Data-driven Efficient Solvers and Predictions of Conformational Transitions for Langevin Dynamics on Manifold in High Dimensions

We work on dynamic problems with collected data $\{\mathsf{x}_i\}$ that distributed on a manifold $\mathcal{M}\subset\mathbb{R}^p$. Through the diffusion map, we first learn the reaction coordinates $\{\mathsf{y}_i\}\subset \mathcal{N}$ where $\mathcal{N}$ is a manifold isometrically embedded into an Euclidean space $\mathbb{R}^\ell$ for $\ell \ll p$. 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 $\mathcal{N}$. Furthermore, we provide a weighted $L^2$ convergence analysis of the upwind scheme to the Fokker-Planck equation. The proposed upwind scheme leads to a Markov chain with transition probability between the nearest neighbor points. We can benefit from such property 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 establish the Fokker-Planck equation, we need to acquire information about the equilibrium potential of the physical system on $\mathcal{N}$. Hence, we apply a Gaussian Process regression algorithm to generate equilibrium potential 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 $\mathcal{N}$ 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.