Challenges in Dynamic Mode Decomposition

The candidate solution reproduces, almost verbatim, the paper’s Appendix A.1–A.2 arguments. For the “diagonal linear system + multinomial observable” construction (Theorem 2 in the paper), the model’s Theorem A is identical to the paper’s statement and proof: lift by monomials up to degree m, note that each monomial evolves with e^{t λ[α]}, and obtain h(exp(tΛ)x) = H·exp(tΛ̃)·Ψ(x) (see eqs. (26)–(30) and the displayed identity immediately after, in Appendix A.2) . For the universality/density result (Theorem 1), the model’s Theorem B follows the same steps as Appendix A.1: (i) approximate F0 by Fε in the generic class Gfix(U) using [63], (ii) invoke principal Koopman eigenfunctions to obtain a C∞ conjugacy to exp(t DFε(0)), (iii) approximate the inverse conjugacy by polynomials via Stone–Weierstrass, and (iv) use a standard Lipschitz stability bound for flows to transfer the error back to F0; the paper reaches inequality (25) and then defines P := Ψ−1_ε∘V^{-1}, S := V·Ψ to conclude . Differences are minor: the model emphasizes approximation on a single trajectory segment K_x (yielding a pointwise-in-x result), whereas the paper takes a uniform-on-Ψ(U) approximation, yielding uniformity in x over U. The model also explicitly notes the P(K_x) ⊂ U safeguard, a detail implicit but not spelled out in the paper. Overall, both arguments are correct and essentially the same.