# Improving the Performance of Echo State Networks Through State Feedback

arXiv: https://arxiv.org/abs/2312.15141 · HTML: https://mlsanity.com/paper/2312.15141/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 48 references that are in the map.
- First posted on arXiv 23 Dec 2023.

## Builds on (most cited first, 9 of 9)

- [Echo state networks are universal](https://mlsanity.com/paper/1806.00797/) (2018)
- [Reservoir Computing Universality With Stochastic Inputs](https://mlsanity.com/paper/1807.02621/) (2018)
- [Temporal Information Processing on Noisy Quantum Computers](https://mlsanity.com/paper/2001.09498/) (2020)
- [Using a reservoir computer to learn chaotic attractors, with applications to chaos synchronisation and cryptography](https://mlsanity.com/paper/1802.02844/) (2018)
- [A Unifying Framework for Information Processing in Stochastically Driven Dynamical Systems](https://mlsanity.com/paper/1906.04608/) (2019)
- [Harnessing disordered quantum dynamics for machine learning](https://mlsanity.com/paper/1602.08159/) (2016)
- [Convergence Properties for Discrete-time Nonlinear Systems](https://mlsanity.com/paper/1612.05327/) (2016)
- [Nonlinear Autoregression with Convergent Dynamics on Novel Computational Platforms](https://mlsanity.com/paper/2108.08001/) (2021)
- [Composite FORCE learning of chaotic echo state networks for time-series prediction](https://mlsanity.com/paper/2207.02420/) (2022)

## Built on by (most cited first, 2 of 2)

- [Optical Echo State Network Reservoir Computing](https://mlsanity.com/paper/2504.08224/) (2025)
- [A Solvable Molecular Switch Model for Stable Temporal Information Processing](https://mlsanity.com/paper/2508.15451/) (2025)

## Abstract (arXiv)

Reservoir computing, using nonlinear dynamical systems, offers a cost-effective alternative to neural networks for complex tasks involving processing of sequential data, time series modeling, and system identification. Echo state networks (ESNs), a type of reservoir computer, mirror neural networks but simplify training. They apply fixed, random linear transformations to the internal state, followed by nonlinear changes. This process, guided by input signals and linear regression, adapts the system to match target characteristics, reducing computational demands. A potential drawback of ESNs is that the fixed reservoir may not offer the complexity needed for specific problems. While directly altering (training) the internal ESN would reintroduce the computational burden, an indirect modification can be achieved by redirecting some output as input. This feedback can influence the internal reservoir state, yielding ESNs with enhanced complexity suitable for broader challenges. In this paper, we demonstrate that by feeding some component of the reservoir state back into the network through the input, we can drastically improve upon the performance of a given ESN. We rigorously prove that, for any given ESN, feedback will almost always improve the accuracy of the output. For a set of three tasks, each representing different problem classes, we find that with feedback the average error measures are reduced by 30%-60%. Remarkably, feedback provides at least an equivalent performance boost to doubling the initial number of computational nodes, a computationally expensive and technologically challenging alternative. These results demonstrate the broad applicability and substantial usefulness of this feedback scheme.