Before revision:  The reaction coordinates enable us to obtain an efficient approximation for the dynamics described by a Fokker-Planck equation on the manifold \mathcal{N} . By using the reaction coordinates , we propose an implementable, unconditionally stable, data-driven upwind scheme which automatically incorporates the manifold structure of \mathcal{N}.
After revision:  The induced Langevin dynamics on \mathcal{N approximation for the dynamics described by a Fokker-Planck equation on the manifold \mathcal{N} . By using the reaction coordinates , we propose an implementable, unconditionally stable, data-driven upwind scheme which automatically incorporates the manifold structure of \mathcal{N}.