Dynamical system analysis of a data-driven model constructed by reservoir computing

The paper reports compelling empirical evidence that a reservoir-computing model reconstructs many dynamical features of Lorenz (fixed points with close Jacobian spectra, periodic-orbit-like trajectories, Lyapunov exponents/DKY, CLV angle distributions), and uses this to estimate laminar lasting times in Navier–Stokes; however, it does not provide rigorous theorems or hypotheses (e.g., no formal C1 output-space map and no uniform Jacobian closeness on the attractor), instead relying on numerically estimated Jacobians and pre-iterate selection via the echo-state property in the reservoir state space . The candidate solution supplies a plausible proof outline but depends on strong, unestablished assumptions (existence of a C1 output-space map ψ(Δt), uniform per-step Jacobian closeness on K, dominated splitting at r=60, and spectral-gap statistical stability for the Navier–Stokes observable). These key hypotheses are neither stated nor justified in the paper, which explicitly positions its findings as empirical reconstructions (including in the non-hyperbolic, tangency case r=60) . Hence, the paper is incomplete theoretically and the model’s proof is incomplete relative to the actual setting.