[Paper Review] Data-driven Efficient Solvers and Predictions of Conformational Transitions for Langevin Dynamics on Manifold in High Dimensions.

Yuan Gao, Jianguo Liu+1 more|arXiv (Cornell University)|May 22, 2020

Gaussian Processes and Bayesian Inference24 references4 citations

TL;DR

This paper proposes a data-driven, unconditionally stable upwind scheme for solving Fokker-Planck equations on manifolds using reaction coordinates derived via diffusion maps. By leveraging Gaussian Process regression to infer equilibrium potentials and projecting trajectories back to high-dimensional space, the method enables efficient, structure-preserving simulation of conformational transitions in high-dimensional systems.

ABSTRACT

We work on dynamic problems with collected data $\{\mathsf{x}_i\}$ that distributed on a manifold $\mathcal{M}\subset\mathbb{R}^p$. Through the diffusion map, we first learn the reaction coordinates $\{\mathsf{y}_i\}\subset \mathcal{N}$ where $\mathcal{N}$ is a manifold isometrically embedded into an Euclidean space $\mathbb{R}^\ell$ for $\ell \ll p$. The reaction coordinates enable us to obtain an efficient approximation for the dynamics described by a Fokker-Planck equation on the manifold $\mathcal{N}$. By using the reaction coordinates, we propose an implementable, unconditionally stable, data-driven upwind scheme which automatically incorporates the manifold structure of $\mathcal{N}$. Furthermore, we provide a weighted $L^2$ convergence analysis of the upwind scheme to the Fokker-Planck equation. The proposed upwind scheme leads to a Markov chain with transition probability between the nearest neighbor points. We can benefit from such property to directly conduct manifold-related computations such as finding the optimal coarse-grained network and the minimal energy path that represents chemical reactions or conformational changes. To establish the Fokker-Planck equation, we need to acquire information about the equilibrium potential of the physical system on $\mathcal{N}$. Hence, we apply a Gaussian Process regression algorithm to generate equilibrium potential for a new physical system with new parameters. Combining with the proposed upwind scheme, we can calculate the trajectory of the Fokker-Planck equation on $\mathcal{N}$ based on the generated equilibrium potential. Finally, we develop an algorithm to pullback the trajectory to the original high dimensional space as a generative data for the new physical system.

Motivation & Objective

• To enable efficient, stable numerical solution of Fokker-Planck dynamics on high-dimensional manifolds using data-driven dimensionality reduction.
• To develop a manifold-aware numerical scheme that preserves geometric structure and ensures unconditional stability.
• To predict conformational transitions in new physical systems with unknown parameters by learning equilibrium potentials via Gaussian Process regression.
• To generate high-dimensional trajectories for new systems by pulling back low-dimensional dynamics to the original space.
• To support coarse-grained modeling tasks such as identifying minimal energy paths and optimal reaction networks.

Proposed method

• Apply diffusion maps to extract reaction coordinates from high-dimensional data, embedding the manifold isometrically into a lower-dimensional Euclidean space.
• Construct an unconditionally stable, data-driven upwind finite difference scheme on the reaction coordinate manifold that respects the underlying geometry.
• Use weighted $L^2$ convergence analysis to establish theoretical stability and accuracy of the upwind scheme for the Fokker-Planck equation.
• Model the equilibrium potential of a new physical system using Gaussian Process regression trained on existing data from similar systems.
• Simulate the Fokker-Planck dynamics on the low-dimensional manifold using the inferred potential and the upwind scheme.
• Pull back the resulting low-dimensional trajectories to the original high-dimensional space to generate synthetic data for new systems.

Experimental results

Research questions

• RQ1 How can we efficiently and stably simulate Fokker-Planck dynamics on a high-dimensional manifold using only observed data?
• RQ2 What is the role of reaction coordinates in enabling accurate and stable numerical approximation of conformational transitions?
• RQ3 How can we generalize to new physical systems with unknown parameters without full re-simulation?
• RQ4 Can a data-driven upwind scheme preserve the geometric structure of the underlying manifold while ensuring numerical stability?
• RQ5 How can we generate high-fidelity, high-dimensional trajectories for new systems based on low-dimensional dynamics and learned potentials?

Key findings

• The proposed upwind scheme is unconditionally stable and automatically incorporates the manifold structure of the reaction coordinate space.
• The weighted $L^2$ convergence analysis confirms the theoretical accuracy and robustness of the numerical scheme for the Fokker-Planck equation.
• Gaussian Process regression enables reliable inference of equilibrium potentials for new physical systems with unknown parameters.
• The method successfully generates high-dimensional trajectories by pulling back low-dimensional dynamics, enabling data generation for new systems.
• The Markov chain formulation based on nearest-neighbor transitions supports direct computation of minimal energy paths and optimal coarse-grained networks.
• The framework enables efficient simulation of conformational transitions without requiring explicit knowledge of the system's potential energy surface.