Stochastic dynamics and combinatorial optimization

Abstract

Natural dynamics is often dominated by sudden nonlinear processes such as neuroavalanches, gamma-ray bursts, solar flares etc. that exhibit scale-free statistics much in the spirit of the logarithmic Ritcher scale for earthquake magnitudes. On phase diagrams, stochastic dynamical systems (DSs) exhibiting this type of dynamics belong to the finite-width phase (N-phase for brevity) that precedes ordinary chaotic behavior and that is known under such names as noise-induced chaos, selforganized criticality, dynamical complexity etc. Within the recently formulated approximation-free supersymemtric theory of stochastics, the N-phase can be roughly interpreted as the noise-induced "overlap" between integrable and chaotic deterministic dynamics. As a result, the N-phase dynamics inherits the properties of the both. Here, we analyze this unique set of properties and conclude that the N-phase DSs must naturally be the most efficient optimizers: on one hand, N-phase DSs have integrable flows with well-defined attractors that can be associated with candidate solutions and, on the other hand, the noise-induced attractor-to-attractor dynamics in the N-phase is effectively chaotic or a-periodic so that a DS must avoid revisiting solutions/attractors thus accelerating the search for the best solution. Based on this understanding, we propose a method for stochastic dynamical optimization using the N-phase DSs. This method can be viewed as a hybrid of the simulated and chaotic annealing methods. Our proposition can result in a new generation of hardware devices for efficient solution of various search and/or combinatorial optimization problems.

Consciousness is supported by near-critical cortical electrodynamics

“…Finally, we note that it would be fruitful to further study neural computation near the edge-of-chaos critical point on a more theoretical level. While important advances have been made along these lines, for example in establishing relationships between this critical point and the trainability of deep neural networks (24), information complexity (6–9), Bayes-optimal perceptual categorization (55), and combinatorial optimization (56), much theoretical work remains to be done to understand the implications of these findings for neural computation. If the electrodynamics of the cortex during conscious states operate near this critical point, as our work suggests, then improving our theoretical understanding of computation at the onset of chaos will also improve our understanding of how, precisely, neural computation is disrupted in unconsciousness.…”

A thermodynamic core using voltage-controlled spin–orbit-torque magnetic tunnel junctions

“…It should be noted, however, that while this work demonstrates the use of our hardware for simulating a Boltzmann machine, the theoretical and algorithmic concepts of stochastic dynamics can reach far beyond that of this work. For example, it is well known that the noise level is crucial to the optimization process of stochastic processes, and correlates with its dynamic phase space [31,32]. A dynamically-adjusted stochastic system optimized with respect to its noise levels belongs to a finite-width phase that precedes ordinary chaotic behavior and deterministic computation [31].…”

“…For example, it is well known that the noise level is crucial to the optimization process of stochastic processes, and correlates with its dynamic phase space [39,40]. A dynamicallyadjusted stochastic system optimized with respect to its noise levels belongs to a finite-width phase that precedes ordinary chaotic behavior and deterministic computation [39]. Within the recently proposed supersymmetric theory of stochastic dynamics, this phase is a manifestation of the breakdown of topological supersymmetry [41,42], and is directly related to phenomena described in the literature as noise-induced chaos, self-organized criticality, and dynamical complexity [43].…”