Gentlest ascent dynamics on manifolds defined by adaptively sampled point-clouds

TL;DR

This paper introduces a data-driven extension of gentlest ascent dynamics (GAD) for finding saddle points on manifolds represented by adaptively sampled point-clouds, without needing explicit constraints.

Contribution

It develops an intrinsic, data-driven GAD method for manifolds defined by point-clouds, enabling saddle point detection without explicit constraint equations.

Findings

Successfully extends GAD to point-cloud manifolds

Does not require explicit constraint equations

Uses adaptive sampling to guide system to saddle points

Abstract

Finding saddle points of dynamical systems is an important problem in practical applications such as the study of rare events of molecular systems. Gentlest ascent dynamics (GAD) is one of a number of algorithms in existence that attempt to find saddle points in dynamical systems. It works by deriving a new dynamical system in which saddle points of the original system become stable equilibria. GAD has been recently generalized to the study of dynamical systems on manifolds (differential algebraic equations) described by equality constraints and given in an extrinsic formulation. In this paper, we present an extension of GAD to manifolds defined by point-clouds, formulated using the intrinsic viewpoint. These point-clouds are adaptively sampled during an iterative process that drives the system from the initial conformation (typically in the neighborhood of a stable equilibrium) to a saddle point. Our method requires the reactant (initial conformation), does not require the explicit constraint equations to be specified, and is purely data-driven.