高维流形上Langevin动力学的数据驱动高效求解器
Data-driven Efficient Solvers for Langevin Dynamics on Manifold in High Dimensions
摘要：
我们研究了具有流形结构$\mathcal{M}\subset\mathbb{R}^p$的物理系统的朗之万动力学，基于探测未知流形$\mathcal{M}$的采样点$\{\mathsf{x}_i\}_{i=1}^n \subset \mathcal{M}$。通过扩散图，我们首先学习对应于$\{\mathsf{x}_i\}_{i=1}^n$的反应坐标$\{\mathsf{y}_i\}_{i=1}^n\subset \mathcal{N}$，其中$\mathcal{N}$是与$\mathcal{M}$同构的流形，并且等距嵌入在具有$\ell \ll p$的$\mathbb{R}^\ell$中。$\mathcal{N}$上的反应坐标的诱导Langevin动力学捕获慢时间尺度动力学，如生化反应中的构象变化。为了构造$\mathcal{N}$上的朗之万动力学的有效且稳定的近似，我们利用了流形$\mathcal{N}$上的对应的Fokker-Planck方程关于反应坐标$\mathsf{y}$。我们提出了一个可实现的，无条件稳定的，数据驱动的有限体积格式的福克-普朗克方程，它自动纳入了流形结构的$\mathcal{N}$。进一步，我们给出了求解Fokker-Planck方程的有限体积格式在$\mathcal{N}$上的加权$L^2$收敛性分析.所提出的有限体积格式导致$\{\mathsf{y}_i\}_{i=1}^n$上的马尔可夫链，具有近似的最近邻点之间的转移概率和跳跃率。在无条件稳定的显式时间离散后，数据驱动的有限体积格式给出了$\mathcal{N}$上Langevin动力学的一个近似Markov过程，该近似Markov过程具有细节平衡性、遍历性等良好性质.
英文摘要：
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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