Staying the course: Locating equilibria of dynamical systems on Riemannian manifolds defined by point-clouds

TL;DR

This paper presents a novel iterative method to locate equilibria of dynamical systems on unknown Riemannian manifolds defined by point-cloud data, extending classical techniques to high-dimensional, data-driven settings.

Contribution

The authors develop a method to find equilibria on manifolds without explicit parametrization, using parallel transport and isoclines, applicable to high-dimensional stochastic systems.

Findings

Successfully locates saddle points in high-dimensional molecular dynamics models.

Extends Newton trajectory methods to unknown Riemannian manifolds.

Integrates manifold learning with sampling techniques for equilibrium detection.

Abstract

We introduce a method to successively locate equilibria (steady states) of dynamical systems on Riemannian manifolds. The manifolds need not be characterized by an a priori known atlas or by the zeros of a smooth map. Instead, they can be defined by point-clouds and sampled as needed through an iterative process. If the manifold is an Euclidean space, our method follows isoclines, curves along which the direction of the vector field $X$ is constant. For a generic vector field $X$, isoclines are smooth curves and every equilibrium lies on isoclines. We generalize the definition of isoclines to Riemannian manifolds through the use of parallel transport: generalized isoclines are curves along which the directions of $X$ are parallel transports of each other. As in the Euclidean case, generalized isoclines of generic vector fields $X$ are smooth curves that connect equilibria of $X$. Our algorithm can be regarded as an extension of the method of Newton trajectories to the manifold setting when the manifold is unknown. This work is motivated by computational statistical mechanics, specifically high dimensional (stochastic) differential equations that model the dynamics of molecular systems. Often, these dynamics concentrate near low-dimensional manifolds and have transitions (saddle points with a single unstable direction) between metastable equilibria. We employ iteratively sampled data and isoclines to locate these saddle points. Coupling a black-box sampling scheme (e.g., Markov chain Monte Carlo) with manifold learning techniques (diffusion maps in the case presented here), we show that our method reliably locates equilibria of $X$.