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REVIEW 4 major objections 4 minor 25 references

Extensions of the Path-integral formula for computation of Koopman eigenfunctions

T0 review · 4 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read This paper claims that finite-time path-integral formulas, with boundary values obtained either from local trajectory data or from a vector-field transformation, compute principal Koopman eigenfunctions for dynamical systems with saddle…

desk verdict Finite-time path-integral extension for saddle equilibria has one solid method (local EDMD) and one overclaimed method (vector-field transformation) whose core eigenfunction-closeness statement is not actually proven. read the letter →

arxiv 2411.16605 v1 pith:MXAEG2BU submitted 2024-11-25 eess.SY cs.SYmath.DS

Shankar A. Deka , Umesh Vaidya This is my paper · ORCID
classification eess.SYcs.SYmath.DS
keywords Koopmanoperatoreigenfunctionspath-integralformulasaddleequilibriumExtendedDynamicModeDecompositionHamiltoniansystemsoptimalcontrolfinite-timehorizon
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

Koopman eigenfunctions turn nonlinear dynamics into linear coordinates, and the principal ones (those tied to the equilibrium's linearization) encode manifolds and stability boundaries. The paper's earlier path-integral formula computed these eigenfunctions as an infinite-time integral, which works only for stable or anti-stable equilibria. This paper claims that for a saddle equilibrium the integral can be truncated to finite time, provided one knows the eigenfunction's nonlinear part on a boundary that every trajectory reaches. It gives two ways to supply that boundary data, proves an error bound for the first, and demonstrates both on numerical examples, including a Hamiltonian optimal-control problem.

What carries the argument

The engine is the path-integral solution to the Koopman-generator PDE $\nabla_x\phi_\lambda\cdot f=\lambda\phi_\lambda$, namely the recurrence $h_\lambda(x)=e^{-\lambda t}h_\lambda(s_t(x))+\int_0^t e^{-\lambda\tau}w_\lambda^\top f_n(s_\tau(x))\,d\tau$. The extension treats the terminal value $e^{-\lambda t}h_\lambda(s_t(x))$ as a boundary condition rather than a term to be killed in the limit $t\to\infty$. Method A feeds it with a local EDMD estimate on a boundary set $S$ and Theorem 4 bounds the resulting error; Method B feeds it with an approximately zero value by using the blending transformation (18), whose smooth step $\sigma(\|x\|-r)$ makes the field linear outside the disk and nonlinear inside, so Proposition 2 justifies the zero boundary condition at a sphere far from the origin.

What would settle it

Take a planar saddle system with a known closed-form principal eigenfunction, build the transformed field (18) with increasing steepness $a$, and compute the transformed system's true eigenfunction inside $\|x\|<r-\epsilon_2$ by a converged independent method: if the gap to the original eigenfunction stays large while the residual (19) falls below $\epsilon_1$, the Method B claim fails.

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Extended reading notes

Core claim

The central claim is that the principal Koopman eigenfunction $\phi_\lambda(x)=w_\lambda^\top x+h_\lambda(x)$ of a saddle-point system can be approximated by the finite-time path-integral formula $h_\lambda(x)=e^{-\lambda t}h_\lambda(s_t(x))+\int_0^t e^{-\lambda\tau}w_\lambda^\top f_n(s_\tau(x))\,d\tau$, once the terminal term $h_\lambda(s_t(x))$ is estimated on a reachable boundary set. Theorem 4 shows that an estimate with error $\epsilon_S$ on the boundary produces an eigenfunction error no larger than $\epsilon_S e^{-\mathrm{Re}(\lambda)\bar t(x)}$ on the domain. Method A obtains the boundary estimate with local EDMD from trajectory data in a thin neighborhood of the boundary, while Method B replaces the vector field by a smoothly blended field $\tilde f(x)=f(x)+\sigma(\|x\|-r)(Ax-f(x))$, making the nonlinear part of the transformed eigenfunction approximately zero on a sphere $\|x\|=R>r$ (Proposition 2) and keeping the eigenfunction close to the original inside $\|x\|<r$ (Proposition 1).

Load-bearing premise

Method B's guarantee rests on the assumption that a small residual in the eigenfunction PDE for the transformed vector field forces the transformed system's eigenfunction to be close to the original eigenfunction inside $\|x\|<r$; the paper's proof bounds the residual, not that eigenfunction gap.

Editorial extensions

If this is right

  • For systems with saddle equilibria, principal Koopman eigenfunctions -- and with them stable and unstable manifolds, stability boundaries, and level sets used in optimal control -- become computable by finite-horizon path integrals instead of infinite-horizon ones.
  • The eigenfunction value at a point is obtained pointwise from a trajectory integral, so global approximation no longer requires global EDMD; in Method A the data need only cover a small neighborhood of the boundary set.
  • The blending field in Method B converts the saddle-point problem into a finite-exit-time problem, so trajectories only need to be integrated until they cross the sphere $\|x\|=R$.
  • Relaxing the spectral-distribution condition (10) and the stable/anti-stable restriction enlarges the class of nonlinear systems to which the closed-form path-integral approach applies, as demonstrated in the numerical example with eigenvalues $-1$ and $2.5$.

Reading between the lines

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

  • Extending beyond the paper, the same boundary-condition idea should apply to other invariant structures -- limit cycles, for instance -- because the only requirement is a hypersurface that forward orbits cross in finite time and on which the nonlinear part of the eigenfunction can be estimated or zeroed.
  • The two methods are complementary rather than exclusive: a natural next step is to use the transformed vector field to generate a crude boundary estimate and then refine it with local EDMD, which could reduce both data demand and residual error.
  • A sharper proof of Proposition 1 would quantify the gap between the original and transformed eigenfunctions in terms of the PDE residual, for example through a smallness condition on the blending layer, rather than only bounding the residual itself.
  • In practice the transformation's transition layer has width roughly $1/a$; numerical integrators will need adaptive stepping there, an implementation detail the paper does not examine.
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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

4 major / 4 minor

Summary. The paper extends the authors' earlier path-integral formula for principal Koopman eigenfunctions from systems with stable or anti-stable equilibria to systems with saddle-type equilibria. Two finite-time strategies are proposed: Method A estimates the terminal nonlinear eigenfunction value on a boundary set using local EDMD, and Method B transforms the vector field outside a radius so that the nonlinear part of the eigenfunction is approximately zero there. The main theoretical result, Theorem 4, is a clean error-propagation statement for a given terminal error. The numerical examples demonstrate the method on a known saddle system and on a Hamiltonian system for optimal control, and the optimal-control example reproduces the known controller from the zero level-set of the computed eigenfunction.

Significance. If the central claim were established rigorously, the paper would be a useful contribution: it makes the path-integral approach applicable to saddle equilibria, replaces infinite-horizon integrals with finite-time ones, and offers a basis-free evaluation of eigenfunctions away from the local EDMD region. The paper's strengths include the clean and correct Theorem 4, the sensible use of local trajectory data to reduce EDMD sampling requirements, and numerical validation against externally known eigenfunctions. However, the theoretical guarantee for Method B is currently overclaimed: Proposition 1 only bounds a PDE residual, not the eigenfunction error, and the bridge from the residual to the closeness of eigenfunctions is missing. Since Method B is advertised as a main contribution, the paper's significance is conditional on fixing or reframing that step.

major comments (4)
  1. [Section IV, Proposition 1 and Eq. (19)] Proposition 1 proves only that the original eigenfunction phi_lambda has small PDE residual with respect to the transformed vector field tilde_f. The sentence immediately after the proof, claiming that this "establishes that the eigenfunctions of the original and the transformed dynamical systems ... can be made arbitrarily close," overstates the result. A small residual does not imply a small eigenfunction error ||tilde_phi_lambda - phi_lambda||. Moreover, in the limit a -> infinity the exterior field is exactly A x, whose C^1 eigenfunctions for a given eigenvalue are nonunique (e.g., in two dimensions they include x_1 h(x_1 x_2) for arbitrary C^1 h), and matching an inner eigenfunction such as phi_lambda = x_1 + x_2^2/3 for f(x) = (x_1 + x_2^2, -x_2), lambda = 1, to any such exterior eigenfunction on ||x|| = r is generically impossible. Thus the theoretical guarantee for Method B is unsupported as stated.
  2. [Section IV, Proposition 2 and the subsequent finite-time formula] Proposition 2 bounds only the PDE residual of w_lambda^T x on the sphere ||x|| = r + epsilon_2; it does not bound the terminal value error ||tilde_h_lambda(s_T(x))|| that actually enters the path-integral formula. Consequently, the derivation sets tilde_h_lambda approximately zero on the boundary without an error estimate, and no theorem connects the finite-time integral along tilde_f to the nonlinear part h_lambda of the original vector field. A quantitative perturbation argument is needed before this can be presented as a method with guarantees.
  3. [Section IV, Assumption 1; Section V, Example B] Assumption 1 is not verified for the numerical example. The construction (18) with r = 4 and a = 10 may introduce spurious equilibria or omega-limit sets; Remark 3 explicitly acknowledges that only a sufficient condition for uniqueness of equilibrium is provided and that omega-limit sets require more analysis. Without verifying Assumption 1, the statement that trajectories from almost every point in ||x|| < r reach the boundary ||x|| = R, and hence that T(x) is well defined, is not justified for the example.
  4. [Section IV, Theorem 4 and Eq. (15)] For a stable principal eigenfunction with Re(lambda) < 0, the error bound (15) grows as e^{-Re(lambda) \bar{t}(x)} with the time needed to reach the boundary, so the finite-time approximation does not have a uniform accuracy guarantee over the domain. The numerical example reports only maximum relative errors; the paper should either state this limitation explicitly or supply bounds on \bar{t}(x) over the computed domain to support the general claim that finite-time path-integrals approximate the principal eigenfunctions.
minor comments (4)
  1. [Section II, Definition 1] The Koopman operator is defined for "the dynamical system (5)", but the system is labelled (1); the cross-reference should be corrected.
  2. [Section V, Example A and Fig. 2] The text refers both to a sphere of radius 3 and to an "annular disk"; the wording should distinguish the boundary set B = {x : ||x|| = 3} from its neighborhood N_B.
  3. [Section IV, Method B, formula after Proposition 2] The struck-through underbrace notation is nonstandard and should be replaced by an explicit statement that the terminal term is neglected as part of the boundary approximation.
  4. [Section V, Example A, Eq. (21)] The vector-field expression is difficult to read; splitting the two components into separate lines would improve readability.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the finite-time path-integral derivation is self-contained; only minor self-citation of the authors' prior path-integral theorem.

full rationale

The derivation chain is not circular. Theorem 4 follows directly from the exact variation-of-constants formula (9): if the boundary estimate is epsilon-close to the true value on S, then equations (16)-(17) give the stated error bound by subtraction. The EDMD boundary estimation in Method A is local (sampled near the sphere B) and is not fitted to the target eigenfunction values on the compact set C; the path integral then propagates this boundary information along the known dynamics, and the numerical example compares against analytically known eigenpairs, so the validation is external. In Method B, the vector-field transformation (18) is used to construct a boundary condition with approximately zero nonlinear part; Propositions 1 and 2 prove only PDE-residual bounds, and the paper's assertion that this makes the eigenfunctions of the original and transformed systems close is a correctness gap, not a circular reduction. The only self-citation is the restatement of Theorems 1-3 from the authors' prior work [16]; these are standard variation-of-constants identities for the Koopman PDE, and Theorem 4's proof is self-contained, so the self-citation is not load-bearing in a circular sense. No fitted parameter is renamed as a prediction, and no uniqueness theorem is imported from the authors' prior work to force a choice.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The central method rests on the prior exact path-integral formula for the nonlinear part of principal eigenfunctions, restated from [16]. The finite-time extension then needs a boundary condition. Method A obtains it by fitting local EDMD models to trajectory data, so its accuracy depends on the unquantified boundary error epsilon_S. Method B obtains it from the linearized far field, but its proof only shows the PDE residual is small; the step from residual to eigenfunction closeness is an unproven assumption. The transformation parameters r and a are chosen by hand. No new physical entities are introduced.

free parameters (3)
  • r (transformation radius) = 4 (Example B)
    Chosen by hand in Method B; defines the disk where the transformed vector field approximates the original. The result depends on r because path integrals are evaluated only inside this disk.
  • a (step-function steepness) = 10 (Example B)
    Chosen by hand; larger a makes the transformed field match the original field inside radius r and the linear field outside, but no bound links a to the eigenfunction approximation error.
  • EDMD basis degree and sampling band = degree 10, band width 0.4, 800 trajectories (Example A)
    Hyperparameters chosen by hand for the local EDMD boundary estimate in Method A; the error epsilon_S on the boundary is not quantified as a function of these choices.
assumptions (4)
  • standard math The path-integral solution formula (Theorem 1) and the convergence result (Theorem 2) from [16] are accepted as background.
    Theorems 1-3 are quoted from the authors' prior work [16] and used without proof to derive the finite-time formulas in Section IV.
  • ad hoc to paper Assumption 1: the transformed vector field tilde_f does not introduce spurious equilibria or omega-limit sets.
    Stated in Section IV, Method B. Needed so that trajectories from almost every starting point reach the boundary sphere ||x||=R, making the finite time T(x) well defined. Not verified for the examples.
  • ad hoc to paper Small PDE residual implies eigenfunction closeness for the transformed system.
    Used implicitly to conclude from Propositions 1 and 2 that eigenfunctions of the transformed system approximate those of the original. Not proven; the propositions only bound the residual, not the eigenfunction difference.
  • domain assumption Hyperbolicity of the origin and existence of smooth principal Koopman eigenfunctions, as in [12].
    This is the standard setting for principal Koopman spectrum. Used throughout Section III.

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Pith. "Pith review of Extensions of the Path-integral formula for computation of Koopman eigenfunctions." pith.science (2026). https://pith.science/paper/MXAEG2BU

@misc{pith2026241116605,
  author       = {Pith},
  title        = {Pith review of: Extensions of the Path-integral formula for computation of Koopman eigenfunctions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MXAEG2BU}},
  note         = {Machine review of arXiv:2411.16605}
}
read the original abstract

Representing nonlinear dynamical systems using the Koopman Operator and its spectrum has distinct advantages in terms of linear interpretability of the model as well as in analysis and control synthesis through the use of well-studied techniques from linear systems theory. As such, efficient computation of Koopman eigenfunctions is of paramount importance towards enabling such Koopman-based constructions. To this end, several approaches have been proposed in literature, including data-driven, convex optimization, and Deep Learning-based methods. In our recent work, we proposed a novel approach based on path-integrals that allowed eigenfunction computations using a closed-form formula. In this paper, we present several important developments such as finite-time computations, relaxation of assumptions on the distribution of the principal Koopman eigenvalues, as well as extension towards saddle point systems, which greatly enhance the practical applicability of our method.

Figures

Figures reproduced from arXiv: 2411.16605 by the authors.

Figure 1
Figure 1. Path-integral formula for computing eigenfunction [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Local-EDMD enabled path-integral computation for [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. (a) Setting path-integral boundary conditions. The [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: Optimal control extracted from the zero level-set [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]

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