On Dominant Manifolds in Reservoir Computing Networks

arXiv cs.LG / 4/8/2026

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Key Points

The paper studies how training recurrent Reservoir Computing (RC) networks shapes the geometry of their dynamics, focusing on the emergence of low-dimensional “dominant manifolds” for temporal forecasting.
For a simplified linear, continuous-time reservoir model, it derives explicit links between the dimensionality/structure of dominant modes and the intrinsic dimensionality and information content of the training data.
When training data come from an autonomous dynamical system, the dominant modes are shown to approximate Koopman eigenfunctions of the underlying system.
The authors clarify a direct relationship between reservoir computing and Dynamic Mode Decomposition (DMD) by connecting dominant reservoir modes to Koopman-based representations.
They illustrate how eigenvalue motion during training generates dominant manifolds in simulation and discuss extending the framework to nonlinear RC using tangent dynamics and differential p-dominance.

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.