(298d) Staying the Course: Locating Fixed Points of Dynamical Systems (and Critical Points of Potentials) on Riemannian Manifolds Defined By Sampling Point-Clouds | AIChE

(298d) Staying the Course: Locating Fixed Points of Dynamical Systems (and Critical Points of Potentials) on Riemannian Manifolds Defined By Sampling Point-Clouds

Authors 

Bello-Rivas, J., Princeton University
Guckenheimer, J., Cornell University
Kevrekidis, I. G., Princeton University
Locating equilibria of dynamical systems is of primary importance to a variety of applications, including the exploration of energy landscapes in computational statistical mechanics [1], traversing energy-based models in deep learning [2]; and for locating the transition states in chemical systems [3]. Many of these high dimensional differential equations have dynamics that concentrate near much lower-dimensional manifolds, and simulations spend most of the time in a single basin of attraction.

We introduce a new method to locate these steady states on smooth Riemannian manifolds that are not necessarily known a priori. If the manifold is in Rn, our method works by following isoclines, smooth curves along where the vector field X is constant, which guarantee passing of equilibrium points. We generalize the definition of isoclines from Rn 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 and connect the equilibria of X. 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 path that joins equilibrium points. The algorithm successfully and reliably locates equilibria 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 M of a high-dimensional Euclidean space and a smooth vector field X on M. Here, we present the technique on a commonly used toy model in computational chemistry, the Müller-Brown potential [4], both mapped onto a sphere and a pseudosphere, and by foregoing a priori manifold knowledge and constructing its atlas on the fly via sampling and dimensionality reduction.

[1] D. J. Wales, Exploring Energy Landscapes, Annual Review of Physical Chemistry 69 (2018), no. 1, 401–425.

[2] Y. LeCun, S. Chopra, R. Hadsell, M. A. Ranzato, and F. J. Huang, A Tutorial on Energy-Based Learning, Predicting structured data (G. Bakir, T. Hofman, B. Sch¨olkopf, A. Smola, and B. Taskar, eds.), MIT Press, 2006 (English (US)).

[3] 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 113 (2013), no. 18, 2110–2142 (en).

[4] K. Müller and L. D. Brown, Location of Saddle Points and Minimum Energy Paths by a Constrained Simplex Optimization Procedure, Theoretica chimica acta 53 (1979), no. 1, 75–93 (en)