperbolic operator, which is not applicable. Consequently, the stability bound (24) and all results depending on the bounded invertibility of D, including Theorem 6, are not established.","section":"Section 4, Theorem 5 (Eqs. (22)-(24))"},{"comment":"The convergence estimate in Theorem 6 assumes that the exact solution h_λ lies in the RKHS H of the chosen kernel and that H embeds continuously into W^{m+1}_2(Ω). This is a strong regularity condition, analogous to analyticity for Gaussian kernels, and it is not verified or quantified anywhere in the paper. Moreover, the proof of Theorem 6 uses the stability bound (24) from Theorem 5, which is false as shown above. Thus the claimed convergence rate is not proven even under the additional RKHS assumptions.","section":"Section 4, Theorem 6"},{"comment":"The numerical experiments rely on hand-tuned kernel bandwidths σ1 and σ2 (Sections 5.1, 5.2, 5.3) and do not specify the regularization parameter η in Eq. (19), although Theorem 6 assumes η=0. The experiments only test cases with known analytical eigenfunctions; they cannot detect the failure of the well-posedness assumption on general bounded domains. The method may work on the particular examples, but the experiments do not substantiate the general theoretical claims made in Sections 4.","section":"Section 5, numerical experiments"}],"minor_comments":[{"comment":"The notation F(z_i)_j conflicts with the function space F introduced in Section 2.1; the vector field f should be used consistently throughout.","section":"Section 3.2, Eq. (18)"},{"comment":"The statement 'k spans H: H = span{k_x | x ∈ X}' should refer to the closure of the span, and the quantifier in item i should be 'for all x ∈ X' rather than 'for all x ∈ H'.","section":"Section 3.1, Definition 4"},{"comment":"The theorem says the constants do not depend on λ or f, but the displayed bound contains the factor (∥f∥_{W^m_2(Ω,R^d)} + |λ|); this wording is misleading and should be clarified.","section":"Section 4, Theorem 6"},{"comment":"The text uses 'principle eigenfunctions' instead of 'principal eigenfunctions'; the terminology should be unified.","section":"Section 2.2"},{"comment":"The distributional pairings K(z, φ~) and K(φ~, φ~) involve products of the kernel with Dirac deltas and derivatives; the paper should specify the precise function-space setting in which these pairings are well defined for Gaussian kernels.","section":"Section 3.2, Eqs. (20)-(21)"}],"recommendation":"reject","confidential_remarks":"The paper contains a promising numerical idea, and the examples with known eigenfunctions are encouraging. However, the central theoretical claim is false as stated: a simple counterexample to Theorem 5 shows that the stability bound and the convergence theorem collapse. Repairing this would require replacing the well-posedness argument with a substantially different approach, such as a careful method-of-characteristics analysis with appropriate boundary conditions on the domain, and it would also require verifying the RKHS regularity assumption. Given that the rigorous guarantees are the main advertised contribution, I cannot recommend acceptance in the current form."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Punchline: this is a usable numerical recipe, but the advertised error analysis is not just unproven—Theorem 5 is false as stated. For the linear saddle f(x)=(-x1,x2) on Ω=(-1,1)^2, take λ=-1, w=(1,0), and h(x)=x1^3 x2^2. Then Dh=0, h(0)=0, ∇h(0)=0, so D has a nonzero kernel element. The proof's Step 1 (injectivity from hyperbolicity) is wrong, and Step 2 invokes elliptic regularity for a first-order transport operator. The stability bound (24) and therefore Theorem 6 rest on the bounded inverse and collapse. The stress-test counterexample is correct; I checked it.\n\nWhat is actually new: the combination of the Deka et al. linear/nonlinear decomposition with RKHS collocation to solve the PDE for h. The ingredients are known, but the packaging is a concrete algorithm and the code is public. The experiments with known analytic solutions in Sections 5.1 and 5.2 genuinely show the method recovering h with small error; the relative-error plots are honest about the zero-crossings. The Duffing and 3D examples are illustrative. Conditional on Theorem 5, the convergence proof is a reasonable use of the Narcowich-Ward-Wendland zero lemma.\n\nThe other soft spots: Theorem 6 assumes the exact solution lies in the RKHS of the chosen kernel, a strong condition not stated in the introduction. The kernel bandwidths are hand-tuned per example and η is not specified, so the theoretical rates are not what the code actually uses. Also, the 'no spectral pollution' claim is earned by taking λ and w as inputs from the linearization; that is a legitimate design choice, but it means the method constructs known eigenvalues rather than discovering them. The abstract oversells this.\n\nWho this is for: readers who want a concrete kernel-collocation implementation for principal Koopman eigenfunctions from a known model. It is not a source for rigorous error estimates. The paper deserves a serious referee because the numerical idea is plausible and the flaw is specific and fixable: prove injectivity on the actual bounded domain (or restrict to domains where the transport problem is well-posed) and state the RKHS regularity assumption honestly. As it stands, I would not cite the error bounds, and the paper needs major revision before acceptance.","headline":"The numerical method works on the tested examples, but the central stability theorem is false as stated, so the paper's rigor claim collapses.","tokens_in":15439,"tokens_out":3589,"would_cite":false,"duration_ms":31311,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["37M10","46E22","47B33","65P99"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper claims that the principal eigenfunctions of the Koopman operator, with eigenvalues inherited from the linearization at a hyperbolic equilibrium, can be approximated directly by solving a linear PDE with kernel methods, with…","keywords":["Koopman operator","principal eigenfunctions","kernel methods","reproducing kernel Hilbert space","spectral pollution","linearization","PDE approximation","error estimates"],"falsifier":"A direct way to test the central claim is to pick a smooth one-dimensional system such as $\\dot x=2x+x^2$ with $\\lambda=2$, and check numerically whether the kernel approximation of $h$ converges at the rate predicted by Theorem 6 on a bounded domain. If the exact $h_\\lambda$ is chosen outside the Gaussian RKHS, for example a $C^\\infty$ function with compact support, the assumption $\\|h_\\lambda\\|_K<\\infty$ fails and the stated right-hand side is not finite, so the predicted rate should not hold; a reader can test whether the method still converges and at what rate.","tokens_in":14443,"feed_emoji":"🧮","tokens_out":9833,"duration_ms":103299,"temperature":0.7,"pith_summary":"The paper aims to turn the computation of principal Koopman eigenfunctions into a kernel-regression problem that never forms the Koopman operator itself. It claims that for a nonlinear system near a hyperbolic equilibrium, each principal eigenfunction decomposes as a linear part, the left eigenvector of the linearization, plus a purely nonlinear part solving the linear PDE $\\nabla h(x)\\cdot f(x)-\\lambda h(x)=-w^\\top G(x)$. The paper argues that solving this PDE in a reproducing kernel Hilbert space yields approximations with explicit convergence rates and avoids the spurious eigenvalues that plague data-driven approximations. A sympathetic reader would care because these eigenfunctions encode stability boundaries, stable and unstable manifolds, and optimal-control value functions, so a direct and provably convergent route to them would make Koopman analysis more reliable.","feed_headline":"Koopman eigenfunctions from one linear PDE, no spectral pollution","feed_subtitle":"Linear part is a left eigenvector; kernel regression solves the nonlinear PDE with proven error bounds.","key_machinery":"The load-bearing object is the decomposition $\\phi_\\lambda(x)=w^\\top x+h(x)$ inherited from the path-integral construction, which turns the nonlinear eigenvalue problem $\\nabla\\phi\\cdot f=\\lambda\\phi$ into the linear PDE $\\nabla h\\cdot f-\\lambda h=-w^\\top G(x)$ with boundary conditions $h(x_e)=0$ and $\\nabla h(x_e)=0$. The second ingredient is the RKHS representer formula $h^*(z)=K(z,\\tilde\\phi)(K(\\tilde\\phi,\\tilde\\phi)+\\eta I)^{-1}Y$, where the constraints include evaluation of the PDE residual at collocation points and vanishing of $h$ and $\\nabla h$ at the equilibrium. The matrix $K(\\tilde\\phi,\\tilde\\phi)$ has entries built by applying the PDE operator to the kernel in each variable, so the method never discretizes the state-space dynamics into a finite matrix approximating the Koopman operator; it solves a constrained least-norm interpolation problem.","core_discovery":"On the paper's own terms, the central claim is that the nonlinear component $h$ of a principal Koopman eigenfunction can be recovered as the minimal-norm RKHS interpolant that satisfies the PDE and the boundary conditions $h(x_e)=0$, $\\nabla h(x_e)=0$, and that this recovery is stable: Theorem 5 asserts the PDE operator $D=\\nabla h\\cdot f-\\lambda h$ is a bounded bijection on Sobolev spaces with those boundary conditions, giving a stability bound, and Theorem 6 asserts that with sufficiently dense collocation points the $L^p$ error between the true $h_\\lambda$ and the kernel approximation decays like $\\rho^{m-d(1/2-1/q)_+}$, provided the true solution lies in the kernel's RKHS. The paper further claims this route is free of spectral pollution because it targets eigenfunctions one at a time rather than approximating the infinite-dimensional operator by a matrix.","pith_inferences":["A purely data-driven version would need to infer $f$, or at least the PDE residuals, from trajectories; the present method takes $f$ as known, so combining it with sparse identification of $f$ is a natural next step.","The paper treats simple real eigenvalues, so extending the decomposition to complex conjugate eigenvalue pairs would cover stable and unstable spiral manifolds and is the immediate formal follow-up.","Because the method computes each eigenfunction individually, it could be parallelized across eigenvalues and does not require storing a large approximating matrix, which matters for high-dimensional systems.","The RKHS-membership assumption on the exact solution is a regularity condition that could be checked or enforced by choosing a kernel matched to the estimated smoothness of $h_\\lambda$."],"forward_implications":["Given the vector field and a left eigenvector of the linearization, one can construct the corresponding principal eigenfunction by kernel regression on collocation points, without forming an extended dynamic mode decomposition matrix.","The error estimate gives a convergence rate in terms of the fill distance, the smoothness $m$ of the vector field, and the RKHS norm of the exact solution, so the method comes with a quantitative certificate rather than a heuristic.","Because eigenvalues are fixed from the linearization and each eigenfunction is computed individually, spurious eigenvalues arising from discretizing the full Koopman operator are bypassed.","The same framework can supply the principal eigenfunctions needed for stability-boundary identification and optimal control, since those applications only require the eigenfunctions themselves."],"supporting_citations":[{"why":"Supplies the linear/nonlinear decomposition $\\phi=w^\\top x+h$ and the PDE for $h$ that the method solves.","marker":"[26]"},{"why":"Defines the principal eigenfunctions and establishes that their eigenvalues match the linearization at the equilibrium.","marker":"[60]"},{"why":"Provides the Gaussian-process conditional-expectation representer formula used to solve the constrained RKHS problem.","marker":"[66]"},{"why":"Supplies the scattered-data zero lemma that converts RKHS interpolation error into the $L^q$ error estimate in Theorem 6.","marker":"[67]"},{"why":"Is the EDMD-style operator approximation whose spurious eigenvalues motivate the direct eigenfunction approach.","marker":"[28]"},{"why":"Introduces the Koopman operator whose principal eigenfunctions are the objects being approximated.","marker":"[19]"}],"fun_headline_variants":["Koopman eigenfunctions from a single PDE, no spectral pollution","No spurious eigenvalues: Koopman eigenfunctions via kernel PDE","Kernel method gets Koopman eigenfunctions without matrix","Eigenfunctions from one PDE: kernel method, no spurious spectra","Solve one PDE for Koopman eigenfunctions, skip the matrix"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the PDE operator $D=\\nabla h\\cdot f-\\lambda h$ with boundary conditions $h(x_e)=0$ and $\\nabla h(x_e)=0$ is a bounded bijection on the Sobolev spaces used for the error estimates; the paper's proof of this point invokes elliptic regularity for a first-order hyperbolic operator and infers injectivity from hyperbolicity without a valid argument.","fun_headline_variants_meta":{"raw":{"variants":["Koopman eigenfunctions from a single PDE, no spectral pollution","No spurious eigenvalues: Koopman eigenfunctions via kernel PDE","Kernel method gets Koopman eigenfunctions without matrix","Eigenfunctions from one PDE: kernel method, no spurious spectra","Solve one PDE for Koopman eigenfunctions, skip the matrix"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000631,"raw_usage":{"total_tokens":2892,"prompt_tokens":898,"completion_tokens":1994,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":514,"completion_tokens_details":{"reasoning_tokens":1904}},"tokens_in":514,"tokens_out":1994,"duration_ms":12010,"temperature":1.0,"reasoning_tokens":1904,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T10:26:52.818371+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A direct way to test the central claim is to pick a smooth one-dimensional system such as $\\dot x=2x+x^2$ with $\\lambda=2$, and check numerically whether the kernel approximation of $h$ converges at the rate predicted by Theorem 6 on a bounded domain. If the exact $h_\\lambda$ is chosen outside the Gaussian RKHS, for example a $C^\\infty$ function with compact support, the assumption $\\|h_\\lambda\\|_K<\\infty$ fails and the stated right-hand side is not finite, so the predicted rate should not hold; a reader can test whether the method still converges and at what rate.","supporting_citations":[{"cited_title":"Path-integral formula for com- puting koopman eigenfunctions,","cited_arxiv_id":null,"evidence_quote":"Supplies the linear/nonlinear decomposition $\\phi=w^\\top x+h$ and the PDE for $h$ that the method solves."},{"cited_title":"Spectrum of the koopman operator, spectral expansions in functional spaces, and state-space geometry,","cited_arxiv_id":null,"evidence_quote":"Defines the principal eigenfunctions and establishes that their eigenvalues match the linearization at the equilibrium."},{"cited_title":"Owhadi and C","cited_arxiv_id":null,"evidence_quote":"Provides the Gaussian-process conditional-expectation representer formula used to solve the constrained RKHS problem."},{"cited_title":"Sobolev bounds on functions with scattered zeros, with applications to radial basis function surface fitting,","cited_arxiv_id":null,"evidence_quote":"Supplies the scattered-data zero lemma that converts RKHS interpolation error into the $L^q$ error estimate in Theorem 6."},{"cited_title":"A data–driven approx- imation of the koopman operator: Extending dynamic mode decomposition,","cited_arxiv_id":null,"evidence_quote":"Is the EDMD-style operator approximation whose spurious eigenvalues motivate the direct eigenfunction approach."},{"cited_title":"Dynamical systems of continuous spec- tra,","cited_arxiv_id":null,"evidence_quote":"Introduces the Koopman operator whose principal eigenfunctions are the objects being approximated."}],"review_version":1}