Accelerating Neural ODEs Using Model Order Reduction

The paper derives the POD–DEIM reduced Neural ODE explicitly: after POD Galerkin projection, x̃'(t) = V_k^T f(A V_k x̃ + b) + V_k^T Z u (their Eq. (3)), and then replaces the nonlinear term via DEIM to obtain x̃'(t) = V_k^T U_m (P^T U_m)^{-1} f_m(Ã x̃ + b) + Z̃ u (their Eq. (5)), followed by selecting only the m rows indexed by the DEIM points to get x̃'(t) = N f_m(Ã_m x̃ + b_m) + Z̃ u (their Eq. (6)), with the explicit statement that “only m nonlinear functions are evaluated” online. These steps are present verbatim in the PDF and match the model’s derivation and notation (Ã = A V_k, Z̃ = V_k^T Z, N = V_k^T U_m (P^T U_m)^{-1}). The model’s added remark that P^T f(y) = f(P^T y) when f is componentwise makes explicit the paper’s implicit assumption behind the m-row compression step (the paper phrases it as ‘Due to the structure of f_m: R^m → R^m…’). Therefore, both the paper and the candidate solution present the same argument and are mutually consistent. Citations: the paper’s Eq. (3) derivation and lifting bottleneck discussion , the DEIM interpolation formula and ROM Eq. (5) with fm definition and Algorithm 1 , and the ‘only m nonlinear functions’ and Eq. (6) statement .