Enabling Local Neural Operators to perform Equation-Free System-Level Analysis

The paper clearly formulates the EF residual ψ(u;λ) = u − ST[u,λ], advocates Jacobian-free Newton–GMRES, Arnoldi stability analysis, and pseudo‑arclength continuation for neural-operator timesteppers, and demonstrates these numerically; however, it does not provide rigorous convergence, error, or perturbation bounds beyond heuristic discussion and standard algorithmic descriptions. See the paper’s definitions and EF setup, including ψ and ST, and the matrix-free JVP via finite differences and Newton–GMRES, as well as the continuation and Arnoldi details in the main text and Appendix E . The homotopy-based embedding and empirical spectral accuracy claims are described but not proved theoretically; the Bauer–Fike rationale appears only as qualitative guidance . The candidate model solution supplies a plausible theoretical framework (Newton–Kantorovich, implicit function/pseudo‑arclength, spectral perturbation) but leans on an overly strong contraction assumption for ST that cannot hold near unstable equilibria and mixes state and derivative consistency in a key composition error bound; thus, it is also incomplete.