Exploring the potential energy landscape of the Thomson problem via Newton homotopies

Mehta, Dhagash; Chen, Tianran; Morgan, John W. R.; Wales, David J.

doi:10.1063/1.4921163

Cited by 12 publications

(8 citation statements)

References 96 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…PRL 117, 028301 (2016) P H Y S I C A L R E V I E W L E T T E R S week ending 8 JULY 2016 028301-4 kinetic transition rates. We also plan to develop more specific algorithms to locate local and global minima by exploiting small-world properties [86][87][88]. Analyzing the appropriately weighted and directed networks of free energy minima [89] and transition states may provide additional insight into the Thomson problem.…”

Section: Prl 117 028301 (2016) P H Y S I C a L R E V I E W L E T T Ementioning

confidence: 99%

Kinetic Transition Networks for the Thomson Problem and Smale’s Seventh Problem

Mehta

Chen

et al. 2016

Phys. Rev. Lett.

Self Cite

View full text Add to dashboard Cite

Readers may view, browse, and/or download material for temporary copying purposes only, provided these uses are for noncommercial personal purposes. Except as provided by law, this material may not be further reproduced, distributed, transmitted, modied, adapted, performed, displayed, published, or sold in whole or part, without prior written permission from the American Physical Society.Additional information: Use policyThe full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. The Thomson problem, arrangement of identical charges on the surface of a sphere, has found many applications in physics, chemistry and biology. Here, we show that the energy landscape of the Thomson problem for N particles with N ¼ 132, 135, 138, 141, 144, 147, and 150 is single funneled, characteristic of a structure-seeking organization where the global minimum is easily accessible. Algorithmically, constructing starting points close to the global minimum of such a potential with spherical constraints is one of Smale's 18 unsolved problems in mathematics for the 21st century because it is important in the solution of univariate and bivariate random polynomial equations. By analyzing the kinetic transition networks, we show that a randomly chosen minimum is, in fact, always "close" to the global minimum in terms of the number of transition states that separate them, a characteristic of small world networks.

show abstract

Section: Prl 117 028301 (2016) P H Y S I C a L R E V I E W L E T T Ementioning

confidence: 99%

Kinetic Transition Networks for the Thomson Problem and Smale’s Seventh Problem

Mehta

Chen

et al. 2016

Phys. Rev. Lett.

Self Cite

View full text Add to dashboard Cite

show abstract

“…The possible applicability of quasi-Newton methods (Liu & Nocedal, 1989) for large problems remains unexplored and is left to future investigation. Newton-based homotopy methods (Mehta et al, 2015) are another promising future direction, based on preliminary results in (Coetzee & Stonick, 1997).…”

Section: Discussionmentioning

confidence: 99%

Numerically Recovering the Critical Points of a Deep Linear Autoencoder

Frye,

Wadia,

DeWeese

et al. 2019

Preprint

View full text Add to dashboard Cite

Numerically locating the critical points of nonconvex surfaces is a long-standing problem central to many fields. Recently, the loss surfaces of deep neural networks have been explored to gain insight into outstanding questions in optimization, generalization, and network architecture design. However, the degree to which recentlyproposed methods for numerically recovering critical points actually do so has not been thoroughly evaluated. In this paper, we examine this issue in a case for which the ground truth is known: the deep linear autoencoder. We investigate two sub-problems associated with numerical critical point identification: first, because of large parameter counts, it is infeasible to find all of the critical points for contemporary neural networks, necessitating sampling approaches whose characteristics are poorly understood; second, the numerical tolerance for accurately identifying a critical point is unknown, and conservative tolerances are difficult to satisfy. We first identify connections between recently-proposed methods and well-understood methods in other fields, including chemical physics, economics, and algebraic geometry. We find that several methods work well at recovering certain information about loss surfaces, but fail to take an unbiased sample of critical points. Furthermore, numerical tolerance must be very strict to ensure that numericallyidentified critical points have similar properties to true analytical critical points. We also identify a recently-published Newton method for optimization that outperforms previous methods as

show abstract

“…satisfies the smoothness condition with plenty choices of (a, b) ∈ R 2 and is capable of finding the singular solution (x, y) = (0, 0) with no difficulties. Parts of these analysis first appeared in [72,73].…”

Section: Strengths Of Homotopy Methods and Case Studiesmentioning

confidence: 99%

Exploring the potential energy landscape of the Thomson problem via Newton homotopies

Cited by 12 publications

References 96 publications

Kinetic Transition Networks for the Thomson Problem and Smale’s Seventh Problem

Kinetic Transition Networks for the Thomson Problem and Smale’s Seventh Problem

Numerically Recovering the Critical Points of a Deep Linear Autoencoder

Homotopy continuation method for solving systems of nonlinear and polynomial equations

Contact Info

Product

Resources

About