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Kernel Methods for the Approximation of the Eigenfunctions of the Koopman Operator

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Pith's one-line read 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…

desk verdict The numerical method works on the tested examples, but the central stability theorem is false as stated, so the paper's rigor claim collapses. read the letter →

arxiv 2412.16588 v1 pith:OIAWXAVM submitted 2024-12-21 math.DS cs.NAmath.NAstat.ML

Jonghyeon Lee , Boumediene Hamzi , Boya Hou , Houman Owhadi , Gabriele Santin , Umesh Vaidya This is my paper · ORCID
classification math.DScs.NAmath.NAstat.ML MSC 37M1046E2247B3365P99
keywords KoopmanoperatorprincipaleigenfunctionskernelmethodsreproducingHilbertspacespectralpollutionlinearizationPDEapproximationerrorestimates
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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 $h_\lambda$.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

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.

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 (3)
  1. [Section 4, Theorem 5 (Eqs. (22)-(24))] 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.
  2. [Section 4, Theorem 6] 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.
  3. [Section 5, numerical experiments] 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.
minor comments (5)
  1. [Section 3.2, Eq. (18)] 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.
  2. [Section 3.1, Definition 4] 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'.
  3. [Section 4, Theorem 6] 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.
  4. [Section 2.2] The text uses 'principle eigenfunctions' instead of 'principal eigenfunctions'; the terminology should be unified.
  5. [Section 3.2, Eqs. (20)-(21)] 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.

Circularity Check

0 steps flagged · score 2.0 of 10

No direct circularity: the core PDE-solving pipeline is self-contained, though the paper relies on self-citations to coauthor work and uses hand-tuned kernel bandwidths.

full rationale

The derivation chain is: (i) define Koopman eigenfunctions by ∂φ/∂x·f = λφ; (ii) decompose φ = wᵀx + h with w a left eigenvector of E = ∂f/∂x(0) and h the nonlinear part; (iii) substitute to obtain the linear PDE ∇h·f − λh = −wᵀG; (iv) solve this PDE by RKHS regression with the representer formula. None of these steps fits the target eigenfunction values or imports the conclusion as an input. The w and λ are obtained from the linearization and G from f, while the collocation constraints in (17) use only this known data, so the approximation is not circular by construction. The cited representer formula (Theorems 3–4, from Owhadi–Scovel, a coauthor text) is standard and is sketched in the paper itself, so the citation is not load-bearing in a circular sense; the decomposition 'following [26]' is elementary algebra verifiable from (8)–(11) within the paper. The main weaknesses are rigor issues, not circularity: Theorem 5's proof claims elliptic regularity and injectivity for the first-order transport operator D, which is mathematically unsupported and in fact false for simple linear saddles on bounded domains; Theorem 6 additionally assumes h_λ lies in the RKHS H, a strong smoothness condition. These are correctness concerns, not cases where a quantity is fitted and then renamed as a prediction. The hand-tuned bandwidths σ in Section 5 affect numerical accuracy but are not hidden fits of the eigenfunction values used as predictions. No circular step can be exhibited, so the score is low.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The method relies on standard RKHS machinery plus several strong domain assumptions. The critical unproven assumptions are the well-posedness of the transport PDE with point boundary conditions and the membership of the true solution in the RKHS. These are loaded into Theorems 5 and 6. No new physical entities are introduced.

free parameters (3)
  • Gaussian kernel bandwidths σ1, σ2 = σ1=σ2=2 (Ex5.1 λ1); σ1=2, σ2=3 (Ex5.1 λ2); σ1=σ2=3 (Ex5.2 λ1); σ1=σ2=7 (Ex5.2 λ2); σ1=σ2=15 (Ex5.3); σ=1.1 (Ex5.4)
    Chosen by hand per example and per eigenvalue; no selection rule or sensitivity analysis is provided.
  • Regularization constant η = not specified
    Introduced in (19) to stabilize matrix inversion; no value or selection criterion is given.
  • Number of collocation points and domain = 3600, 2500, 2500, 3379 over specified domains
    Chosen per example; no convergence study varying N.
assumptions (5)
  • domain assumption The vector field f is C^m and the equilibrium is hyperbolic.
    Assumed in Theorem 5 and throughout; needed for existence of principal eigenfunctions with the linearization eigenvalues.
  • domain assumption The principal eigenfunctions exist on the domain Ω and can be decomposed as w^T x + h(x).
    Taken from [60, 26]; the paper does not prove the decomposition on bounded domains.
  • ad hoc to paper The operator D = ∇h·f - λh is a bounded bijection between W^{m+1}_2(Ω) and L_q(Ω).
    This is Theorem 5's conclusion, but its proof is invalid; the paper's error estimates depend on it.
  • ad hoc to paper The true nonlinear part h_λ belongs to the RKHS H of the chosen kernel, and H embeds continuously into W^{m+1}_2(Ω).
    Required for Theorem 6's error bound; for Gaussian kernels H contains analytic functions, while h_λ from a C^m vector field need not be analytic.
  • standard math Standard RKHS representer theorems and Sobolev embedding theorems.
    Used without proof; standard.

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Pith. "Pith review of Kernel Methods for the Approximation of the Eigenfunctions of the Koopman Operator." pith.science (2026). https://pith.science/paper/OIAWXAVM

@misc{pith2026241216588,
  author       = {Pith},
  title        = {Pith review of: Kernel Methods for the Approximation of the Eigenfunctions of the Koopman Operator},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OIAWXAVM}},
  note         = {Machine review of arXiv:2412.16588}
}
read the original abstract

The Koopman operator provides a linear framework to study nonlinear dynamical systems. Its spectra offer valuable insights into system dynamics, but the operator can exhibit both discrete and continuous spectra, complicating direct computations. In this paper, we introduce a kernel-based method to construct the principal eigenfunctions of the Koopman operator without explicitly computing the operator itself. These principal eigenfunctions are associated with the equilibrium dynamics, and their eigenvalues match those of the linearization of the nonlinear system at the equilibrium point. We exploit the structure of the principal eigenfunctions by decomposing them into linear and nonlinear components. The linear part corresponds to the left eigenvector of the system's linearization at the equilibrium, while the nonlinear part is obtained by solving a partial differential equation (PDE) using kernel methods. Our approach avoids common issues such as spectral pollution and spurious eigenvalues, which can arise in previous methods. We demonstrate the effectiveness of our algorithm through numerical examples.

Figures

Figures reproduced from arXiv: 2412.16588 by the authors.

Figure 1
Figure 1. Learned ϕ ∗ λ1 (left), true ϕλ1 (center) and relative error (right) [PITH_FULL_IMAGE:figures/full_fig_p013_1.png] view at source ↗
Figure 2
Figure 2. Learned ϕ ∗ λ2 (left), true ϕλ2 (center) and relative error (right) 5.2 Second Analytical Example x˙ =     (7.5x 2 2+5.0)(x 3 1+x1+sin(x2))+(−x1+x 3 2+2x2) cos(x2) 9x 2 1 x 2 2+6x 2 1+3x 2 2+cos(x2)+2 2.5x 3 1+2.5x1−(3x 2 1+1)(−x1+x 3 2+2x2)+2.5 sin(x2) 9x 2 1 x 2 2+6x 2 1+3x 2 2+cos(x2)+2     . (30) The vector field for this system is smooth everywhere on R 2 , with a saddle point at the origin. One can ver… view at source ↗
Figure 3
Figure 3. Learned ϕ ∗ λ1 (left), true ϕλ1 (center) and relative error (right) [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: Learned ϕ ∗ λ2 (left), true ϕλ2 (center) and relative error (right) 5.3 The Duffing Oscillator Consider the unforced Duffing oscillator, described by x˙ 1 = x2 x˙ 2 = −δx2 − x1(β + αx2 1 ) (31) with δ = 0.5, β = −1, and α = 1, where z ∈ R and ˙z ∈ R are the scalar posi…
Figure 5
Figure 5. Figure 5: Learned ϕ ∗ λ1 where the potential function V is given by V (x) = x ⊤P x + e −(x1−x2) 2 , (33) where P =   0.2 0.1 0.05 0.1 0.3 0.05 0.05 0.05 0.2   is a positive definite matrix. The system admits an unstable equilibrium at the origin and two stable minima at x = …
Figure 6
Figure 6. Figure 6: Learned ϕ ∗ λ1 15 [PITH_FULL_IMAGE:figures/full_fig_p015_6.png]

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Forward citations

Cited by 1 Pith paper

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Pith tools

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