Data-driven Efficient Solvers for Langevin Dynamics on Manifold in High Dimensions

We study the Langevin dynamics of a physical system with manifold structure $\mathcal{M}\subset\mathbb{R}^p$ based on collected sample points ${\mathsf{x}i}{i=1}ⁿ \subset \mathcal{M}$ that probe the unknown manifold $\mathcal{M}$. Through the diffusion map, we first learn the reaction coordinates ${\mathsf{y}i}{i=1}^n\subset \mathcal{N}$ corresponding to ${\mathsf{x}i}{i=1}^n$, where $\mathcal{N}$ is a manifold diffeomorphic to $\mathcal{M}$ and isometrically embedded in $\mathbb{R}^\ell$ with $\ell \ll p$. The induced Langevin dynamics on $\mathcal{N}$ in terms of the reaction coordinates captures the slow time scale dynamics such as conformational changes in biochemical reactions. To construct an efficient and stable approximation for the Langevin dynamics on $\mathcal{N}$, we leverage the corresponding Fokker-Planck equation on the manifold $\mathcal{N}$ in terms of the reaction coordinates $\mathsf{y}$. We propose an implementable, unconditionally stable, data-driven finite volume scheme for this Fokker-Planck equation, which automatically incorporates the manifold structure of $\mathcal{N}$. Furthermore, we provide a weighted $L^2$ convergence analysis of the finite volume scheme to the Fokker-Planck equation on $\mathcal{N}$. The proposed finite volume scheme leads to a Markov chain on ${\mathsf{y}i}{i=1}^n$ with an approximated transition probability and jump rate between the nearest neighbor points. After an unconditionally stable explicit time discretization, the data-driven finite volume scheme gives an approximated Markov process for the Langevin dynamics on $\mathcal{N}$ and the approximated Markov process enjoys detailed balance, ergodicity, and other good properties.