(241a) Locating Saddle Points of Dynamical Systems: Gentlest Ascent Dynamics & Gradient Extremals on Manifolds Defined By Adaptively Sampled Point Clouds

Locating saddle points of dynamical systems is of primary importance to a variety of applications, including the location of transition states of chemical systems described at an atomistic level [1]. Multiple algorithms to locate saddle points are currently in existence: Gentlest Ascent Dynamics (GAD) works by deriving a new dynamical system in which saddle points of the original system become stable equilibria [2] and Gradient Extremals (GE) define paths in which the gradient of a potential surface is an eigenvector of the Hessian of the potential surface, which traverse saddle points [3].

GAD has been recently generalized to manifolds described by equality constraints [4] and given an extrinsic formulation. Here, we extend both GAD and GE to locate saddle points on smooth manifolds defined by point clouds and formulated intrinsically. The point clouds are adaptively sampled during an iterative process that drives the system from an initial conformation (typically a stable equilibrium) to a saddle point. Further, when the manifold is unknown a priori and defined only by point clouds, we couple the method with manifold learning techniques (here, diffusion maps) and Gaussian process regression to obtain the sought-after paths to the saddle points. Our methodology successfully and reliably locates saddle points using a single initial point and without need for a priori knowledge of a set collective variables.

The technique addresses applications in which have a compact low-dimensional sub-manifold Μ of a high-dimensional Euclidean space and a smooth vector field X on Μ. Here, we present our method on a simple potential mapped onto a sphere and by foregoing a priori manifold knowledge and constructing its atlas on the fly via sampling and dimensionality reduction.

[1] A. D. Bochevarov, E. Harder, T. F. Hughes, J. R. Greenwood, D. A. Braden, D. M. Philipp, D. Rinaldo, M. D. Halls, J. Zhang, and R. A. Friesner, Jaguar: A HighPerformance Quantum Chemistry Software Program with Strengths in Life and Materials Sciences, International Journal of Quantum Chemistry, 2013.

[2] E, W.; Zhou, X. The Gentlest Ascent Dynamics. Nonlinearity, 2011.

[3] Hoffman, D., Nord, R., Ruedenbergy, K. Gradient extremals. Theoretica chimica acts, 1986.

[4] Yin, J.; Huang, Z.; Zhang, L. Constrained High-Index Saddle Dynamics for the Solution Landscape with Equality Constraints. Journal of Scientific Computing 2022, 91, 62.