We study the Langevin dynamics of a physical system with manifold structure $\mathcal{M}\subset {\mathbb{R}}^p$ based on collected sample points ${\{{\mathbf{x}}_i\}}_{i=1}^n\subset \mathcal{M}$ that probe the unknown manifold $\mathcal{M}$. Through the diffusion map, we first learn the reaction coordinates ${\{{\mathbf{y}}_i\}}_{i=1}^n\subset \mathcal{N}$ corresponding to ${\{{\mathbf{x}}_i\}}_{i=1}^n$, 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 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 ${\{{\mathbf{y}}_i\}}_{i=1}^n$ 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.

我们研究了具有多种结构的物理系统的朗之万动力学$\mathcal{米}\subset {\mathbb{R}}^p$基于收集的样本点${\{{\mathbf{X}}_{\mathrm{一世}}\}}_{\mathrm{一世}=1}^n\subset \mathcal{米}$探索未知的流形$\mathcal{米}$. 通过扩散图，我们首先学习反应坐标${\{{\mathbf{是的}}_{\mathrm{一世}}\}}_{\mathrm{一世}=1}^n\subset \mathcal{ñ}$对应于${\{{\mathbf{X}}_{\mathrm{一世}}\}}_{\mathrm{一世}=1}^n$， 在哪里$\mathcal{ñ}$是流形微分同胚于$\mathcal{米}$并等距嵌入${\mathbb{R}}^{\ell }$和$\ell \ll p$. 诱导朗之万动力学$\mathcal{ñ}$就反应坐标而言，它捕获了慢时间尺度的动力学，例如生化反应中的构象变化。为朗之万动力学构建一个有效且稳定的近似$\mathcal{ñ}$，我们在流形上利用相应的 Fokker-Planck 方程$\mathcal{ñ}$根据反应坐标y。我们为这个 Fokker-Planck 方程提出了一个可实现的、无条件稳定的、数据驱动的有限体积方案，它自动结合了$\mathcal{ñ}$. 此外，我们提供加权${\mathrm{大号}}^2$有限体积方案对 Fokker-Planck 方程的收敛性分析$\mathcal{ñ}$. 所提出的有限体积方案导致马尔可夫链${\{{\mathbf{是的}}_{\mathrm{一世}}\}}_{\mathrm{一世}=1}^n$在最近邻点之间具有近似的转移概率和跳跃率。在无条件稳定的显式时间离散化之后，数据驱动的有限体积方案给出了一个近似于朗之万动力学的马尔可夫过程$\mathcal{ñ}$近似马尔可夫过程具有详细的平衡性、遍历性和其他良好性质。