International Workshop on Interacting Particle Systems

## Data-driven efficient solvers and predictions of conformational transitions for Langevin dynamics on manifold in high dimensions

### Speaker

Jian-Guo Liu , Duke University, USA

### Time

29 Mar, 09:30 - 10:00

### Abstract

Yuan Gao, Jian-Guo Liu, and Nan Wu, Duke University, USA

We work on dynamic problems with collected data ${\mx_i}$ that well-distributed on manifold $\mm\subset\bR^p$, $p\gg 1$. Using the diffusion map, we first learn the reaction coordinates $\my$ such that dataset ${\my_i}\subset\nn$ is isometric embedded into a low dimensional Euclidean space $\bR^\ell$. This enables us to obtain an efficient approximation for the dynamics, described on a Fokker-Plank equation on manifold $\nn$. Thanks to this, we proposed an implementable data-driven upwind scheme which automatically incorporates the manifold structure and give the convergence analysis to the Fokker-Plank equation on $\nn$. The proposed upwind scheme also gives a Markov chain with transition probability between the nearest neighbor points, which enables us to conduct manifold-related computations directly such as finding optimal coarse-grained network and minimal energy path representing chemical reactions or conformational changes. As a byproduct, we also give the algorithms for generating equilibrium potential for new physical system with new parameters. Using the proposed upwind scheme, we calculate the trajectory of the Fokker-Plank equation on $\nn$ with new equilibrium and then pullback to the original high dimensional space as an effective generative data for the new physical system.