REDUCED-ORDER MODELLING BASED ON KOOPMAN OPERATOR THEORY

The paper defines a Koopman/DMD reconstruction u_DMD(x, t_i) = sum_j a_j(t_i) λ_j^(i−1) φ_j(x), introduces a per-mode weight w_Kj = ∫_{Δt}^{t_Nt} ∑_{i=1}^{Nt} a_j(t) λ_j^(i−1) dt, and poses the model-selection problem: minimize N_DMD subject to selecting modes in strictly decreasing weight and achieving a relative error ≤ ε (eqs. (21)–(24)). It then describes choosing modes in descending order of this weight until the error constraint is satisfied, i.e., scanning prefixes of the globally weight-sorted list of modes . The candidate solution implements exactly this selection rule—compute w_Kj (with a practical quadrature), sort by |w_Kj|, and scan prefixes until Er_DMD ≤ ε—together with an explicit argument that the first such prefix is minimal among all ordered selections. The extra details (handling complex weights by magnitude, specifying an L2(Ω) space–time error, and discretizing the weight integral) fill gaps the paper leaves implicit and do not alter the substance. Hence both the paper’s procedure and the model’s solution are aligned in approach and justification. Minor issues in the paper remain (e.g., ambiguous handling of complex weights and imprecise error norm), but they do not contradict the model; rather, the model supplies reasonable, standard assumptions to make the optimization well-posed. The DMD background and data setup in the paper support the selection mechanism , and the experimental sections apply the same thresholding-by-ε strategy .