Contraction and Synchronization in Reservoir Systems

TL;DR

This paper investigates the conditions for contraction and synchronization in reservoir systems, providing guidelines for their construction and exploring their universal approximation capabilities for modeling dynamical systems.

Contribution

It introduces new conditions for global contraction in continuous-time reservoirs and links their universal approximation properties to neural ODEs and activation functions.

Findings

Contraction conditions are derived using the logarithmic norm.

Guidelines for constructing connectivity matrices are provided.

Reservoirs can universally approximate topological conjugates.

Abstract

This paper explores the conditions under which global contraction manifests in the leaky continuous time reservoirs, thus guaranteeing generalized synchronization. Results on continuous time reservoirs make use of the logarithmic norm of the connectivity matrix. Further analysis yields some simple guidelines on how to better construct the connectivity matrix in these systems. Additionally, we outline how the universal approximation property of discrete time reservoirs is readily satisfied by virtue of the activation function being contracting, and how continuous time reservoirs may inherit a limited form of universal approximation due to their overlap with neural ordinary differential equations. The ability of the reservoir computing framework to universally approximate topological conjugates, along with their fast training, make them a compelling data-driven, black-box surrogate of dynamical systems, and a potential candidate for a component of digital twins.