Abstract: Consider Langevin dynamics with collected dataset that distributed on a manifold M in a high dimensional Euclidean space. Through the diffusion map, we learn the reaction coordinates for N which is a manifold isometrically embedded into a low dimensional Euclidean space. This enables us to efficiently approximate the dynamics described by a Fokker-Planck equation on the manifold N. Based on this, we propose an implementable, unconditionally stable, data-driven upwind scheme which automatically incorporates the manifold structure of N and enjoys the weighted l^2 convergence to the Fokker-Planck equation. The proposed upwind scheme leads to a Markov chain  with transition probability between the nearest neighbor points, which enables us to directly conduct manifold-related computations such as finding the optimal coarse-grained network and the minimal energy path that represents chemical reactions or conformational changes. 
To acquire information about the equilibrium potential on manifold N, we apply a Gaussian Process regression algorithm to generate equilibrium potentials for a new physical system with new parameters. Combining with the proposed upwind scheme, we can calculate the trajectory of the Fokker-Planck equation on N based on the generated equilibrium potential. Finally, we develop an algorithm to pullback the trajectory to the original high dimensional space as a generative data for the new physical system. This is a joint work with Nan Wu and Jian-Guo Liu.