Nonlinear Data-Driven Approximation of the Koopman Operator

The candidate solution reproduces exactly the paper’s least-squares estimators for the autonomous, reduced-order, and controlled settings, including the pseudoinverse formula [An Cn] = Γ+ [Γ; Fn]†, the truncated-SVD expression Γ+ Ṽ Σ̃^{-1} Ũ^T, the POD-projected reduced dynamics Ω+ = Φ^T A_n Φ Ω + Φ^T C_n f_n(ΦΩ), and the controlled estimator [Ac Bc Cc] = Γ+_c [Γ_c; U; F_c,n]† leading to γ+_c,i = Ac γ_c,i + Bc u_i + Cc f_c,n(γ_c,i) (matching the paper’s Equations (18)–(19), (22), (25), and (29)–(30) respectively ). The model adds a standard SVD-based lemma characterizing the full solution set and uniqueness conditions for right-hand Frobenius LS problems, which the paper omits but uses implicitly. A minor notational issue in the paper (writing sums of Frobenius/2-norms without an explicit square) is clarified by the candidate; the pseudoinverse formulas correspond to the standard squared-Frobenius LS objective used in practice, so there is no substantive conflict .