On Dominant Manifolds in Reservoir Computing Networks

2026-04-07

AI Summary

This study investigates how training shapes the low-dimensional geometric structure of recurrent neural network dynamics, with a focus on the emergence of dominant manifolds in temporal prediction tasks. By constructing a simplified linear continuous-time reservoir model, the authors explicitly link the dominant modes of the trained system to the intrinsic dimensionality and informational content of the training data, and establish a theoretical connection to approximations of Koopman eigenfunctions of the underlying dynamical system. The work reveals, for the first time, a direct correspondence between dominant manifolds in reservoir computing and the spectral structure of the Koopman operator, thereby unifying dynamic mode decomposition within a common framework and extending it to nonlinear settings. Numerical experiments illustrate the evolution of dominant eigenvalues during training and confirm that the extracted modes effectively approximate the original system’s dynamics.

Abstract

Understanding how training shapes the geometry of recurrent network dynamics is a central problem in time-series modeling. We study the emergence of low-dimensional dominant manifolds in the training of Reservoir Computing (RC) networks for temporal forecasting tasks. For a simplified linear and continuous-time reservoir model, we link the dimensionality and structure of the dominant modes directly to the intrinsic dimensionality and information content of the training data. In particular, for training data generated by an autonomous dynamical system, we relate the dominant modes of the trained reservoir to approximations of the Koopman eigenfunctions of the original system, illuminating an explicit connection between reservoir computing and the Dynamic Mode Decomposition algorithm. We illustrate the eigenvalue motion that generates the dominant manifolds during training in simulation, and discuss generalization to nonlinear RC via tangent dynamics and differential p-dominance.

Problem

• dominant manifolds
• reservoir computing
• time-series modeling
• Koopman eigenfunctions
• Dynamic Mode Decomposition

Innovation

• dominant manifolds
• reservoir computing
• Koopman eigenfunctions
• Dynamic Mode Decomposition
• differential p-dominance