Exact Leakage of Graded Polynomial Koopman Dictionaries: Geometric Decay of the Edmd Residual and Its Basis Dependence
Year: 2026
Type: preprint
Abstract: A finite-dimensional Koopman-invariant subspace yields an exact linear predictor of a nonlinear system, but the graded polynomial dictionaries used by EDMD (all monomials up to degree d) never provide one when the vector field has degree p ≥ 2: the Lie derivative always raises the degree (Proposition 1). This qualitative obstruction is essentially known; we make it quantitative. For every polynomial vector field in any dimension we give an exact closed-form expression for the population EDMD residual R(d) on the cube [-1,1]^n in terms of Legendre coefficients of monomials (Theorem 1), which collapses to a diagonal sum of squares when p ≤ 3. We prove that R(d) decays geometrically at exactly the rate 2^(-d) for every p ≥ 2, with polynomial prefactor Θ(d^((n+1)/2)) only for p ∈ {2, 3} (for p ≥ 4 the prefactor is bracketed between d^((n+1)/2) and d^((n+1)/2 + floor((p-2)/2)); Theorem 2), and we compute the exact asymptotic constants for the scalar cubic model, the damped double-well Duffing oscillator, the Van der Pol oscillator and the Lorenz system (Theorem 3); every formula agrees with independent quadrature computations to better than 1e-12 (typically 1e-14). We then show that this geometric decay is a property of the monomial basis, not of the subspace: the basis-independent (Hilbert-Schmidt) leakage of V_d grows at least linearly in d for every such field (Proposition 2). An EDMD experiment on the Duffing oscillator confirms both statements quantitatively: the monomial residual falls while the residual in a whitened basis rises, matching the theory to within 1% at small time step for d ≤ 10 (the normalisations and the Δt → 0 comparison are derived explicitly). We discuss what this implies for residual certification and for the choice of dictionary, comment on the scaling with the dimension n, and list open problems.
Author: Garcia Guerrero, Lucas
Language: English
Subfield: Statistical and Nonlinear Physics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225750837
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