{"id":"993e9be5-885e-402c-a9c9-dbf10b6aa214","arxiv_id":"2411.16605","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"Finite-time path-integral formulas with boundary data from local EDMD or a linearized far-field transformation compute Koopman principal eigenfunctions for saddle point systems.","lead":"This paper extends a path-integral formula for computing Koopman eigenfunctions, previously usable only for stable or anti-stable equilibria, to saddle-type equilibria using finite-time integration. Boundary values are supplied either by local trajectory data or by a vector-field transformation that makes the far field linear.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 1 bounds only the PDE residual, not the eigenfunction error; the claimed closeness of eigenfunctions under the transformation is therefore unproven and likely false.","rationale":"The reader's weakest_assumption identifies exactly the same load-bearing concern: Proposition 1's residual bound is used to assert eigenfunction closeness without proof. My analysis goes one step further and shows that the implication is not only unproven but likely false, via a concrete two-dimensional example where the exterior linear system admits many eigenfunctions and matching with the inner eigenfunction is impossible. This is the most load-bearing issue because the paper's headline contribution is to provide conditions under which the finite-time path-integral approximates principal eigenfunctions for saddle systems; Method B supplies one of the two proposed conditions, and without a valid closeness guarantee it has no theoretical support. The manuscript still merits conditional acceptance because Method A has a rigorous error-propagation theorem (Theorem 4), and the numerical examples are promising. The authors should either prove an eigenfunction-closeness result under a suitable non-resonance or hyperbolicity condition, or significantly moderate the claims about Method B and verify Assumption 1. Since the reader's conditional verdict already captures this, I recommend no change to the verdict.","tokens_in":11240,"tokens_out":17211,"duration_ms":168869,"concrete_test":"Take f(x)=(x_1+x_2^2, -x_2), for which phi_lambda=x_1+x_2^2/3 is the exact principal eigenfunction for lambda=1. For r>0, examine the a->infinity limit of transformation (18), where tilde_f=Ax outside ||x||=r. Verify that no C^1 function h exists satisfying the matching condition x_1 h(x_1 x_2)=x_1+x_2^2/3 on the circle ||x||=r (e.g., substitute theta and theta+pi to obtain sin^2(theta)=0). In addition, for finite large a (e.g., a=10, 100, 1000), compute the principal eigenfunction of tilde_f using a high-order PDE solver or an independent characteristic method that does not impose w_lambda^T x terminal data, and measure sup_{||x||<r-epsilon_2} |tilde_phi_lambda - phi_lambda|; if this error does not decay with increasing a, the claimed closeness in Proposition 1 is refuted.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central claim depends on Method B, whose only theoretical bridge is the sentence after Proposition 1: 'establishes that the eigenfunctions of the original and the transformed dynamical systems within some domain inside ||x||=r can be made arbitrarily close.' Proposition 1 proves only inequality (19), i.e., that phi_lambda has small PDE residual for the transformed vector field. A small residual does not imply that the actual principal eigenfunction tilde_phi_lambda of the transformed system is close to phi_lambda; the proof never bounds ||tilde_phi_lambda - phi_lambda||. This is not merely a missing lemma. Outside ||x||=r, the transformed field is Ax, and the linear PDE Ax·grad(psi)=lambda*psi has infinitely many C^1 eigenfunctions for a given eigenvalue; in 2D they take the form x_1 h(x_1 x_2) for arbitrary C^1 h. Matching such an exterior solution to an inner eigenfunction across the circle is generically impossible. For example, with f(x)=(x_1+x_2^2, -x_2), the exact principal eigenfunction for lambda=1 is phi_lambda=x_1+x_2^2/3. In the a->infinity limit, if the inner eigenfunction were phi_lambda, matching on ||x||=r would require x_1 h(x_1 x_2)=x_1+x_2^2/3 on the circle, which forces sin(theta)=0 for all theta and hence no such h exists. Thus the transformed eigenfunction can differ from phi_lambda even for arbitrarily large a, so the finite-time path-integral computes a function that is not guaranteed to approximate phi_lambda. Assumption 1 is also asserted without verification. Only Method A has a rigorous conditional guarantee (Theorem 4), leaving a real gap in the main contribution.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":11595,"tokens_out":6372,"duration_ms":58742,"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":[{"comment":"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.","section":"Section IV, Proposition 1 and Eq. (19)"},{"comment":"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.","section":"Section IV, Proposition 2 and the subsequent finite-time formula"},{"comment":"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.","section":"Section IV, Assumption 1; Section V, Example B"},{"comment":"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.","section":"Section IV, Theorem 4 and Eq. (15)"}],"minor_comments":[{"comment":"The Koopman operator is defined for \"the dynamical system (5)\", but the system is labelled (1); the cross-reference should be corrected.","section":"Section II, Definition 1"},{"comment":"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.","section":"Section V, Example A and Fig. 2"},{"comment":"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.","section":"Section IV, Method B, formula after Proposition 2"},{"comment":"The vector-field expression is difficult to read; splitting the two components into separate lines would improve readability.","section":"Section V, Example A, Eq. (21)"}],"recommendation":"major_revision","confidential_remarks":"The paper's central new idea, Method B, currently lacks a valid proof: Proposition 1 establishes only a residual bound, not eigenfunction closeness, and the matching problem across the boundary is real. The authors should either add a genuine perturbation estimate or openly reposition Method B as a numerical heuristic. The rest of the paper, especially Theorem 4 and the local-EDMD example, is solid enough to merit a revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: the paper is worth reading for Theorem 4 and the local-EDMD idea, but the theoretical guarantee claimed for Method B does not follow from the stated propositions. The stress-test note is correct on the merits. Proposition 1 only shows that the transformed vector field makes phi_lambda nearly satisfy the PDE inside ||x||<r. That says nothing about whether the true principal eigenfunction of the transformed system is close to phi_lambda. The same gap appears in Proposition 2. Outside ||x||>r the transformed field is linear, and the linear PDE has many C^1 eigenfunctions for the same eigenvalue; matching an inner eigenfunction to an outer one across the circle is generically impossible. So the path-integral on the transformed system may compute an eigenfunction of the modified system that differs from the original principal eigenfunction, even as the smoothing parameter a goes to infinity. The authors' sentence claiming \"eigenfunctions ... can be made arbitrarily close\" is an overstatement.\n\nWhat is genuinely new: extending the path-integral to finite horizon and to saddle equilibria is a natural and useful step. Theorem 4 is a clean error-propagation result, and the numerical experiments for Method A are convincing, including the comparison with full-domain EDMD. The optimal-control example is a nice illustration, but it rests on Method B, so its theoretical grounding is shaky. Assumption 1 is also stated but not verified for the examples.\n\nIn short: this is a solid submission with one rigorous method and one heuristic method presented as rigorous. A serious referee should see it, and the authors should be asked to either prove the missing eigenfunction-closeness step (under additional assumptions like hyperbolicity of the characteristic foliation) or explicitly reframe Method B as a heuristic with numerical support. I would cite it for Method A and for the finite-time formulation.","headline":"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.","tokens_in":12140,"tokens_out":2835,"would_cite":true,"duration_ms":26649,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"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…","keywords":["Koopman operator","Koopman eigenfunctions","path-integral formula","saddle equilibrium","Extended Dynamic Mode Decomposition","Hamiltonian systems","optimal control","finite-time horizon"],"falsifier":"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.","tokens_in":10999,"feed_emoji":"⚙️","tokens_out":9421,"duration_ms":81467,"temperature":0.7,"pith_summary":"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.","feed_headline":"Path integrals now give Koopman eigenfunctions for saddle equilibria","feed_subtitle":"Local data or a linearized field give the boundary values that make finite-time path integrals work at saddles.","key_machinery":"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.","core_discovery":"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).","pith_inferences":["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."],"forward_implications":["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 $2.5$."],"supporting_citations":[{"why":"Supplies the original path-integral formula and convergence machinery that this paper extends to finite horizon and saddle equilibria.","marker":"[16]"},{"why":"Provides the Extended Dynamic Mode Decomposition algorithm used in Method A to estimate eigenfunction values locally near the boundary set.","marker":"[21]"},{"why":"Connects principal Koopman eigenfunctions of Hamiltonian systems to Lagrangian submanifolds and nonlinear optimal control, motivating Example B.","marker":"[13]"},{"why":"Defines the infinite-horizon optimal control benchmark and its analytic solution used to validate the controller derived from the eigenfunction.","marker":"[22]"},{"why":"Supplies the non-recurrence assumption on the boundary set S that keeps the path-integral boundary-value formulation well posed.","marker":"[20]"},{"why":"Establishes that Koopman eigenfunctions recover stable and unstable manifolds as zero level sets, explaining why saddle-point eigenfunctions matter.","marker":"[12]"}],"fun_headline_variants":["Path integrals extend Koopman eigenfunctions to saddle points","Finite-time path integrals ease Koopman eigenfunction computation at saddles","Saddle-point Koopman eigenfunctions via finite-time path integrals","Path-integral formula for Koopman eigenfunctions now covers saddles","Path-integral eigenfunctions handle saddles with local boundary data"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Path integrals extend Koopman eigenfunctions to saddle points","Finite-time path integrals ease Koopman eigenfunction computation at saddles","Saddle-point Koopman eigenfunctions via finite-time path integrals","Path-integral formula for Koopman eigenfunctions now covers saddles","Path-integral eigenfunctions handle saddles with local boundary data"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001116,"raw_usage":{"total_tokens":4648,"prompt_tokens":945,"completion_tokens":3703,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":561,"completion_tokens_details":{"reasoning_tokens":3612}},"tokens_in":561,"tokens_out":3703,"duration_ms":23583,"temperature":1.0,"reasoning_tokens":3612,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T12:56:30.866862+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Path-integral formula for computing Koopman eigenfunctions,","cited_arxiv_id":null,"evidence_quote":"Supplies the original path-integral formula and convergence machinery that this paper extends to finite horizon and saddle equilibria."},{"cited_title":"A data– driven approximation of the Koopman operator: Extending dynamic mode decomposition,","cited_arxiv_id":null,"evidence_quote":"Provides the Extended Dynamic Mode Decomposition algorithm used in Method A to estimate eigenfunction values locally near the boundary set."},{"cited_title":"Spectral analysis of Koopman operator and nonlinear optimal control,","cited_arxiv_id":null,"evidence_quote":"Connects principal Koopman eigenfunctions of Hamiltonian systems to Lagrangian submanifolds and nonlinear optimal control, motivating Example B."},{"cited_title":"A tutorial on Pontryagin- Koopman operators for infinite horizon optimal control,","cited_arxiv_id":null,"evidence_quote":"Defines the infinite-horizon optimal control benchmark and its analytic solution used to validate the controller derived from the eigenfunction."},{"cited_title":"Optimal construction of Koopman eigen- functions for prediction and control,","cited_arxiv_id":null,"evidence_quote":"Supplies the non-recurrence assumption on the boundary set S that keeps the path-integral boundary-value formulation well posed."},{"cited_title":"Spectrum of the Koopman operator, spectral expansions in functional spaces, and state-space geometry,","cited_arxiv_id":null,"evidence_quote":"Establishes that Koopman eigenfunctions recover stable and unstable manifolds as zero level sets, explaining why saddle-point eigenfunctions matter."}],"review_version":1}