Emulating complex networks with a single delay differential equation

Abstract

A single dynamical system with time-delayed feedback can emulate networks. This property of delay systems made them extremely useful tools for Machine-Learning applications. Here, we describe several possible setups, which allow emulating multilayer (deep) feed-forward networks as well as recurrent networks of coupled discrete maps with arbitrary adjacency matrix by a single system with delayed feedback. While the network’s size can be arbitrary, the generating delay system can have a low number of variables, including a scalar case.

Dynamical phenomena in complex networks: fundamentals and applications

“…Among the possible further developments, the applications of complex dynamical networks to machine learning can play an important role. For example, the presented approach for constructing weighted networks with arbitrary topology from a single dynamical node with feedback [14] may increase the synergy between the community working in dynamical systems and machine learning.…”

“…Stelzer and Yanchuk [14] show how a single dynamical system with time-delayed feedback can emulate networks. This property of delay systems made them extremely useful tools for Machine Learning applications.…”

“…We subdivide the contributions into three main fields: theory and methods, applications to neuroscience, and Earth science applications. The theoretical aspects considered here cover a broad spectrum of topics such as collective dynamics of excitable systems, cluster dynamics [1], interplay of noise and feedback [2], subdiffusive behavior [5], spreading phenomena on networks [8], coarsening [9], multipartite networks, partial synchrony [11], and time-delayed interactions [13,14]. In a series of works, complex networks are successfully employed as tools for detecting artificially inserted data [3], community detection [4,12], identification of time series patterns [6], and modeling a voter dynamics [15].…”

Infinite-dimensional Folded-in-time Deep Neural Networks

“…They have been recently used for finite-dimensional Fit-DNNs [1]. In fact, delay-differential equations have been shown to emulate arbitrary networks of coupled maps [33,62,63] as well as certain types of partial differential equations [43,[64][65][66][67]. Such rich spatiotemporal properties and their infinite-dimensionality make such systems a useful instrument in machine learning applications.…”

Master memory function for delay-based reservoir computers with single-variable dynamics

“…In timedelayed reservoir computing, a single dynamical node with delayed feedback is employed as a reservoir instead of the network [33]. The time-multiplexing procedure allows for such a single-element system to implement a recurrent ring network [33][34][35], see Fig. 1.…”