Out-of-Domain Generalization in Dynamical Systems Reconstruction

The paper’s Theorem 4.2 constructs an infinite family of C^1 vector fields that match the ground-truth vector field on the training basin but differ on the test basin, guaranteeing strictly positive test-domain error by appealing to their general error bound (Theorem 3.3) and a smooth bump-function replacement on a compact subset of the test basin; see theorem statement and proof sketch in the main text and Appx. E.3 , together with the definitions of E_stat and E_top in Defs. 3.1–3.2 . The candidate solution reaches the same conclusion via a more constructive modification (local hyperbolic sink plus a transport corridor) that yields explicit uniform lower bounds for both the topological and statistical (SW1) errors on M_test. Minor fixable technical gaps appear in the model’s Step 3 (where cancellation of f should be guaranteed on the whole contraction region), and in the paper’s phrasing that the whole new basin lies inside V; neither affects the main conclusion. Net: both arguments are valid and consistent, with different proof techniques.