Staying the course: iteratively locating equilibria of dynamical systems on Riemannian manifolds defined by point-clouds

Bello-Rivas, Juan M.; Georgiou, Anastasia S.; Kevrekidis, Ioannis G.

doi:10.1007/s10910-022-01425-9

Cited by 3 publications

(3 citation statements)

References 61 publications

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“…Our contribution is for-mulated intrinsically and is valid on arbitrary manifolds, not necessarily explicitly defined by an atlas or by the zeros of maps. Algorithms like the one presented here or our previous work 19 do not rely on a priori knowledge of good collective coordinates, but rather use manifold learning to find them on the fly. In our method, there is a feedback loop of data collection that drives progress towards a saddle point.…”

Section: Introductionmentioning

confidence: 99%

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

Bello-Rivas¹,

Georgiou²,

Vandecasteele³

et al. 2023

Preprint

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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) 1 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 2 and given an extrinsic formulation.In this paper, we present an extension of GAD to manifolds defined by point-clouds and formulated intrinsically. 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

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Section: Introductionmentioning

confidence: 99%

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

Bello-Rivas¹,

Georgiou²,

Vandecasteele³

et al. 2023

Preprint

View full text Add to dashboard Cite

show abstract

“…Our contribution is formulated intrinsically and is valid on arbitrary manifolds, not necessarily explicitly defined by an atlas or by the zeros of maps. Importantly, algorithms like the one presented here or our previous work 19 do not rely on a priori knowledge of good collective coordinates but rather use manifold learning to find them on the fly. In our method, there is a feedback loop of data collection that drives progress toward a saddle point.…”

Section: ■ Introductionmentioning

confidence: 99%

“…Since the saddle point in general is not expected to lie in the vicinity of the reactant, our algorithm works by iteratively sampling the manifold on the fly, resolving the path on the local chart, and repeatedly switching charts until convergence. Our approach shares algorithmic elements with our previous work 19 which, however, instead of GAD dynamics on manifolds, was following isoclines on manifolds.…”

Section: ■ Introductionmentioning

confidence: 99%

Gentlest Ascent Dynamics on Manifolds Defined by Adaptively Sampled Point-Clouds

et al. 2023

Self Cite

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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) (10.1088/0951-7715/24/6/008) is one of a number of algorithms in existence that attempt to find saddle points. 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 (10.1007/s10915-022-01838-3) and given in an extrinsic formulation. In this paper, we present an extension of GAD to manifolds defined by point-clouds, formulated by using an 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.

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

Georgiou,

Vandecasteele,

Bello-Rivas

et al. 2023

Chaos: An Interdisciplinary Journal of Nonlinear Science

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Steady states are invaluable in the study of dynamical systems. High-dimensional dynamical systems, due to separation of time scales, often evolve toward 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 Müller–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.

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Staying the course: iteratively locating equilibria of dynamical systems on Riemannian manifolds defined by point-clouds

Cited by 3 publications

References 61 publications

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

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

Gentlest Ascent Dynamics on Manifolds Defined by Adaptively Sampled Point-Clouds

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

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