Path integral approach to universal dynamics of reservoir computers

TL;DR

This paper uses a path integral approach to classify reservoir computer networks into universality classes based on their structure and coupling distributions, revealing how these classes influence computational power and phase transitions.

Contribution

It introduces a universal characterization of reservoir networks via path integrals and links network structure to computational performance and phase behavior.

Findings

Classified networks into universality classes based on coupling distributions

Identified a close relationship between eigenvalue distributions and network universality

Demonstrated computational performance peaks near phase transition boundaries

Abstract

In this work, we give a characterization of the reservoir computer (RC) by the network structure, especially the probability distribution of random coupling constants. First, based on the path integral method, we clarify the universal behavior of the random network dynamics in the thermodynamic limit, which depends only on the asymptotic behavior of the second cumulant generating functions of the network coupling constants. This result enables us to classify the random networks into several universality classes, according to the distribution function of coupling constants chosen for the networks. Interestingly, it is revealed that such a classification has a close relationship with the distribution of eigenvalues of the random coupling matrix. We also comment on the relation between our theory and some practical choices of random connectivity in the RC. Subsequently, we investigate the relationship between the RC's computational power and the network parameters for several universality classes. We perform several numerical simulations to evaluate the phase diagrams of the steady reservoir states, common-signal-induced synchronization, and the computational power in the chaotic time series inference tasks. As a result, we clarify the close relationship between these quantities, especially a remarkable computational performance near the phase transitions, which is realized even near a non-chaotic transition boundary. These results may provide us with a new perspective on the designing principle for the RC.