Locating saddle points using gradient extremals on manifolds adaptively revealed as point clouds

TL;DR

This paper presents a novel method for locating saddle points in high-dimensional dynamical systems by using gradient extremals on unknown manifolds represented by point clouds, combining manifold learning with adaptive sampling.

Contribution

It introduces an adaptive approach that locates saddle points on unknown manifolds using gradient extremals and manifold learning, requiring only a single minimum and local sampling.

Findings

Successfully applied to the Muller-Brown potential on an unknown surface

Efficiently biases the system along curves to find saddle points

Compared favorably with Newton trajectories and gentlest ascent methods

Abstract

Steady states are invaluable in the study of dynamical systems. High-dimensional dynamical systems, due to a separation of time-scales, often evolve towards a lower dimensional manifold $M$. We introduce an approach to locate saddle points (and other fixed points) that utilizes gradient extremals on such a priori unknown (Riemannian) manifolds, defined by adaptively sampled point clouds, with local coordinates discovered on-the-fly through manifold learning. The technique, which efficiently biases the dynamical system along a curve (as opposed to exhaustively exploring the state space), requires knowledge of a single minimum and the ability to sample around an arbitrary point. We demonstrate the effectiveness of the technique on the M\"uller-Brown potential mapped onto an unknown surface (namely, a sphere). Previous work employed a similar algorithmic framework to find saddle points using Newton trajectories and gentlest ascent dynamics; we therefore also offer a brief comparison with these methods.