Projective embedding of dynamical systems: Uniform mean field equations

Journal Article · 26 April 2023 · Physica. D, Nonlinear Phenomena

DOI: https://doi.org/10.1016/j.physd.2023.133747 · OSTI ID:1972980

Caravelli, Francesco  ^[1]; Traversa, Fabio Lorenzo ^[2]; Bonnin, Michele ^[3]; Bonani, Fabrizio ^[3]

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1. Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
2. MemComputing, Inc., San Diego, CA (United States)
3. Politecnico di Torino, Turin (Italy)

Herein we study embeddings of continuous dynamical systems in larger dimensions via projector operators. We call this technique PEDS, projective embedding of dynamical systems, as the stable fixed point of the original system dynamics are recovered via projection from the higher dimensional space. In this paper we provide a general definition and prove that for a particular type of rank-1 projector operator, the uniform mean field projector, the equations of motion become a mean field approximation of the dynamical system. While in general the embedding depends on a specified variable ordering, the same is not true for the uniform mean field projector. We prove a variety of results on the relationship between the spectrum of the Jacobian for fixed points in the original and in the embedded system. Direct applications of PEDS can be non-convex optimization and machine learning.

• Research Organization: Los Alamos National Laboratory (LANL), Los Alamos, NM (United States)
• Sponsoring Organization: USDOE Laboratory Directed Research and Development (LDRD) Program; USDOE National Nuclear Security Administration (NNSA)
• Grant/Contract Number: 89233218CNA000001; AC52-06NA25396
• OSTI ID: 1972980
• Report Number(s): LA-UR-21-29420
• Journal Information: Physica. D, Nonlinear Phenomena, Journal Name: Physica. D, Nonlinear Phenomena Vol. 450; ISSN 0167-2789
• Publisher: ElsevierCopyright Statement
• Country of Publication: United States
• Language: English