A Dynamical Systems Perspective on the Analysis of Neural Networks

The paper states that augmented neural ODEs with m ≥ d + q have the universal embedding property with respect to C^k(R^d, R^q) (Theorem 2.6), under a non-parameterized vector field f ∈ C^{0,k} and full-rank affine lift/readout layers, exactly as defined in its architecture section. This directly matches the candidate’s explicit construction: embed x as (x,0,0), run the triangular shear flow (u,v,r) -> (u, v + Ψ(u), r) generated by f(t,(u,v,r)) = (0, Ψ(u), 0), and read out v, yielding Φθ(x) = Ψ(x). The paper’s result is cited as established in [86], while the candidate gives a concrete shear-flow proof; these are substantially the same idea. Minor formal nuances (k ≥ 0 vs the earlier definition for k ≥ 1) do not affect correctness here, since the special triangular field guarantees global existence and uniqueness even for k=0. Key assumptions (augmentation m ≥ d + q; full-rank lift/readout; f ∈ C^{0,k}) are satisfied in both. Citations: the architecture and lift/readout setup (equations (9)–(10) and non-parameterized f) appear in the paper’s Section 2.2–2.3, and Theorem 2.6 is stated in Section 2.3–2.4.