Before revision: We work on dynamic problems with collected data\subsetR^p based on collected sample points } \{x_i\} that distributed on a manifold \mathcal{M\subsetR^p} ^n \subset M that probe the unknown manifold M} . Through the diffusion map, we first learn the reaction coordinates \{y_i\} \subset \mathcal{N^n\subset N corresponding to \{x_i\}_{i=1}^n, } where N is a manifold isometrically embedded into an Euclidean space \mathbb{R^\ell for }  and isometrically embedded in \mathbb{R}^\ell with } \ell \ll p.
After revision: We work on dynamic problems with collected data\subsetR^p based on collected sample points } \{x_i\} \subsetR^p} ^n \subset M that probe the unknown manifold M} . Through the diffusion map, we first learn the reaction coordinates \{y_i\} \subset \mathcal{N^n\subset N corresponding to \{x_i\}_{i=1}^n, } where N is a manifold isometrically embedded into an Euclidean space \mathbb{R^\ell for } and isometrically embedded in \mathbb{R}^\ell with } \ell \ll p.