# Interpretable Design of Reservoir Computing Networks using Realization Theory

arXiv: https://arxiv.org/abs/2112.06891 · HTML: https://mlsanity.com/paper/2112.06891/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 41 references that are in the map.
- First posted on arXiv 13 Dec 2021.

## Builds on (most cited first, 11 of 11)

- [Towards A Rigorous Science of Interpretable Machine Learning](https://mlsanity.com/paper/1702.08608/) (2017)
- [The Mythos of Model Interpretability](https://mlsanity.com/paper/1606.03490/) (2016)
- [Universal discrete-time reservoir computers with stochastic inputs and linear readouts using non-homogeneous state-affine systems](https://mlsanity.com/paper/1712.00754/) (2017)
- [Differentiable reservoir computing](https://mlsanity.com/paper/1902.06094/) (2019)
- [Using a reservoir computer to learn chaotic attractors, with applications to chaos synchronisation and cryptography](https://mlsanity.com/paper/1802.02844/) (2018)
- [Memory and forecasting capacities of nonlinear recurrent networks](https://mlsanity.com/paper/2004.11234/) (2020)
- [Dimension reduction in recurrent networks by canonicalization](https://mlsanity.com/paper/2007.12141/) (2020)
- [The difference between memory and prediction in linear recurrent networks](https://mlsanity.com/paper/1706.09382/) (2017)
- [Echo State Networks with Self-Normalizing Activations on the Hyper-Sphere](https://mlsanity.com/paper/1903.11691/) (2019)
- [Input-to-State Representation in linear reservoirs dynamics](https://mlsanity.com/paper/2003.10585/) (2020)
- [Nonlinear memory capacity of parallel time-delay reservoir computers in the processing of multidimensional signals](https://mlsanity.com/paper/1510.03891/) (2015)

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

- [Quantum Observers: A NISQ Hardware Demonstration of Chaotic State Prediction Using Quantum Echo-state Networks](https://mlsanity.com/paper/2505.06799/) (2025)

## Abstract (arXiv)

The reservoir computing networks (RCNs) have been successfully employed as a tool in learning and complex decision-making tasks. Despite their efficiency and low training cost, practical applications of RCNs rely heavily on empirical design. In this paper, we develop an algorithm to design RCNs using the realization theory of linear dynamical systems. In particular, we introduce the notion of α-stable realization, and provide an efficient approach to prune the size of a linear RCN without deteriorating the training accuracy. Furthermore, we derive a necessary and sufficient condition on the irreducibility of number of hidden nodes in linear RCNs based on the concepts of controllability and observability matrices. Leveraging the linear RCN design, we provide a tractable procedure to realize RCNs with nonlinear activation functions. Finally, we present numerical experiments on forecasting time-delay systems and chaotic systems to validate the proposed RCN design methods and demonstrate their efficacy.