Statistics of attractor embeddings in reservoir computing

A recent branch of AI or Neural Networks that can handle time-varying signals often in real time has emerged as a new direction for signal analysis. These dynamical systems are usually referred to as reservoir computers. A central question in the operation of these systems is whether a reservoir computer (RC) when driven by only one time series from a driving or source system is internally recreating all the drive dynamics or attractor itself., i.e. an embedding of the drive attractor in the RC dynamics. There are some mathematical advances that move that argument closer to a general theorem. However, for RCs constructed from actual physical systems like interacting lasers or analog circuits, the RC dynamics may not be known well or at all. We present a statistic that can help test for homeomorphisms between a drive system and the RC by using the time series from both systems. This statistic is called the continuity statistic and it is modeled on the mathematical definition of a continuous function. We show the interplay of dynamical quantities (e.g. Lyapunov exponents, Kaplan-Yorke dimensions, generalized synchronization, etc.) and embeddings as exposed by the continuity statistic and other statistics based on ideas from nonlinear dynamical systems theory. These viewpoints and results lead to a clarification of various currently vague concepts about RCs, such as fading memory, stability, and types of dynamics that are useful.

• Affiliation : Institute for Research in Electronics and Applied Physics, University of Maryland, College Park, MD