Data-driven Efficient Solvers for Langevin Dynamics on Manifold in High Dimensions
This work addresses the challenge of modeling slow conformational changes in biochemical systems for researchers in computational physics and chemistry, though it appears incremental as it builds on existing diffusion map and finite volume methods.
The paper tackles the problem of simulating Langevin dynamics on high-dimensional manifolds by proposing a data-driven finite volume scheme for the Fokker-Planck equation, which results in an unconditionally stable and efficient Markov chain approximation with properties like detailed balance and ergodicity, supported by a weighted L2 convergence analysis.
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.