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.

翻译：寻找动力系统的鞍点是分子系统稀有事件研究等实际应用中的重要问题。最缓上升动力学（GAD）是现有众多试图在动力系统中寻找鞍点的算法之一。其核心思想是构造一个新的动力系统，使得原系统的鞍点在该新系统中变为稳定平衡点。GAD近期已被推广至由等式约束描述并以外部公式形式给出的流形（微分代数方程）上的动力系统研究。本文提出将GAD扩展至由点云定义的流形，并采用内在视角进行公式化表述。这些点云在迭代过程中自适应采样，驱动系统从初始构象（通常位于稳定平衡点邻域）演化至鞍点。本方法仅需反应物（初始构象），无需显式指定约束方程，且完全基于数据驱动。