A Physics-Informed Neural Network Method for the Approximation of Slow Invariant Manifolds for the General Class of Stiff Systems of ODEs

The paper frames the SIM as a graph x=h(y) in coordinates (x,y) induced by linear maps X(z)=Cz and Y(z)=Dz, and enforces membership and invariance via L1: Cz−N(Dz)=0 and L2: (C−∇yN(Dz)D)F(z)=0, together with pinning constraints on (C,D), see Eqs. (12–16) and (18,21) for the SLFNN and its derivatives, and the invariance equation in Eq. (11) . It also estimates a constant M over the domain of interest using the CSP diagnostic, Eq. (7) , and explicitly emphasizes that the learned SIM is an explicit functional that avoids root-finding . The candidate solution formalizes the same argument: it picks admissible (C,D) satisfying the paper’s pinning constraints, defines H={z: Cz=h(Dz)}, shows this is a C^1 embedded manifold of dimension N−M and is invariant by differentiating Cz−h(Dz) along trajectories to obtain (C−∇yh(Dz)D)F(z), and derives the reduced y-dynamics; it also notes that an SLFNN h with L1≡0 and L2≡0 yields an explicit reduced model. The only overstatement is the claim that the CSP diagnostic itself guarantees M is constant on an arbitrary Ω; in the paper, constancy is ensured in practice by restricting to time windows/domains where the diagnostic reports a constant M during data collection (Sec. 2.3.1) . Aside from this nuance, the logic is aligned and essentially the same invariance-based proof underlies both.