Deficiency of equation-finding approach to data-driven modeling of dynamical systems

The paper convincingly documents—via multiple data-driven reconstructions restricted to the monomial library {1, x, y, z, xy, xz, yz}—that many algebraically distinct quadratic vector fields generate essentially the same Lorenz-like attractor, with near-identical leading Koopman eigenvalues computed by Ulam’s method, close Lyapunov exponents, small KL divergence, and on–off intermittency in velocity-field discrepancies, see their pipeline and results in Figs. 1–3 and related text (Ulam/Koopman setup; library choice; three recovered equation sets; on–off behavior; “in principle an infinite number” claim) . However, the paper provides no theorem-level proof or general structural hypotheses ensuring persistence of the phenomenon; it offers strong empirical evidence and interpretive arguments. In contrast, the model supplies an explicit, mathematically supported construction: an affine-conjugate base family (exactly matching Koopman spectra and Lyapunov exponents under the paper’s normalization/Ulam protocol) plus small C^1 perturbations within the same monomial library, invoking robustness/statistical stability of geometric Lorenz flows and continuity of finite-dimensional spectra to obtain ε-closeness of the first N Koopman eigenvalues and Lyapunov exponents, and proving the observed on–off pattern via rare-event statistics. Hence the model provides a correct constructive proof of the reported phenomenon, while the paper’s argument—though compelling—remains incomplete.