Before revision:  The intrusive reduced-order model (ROM) is constructed with the proper orthogonal decomposition (POD) with the Galerkin projection (POD-G)  . We apply the operator inference (OpInf) with re-projection for non-intrusive reduced-order modeling.
After revision:  The full-order model (FOM) is obtained by applying linearly implicit Kahan's method in time. Applying proper orthogonal decomposition (POD) with the Galerkin projection (POD-G) . We apply the operator inference (OpInf) with re-projection for non-intrusive reduced-order modeling.