Phase2vec: Dynamical systems embedding with a physics-informed convolutional network

The paper defines the fixed‑point–normalized loss L1(Ẋ, Ẋrecon) = ||Ẋ − Ẋrecon||^2 / (||Ẋ||^2 + ε) and explicitly motivates that it is especially large where Ẋ vanishes, thereby emphasizing fixed points and slow regions; the candidate formalizes this with a simple and correct inequality argument that quantifies the weighting effect over regions with ||Ẋ|| ≤ δ and shows dominance as δ→0 . The paper uses a cubic monomial dictionary with 10 basis terms for 2D vector fields and studies “Conservativity,” “Incompressibility,” and “Linear stability” classification tasks; the candidate derives the corresponding coefficient‑space linear constraints (curl=0 and div=0) and the trace–determinant–discriminant characterization for linear systems. These derivations are mathematically correct and consistent with the paper’s setup, which generates/labels these classes (Appendix A.3) but does not spell out the linear constraints in coefficient space . In short: the paper’s claims are correct and empirically supported, while the model provides rigorous proofs for the specific subclaims posed in the solver question.