# On Dominant Manifolds in Reservoir Computing Networks

arXiv: https://arxiv.org/abs/2604.05967 · HTML: https://mlsanity.com/paper/2604.05967/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 25 references that are in the map.
- First posted on arXiv 7 Apr 2026.

## Builds on (most cited first, 8 of 8)

- [Forecasting Chaotic Systems with Very Low Connectivity Reservoir Computers](https://mlsanity.com/paper/1910.00659/) (2019)
- [Fading memory echo state networks are universal](https://mlsanity.com/paper/2010.12047/) (2020)
- [Online dynamic mode decomposition for time-varying systems](https://mlsanity.com/paper/1707.02876/) (2017)
- [Dynamics of random recurrent networks with correlated low-rank structure](https://mlsanity.com/paper/1909.04358/) (2019)
- [Quality of internal representation shapes learning performance in feedback neural networks](https://mlsanity.com/paper/2011.06066/) (2020)
- [Forward-Backward Extended DMD with an Asymptotic Stability Constraint](https://mlsanity.com/paper/2403.10623/) (2024)
- [A dissipativity theorem for p-dominant systems](https://mlsanity.com/paper/1709.06155/) (2017)
- [Differential dissipativity theory for dominance analysis](https://mlsanity.com/paper/1710.01721/) (2017)

## Built on by (most cited first, 1 of 1)

- [Koopman Identification of Nonlinear Systems via Reservoir Liftings](https://mlsanity.com/paper/2605.04917/) (2026)

## Abstract (arXiv)

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 general linear continuous-time reservoir in the infinite-data limit, we show that the training data generate an invariant subspace of the trained reservoir, whose dimension equals the number of dominant modes. We then specialize to a simplified diagonal linear reservoir, where we link the dominant eigenvalues and eigenvectors to the spectrum of a backward Dynamic Mode Decomposition matrix, which yields a finite-dimensional approximation of the backward-time Koopman operator of the system generating the training data. We illustrate the emergence of these dominant modes during training in simulation, and discuss how the analysis may be extended to nonlinear RC via tangent dynamics and differential p-dominance.