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

• URL: http://arxiv.org/abs/2005.12787v3
• Date: Tue, 27 Sep 2022 06:52:48 GMT
• Title: Data-driven Efficient Solvers for Langevin Dynamics on Manifold in High Dimensions
• Authors: Yuan Gao, Jian-Guo Liu, Nan Wu
• Abstract summary: We study the Langevin dynamics of a physical system with manifold structure $mathcalMsubsetmathbbRp$ We leverage the corresponding Fokker-Planck equation on the manifold $mathcalN$ in terms of the reaction coordinates $mathsfy$. We propose an implementable, unconditionally stable, data-driven finite volume scheme for this Fokker-Planck equation.
• Score: 12.005576001523515
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

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