arxiv: 2604.21825 · v1 · submitted 2026-04-23 · 🧮 math.DS · cs.LG· cs.NA· math.NA

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On the algebra of Koopman eigenfunctions and on some of their infinities

Zahra Monfared , Saksham Malhotra , Sekiya Hajime , Ioannis Kevrekidis , Felix Dietrich

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Pith reviewed 2026-05-08 13:37 UTC · model grok-4.3

classification 🧮 math.DS cs.LGcs.NAmath.NA

keywords Koopman operatoreigenfunctionsmultiplicative groupreversible trajectoriespolynomial constructionssingularitiesmultistable systemsdynamical systems

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The pith

Nowhere-vanishing Koopman eigenfunctions of reversible dynamical systems form a multiplicative group that polynomials of principal eigenfunctions can generate in full.

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

The paper shows that in continuous-time systems whose trajectories run both forward and backward, the Koopman eigenfunctions that never hit zero close under multiplication and therefore form a group. From a small number of conventionally computed principal eigenfunctions one can then build polynomials that automatically satisfy the eigenfunction equation, producing a much richer collection without new data fits. The construction also supplies a way to match and continue eigenfunctions across the localized or extended singularities that appear in multistable systems or near limit cycles and separatrices, so that globally consistent representations can be learned from patchy local observations.

Core claim

For continuous-time dynamical systems with reversible trajectories, the nowhere-vanishing eigenfunctions of the Koopman operator form a multiplicative group. Given a small set of principal eigenfunctions approximated conventionally, polynomials of these eigenfunctions belong to the same eigenspaces and therefore enlarge the available basis for representing observables. The group property further permits eigenfunction matching and continuation across singularities, supporting consistent global models from locally sampled data.

What carries the argument

The multiplicative group of nowhere-vanishing Koopman eigenfunctions under pointwise multiplication, which turns polynomial combinations of a few principal eigenfunctions into new eigenfunctions.

Load-bearing premise

The trajectories of the system must be reversible and the eigenfunctions under consideration must never vanish anywhere on the state space.

What would settle it

Exhibit a reversible continuous-time system together with two nowhere-vanishing eigenfunctions whose pointwise product fails to be an eigenfunction of the Koopman operator.

read the original abstract

For continuous-time dynamical systems with reversible trajectories, the nowhere-vanishing eigenfunctions of the Koopman operator of the system form a multiplicative group. Here, we exploit this property to accelerate the systematic numerical computation of the eigenspaces of the operator. Given a small set of (so-called ``principal'') eigenfunctions that are approximated conventionally, we can obtain a much larger set by constructing polynomials of the principal eigenfunctions. This enriches the set, and thus allows us to more accurately represent application-specific observables. Often, eigenfunctions exhibit localized singularities (e.g. in simple, one-dimensional problems with multiple steady states) or extended ones (e.g. in simple, two-dimensional problems possessing a limit cycle, or a separatrix); we discuss eigenfunction matching/continuation across such singularities. By handling eigenfunction singularities and enabling their continuation, our approach supports learning consistent global representations from locally sampled data. This is particularly relevant for multistable systems and applications with sparse or fragmented measurements.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper shows how the multiplicative group property of Koopman eigenfunctions allows polynomial enrichment of the eigenspace and provides a way to handle singularities for global representations from local data.

read the letter

The punchline for this paper is that nowhere-vanishing Koopman eigenfunctions form a multiplicative group for reversible flows. This lets you construct many additional eigenfunctions as polynomials from just a few principal ones obtained by standard methods. The work is new in its systematic use of this group property for enriching the eigenfunction set numerically. It also develops an approach to eigenfunction matching and continuation across singularities, such as those near multiple steady states or around limit cycles. By doing so, it aims to produce consistent global representations even from locally sampled data. This is useful for multistable systems where data might be sparse in some basins. The algebra is solid and follows directly from the Koopman operator acting on products and inverses. The stress-test note is right that there's no internal inconsistency here. The paper earns credit for highlighting how this algebraic closure can help with application-specific observables. Soft spots are in the implementation details. Without explicit error analysis or numerical case studies in the provided abstract, the practical benefits remain to be confirmed in the manuscript. The singularity continuation might require careful numerical treatment to avoid artifacts. This paper is for specialists in Koopman-based data-driven modeling of dynamical systems. Readers interested in extending limited eigenfunction approximations algebraically will get value from it. The paper deserves a serious referee because the idea is logically grounded and addresses a relevant computational challenge. I would recommend it for peer review.

Referee Report

1 major / 2 minor

Summary. The paper claims that for continuous-time dynamical systems with reversible trajectories, the nowhere-vanishing eigenfunctions of the Koopman operator form a multiplicative group. This property is used to generate a larger collection of eigenfunctions via polynomials in a small set of conventionally approximated principal eigenfunctions, thereby enriching the representation of observables. The manuscript further treats the practical problem of eigenfunction singularities (localized or extended) and their matching/continuation to support consistent global models from locally sampled data, with emphasis on multistable systems.

Significance. If the central algebraic claim holds under the stated conditions, the work supplies a direct, parameter-free mechanism for expanding Koopman eigenspaces that could materially reduce the data or computational cost of obtaining accurate observable representations. The singularity-handling discussion adds concrete value for applying the framework to systems with sparse measurements or multiple attractors.

major comments (1)

[Algebra section / singularity discussion] The central group property (nowhere-vanishing eigenfunctions closed under multiplication and inversion) is load-bearing for the polynomial-enrichment claim, yet the manuscript must clarify how this property is preserved when eigenfunctions exhibit the localized or extended singularities treated in the later sections; a local violation of nowhere-vanishing would appear to exclude the product from the group.

minor comments (2)

[Abstract] The abstract invokes 'infinities' in the title but does not indicate whether this refers to infinite-dimensional function spaces, infinite products, or another notion; a single clarifying sentence would remove ambiguity.
[Introduction] Standard citations to the foundational Koopman-operator literature (e.g., works establishing the spectral properties for flows) are needed in the introduction to situate the algebraic observation.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive assessment and the recommendation for minor revision. The single major comment is addressed below with a targeted clarification that strengthens the manuscript without altering its core claims.

read point-by-point responses

Referee: [Algebra section / singularity discussion] The central group property (nowhere-vanishing eigenfunctions closed under multiplication and inversion) is load-bearing for the polynomial-enrichment claim, yet the manuscript must clarify how this property is preserved when eigenfunctions exhibit the localized or extended singularities treated in the later sections; a local violation of nowhere-vanishing would appear to exclude the product from the group.

Authors: We agree that explicit clarification is required. The multiplicative group is formed by the restrictions of the eigenfunctions to open sets on which they are nowhere-vanishing (i.e., the state space minus the singular loci such as steady states, separatrices, or limit-cycle boundaries). On each such punctured domain the product and inversion operations remain well-defined and map back into the same class. Polynomial extensions are constructed locally within these domains using the principal eigenfunctions. The matching and continuation procedures of Section 4 then glue the local representations across the singularities by enforcing consistency on overlapping regular regions, thereby extending the algebraic structure globally without requiring the eigenfunctions to be nowhere-vanishing on the entire state space. We have inserted a new paragraph at the end of Section 2 that states this domain restriction explicitly and adds forward references to the singularity-handling discussion. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation follows directly from operator definition

full rationale

The paper's core algebraic claim—that nowhere-vanishing Koopman eigenfunctions form a multiplicative group under the stated reversibility condition—is a direct, standard consequence of the operator definition: if U^t ψ_i = e^{λ_i t} ψ_i and ψ_i ≠ 0, then U^t (ψ_1 ψ_2) = e^{(λ_1+λ_2)t} (ψ_1 ψ_2) and similarly for inverses and finite products. Polynomial enrichment of a small principal set is therefore obtained by repeated application of this identity, with no fitted parameters, self-referential predictions, or load-bearing self-citations required. The discussion of singularities and continuation is framed as a numerical/practical extension rather than part of the algebraic derivation itself. No step reduces by construction to its own inputs or to an unverified prior result by the same authors.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the reversibility of trajectories and the existence of nowhere-vanishing eigenfunctions; these are domain assumptions from dynamical systems theory rather than new inventions or fitted quantities.

axioms (2)

domain assumption Trajectories of the dynamical system are reversible
Explicitly required in the abstract for the nowhere-vanishing eigenfunctions to form a multiplicative group.
domain assumption Eigenfunctions are nowhere-vanishing
Stated as the condition enabling the group structure under multiplication.

pith-pipeline@v0.9.0 · 5494 in / 1278 out tokens · 21642 ms · 2026-05-08T13:37:49.542150+00:00 · methodology

discussion (0)

Reference graph

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