# Embedding Theory of Reservoir Computing and Reducing Reservoir Network Using Time Delays

arXiv: https://arxiv.org/abs/2303.09042 · HTML: https://mlsanity.com/paper/2303.09042/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- Builds on 6 of its 54 references that are in the map.
- First posted on arXiv 16 Mar 2023.

## Builds on (most cited first, 6 of 6)

- [Next Generation Reservoir Computing](https://mlsanity.com/paper/2106.07688/) (2021)
- [Forecasting Chaotic Systems with Very Low Connectivity Reservoir Computers](https://mlsanity.com/paper/1910.00659/) (2019)
- [Model-free prediction of spatiotemporal dynamical systems with recurrent neural networks: Role of network spectral radius](https://mlsanity.com/paper/1910.04426/) (2019)
- [Echo State Networks trained by Tikhonov least squares are L2(μ) approximators of ergodic dynamical systems](https://mlsanity.com/paper/2005.06967/) (2020)
- [Model-Size Reduction for Reservoir Computing by Concatenating Internal States Through Time](https://mlsanity.com/paper/2006.06218/) (2020)
- [Effect of shapes of activation functions on predictability in the echo state network](https://mlsanity.com/paper/1905.09419/) (2019)

## Built on by (most cited first, 10 of 10)

- [Attractor reconstruction with reservoir computers: The effect of the reservoir's conditional Lyapunov exponents on faithful attractor reconstruction](https://mlsanity.com/paper/2401.00885/) (2023)
- [Controlling Chaotic Maps using Next-Generation Reservoir Computing](https://mlsanity.com/paper/2307.03813/) (2023)
- [Position: A Dynamical Systems Perspective is Needed to Advance Time Series Modeling](https://mlsanity.com/paper/2602.16864/) (2026)
- [Modeling Nonlinear Oscillator Networks Using Physics-Informed Hybrid Reservoir Computing](https://mlsanity.com/paper/2411.05867/) (2024)
- [Mixed Delay/Nondelay Embeddings Based Neuromorphic Computing with Patterned Nanomagnet Arrays](https://mlsanity.com/paper/2412.04622/) (2024)
- [Emergence Transformer: Dynamical Temporal Attention Matters](https://mlsanity.com/paper/2604.19816/) (2026)
- [Forecasting the Forced van der Pol Equation with Frequent Phase Shifts Using Reservoir Computing](https://mlsanity.com/paper/2404.14651/) (2024)
- [Spiking Neural Networks with Random Network Architecture](https://mlsanity.com/paper/2505.13622/) (2025)
- [Embedding of Low-Dimensional Sensory Dynamics in Recurrent Networks: Implications for the Geometry of Neural Representation](https://mlsanity.com/paper/2601.19019/) (2026)
- [Effective Training Principles of Physical Reservoirs](https://mlsanity.com/paper/2606.10130/) (2026)

## Abstract (arXiv)

Reservoir computing (RC), a particular form of recurrent neural network, is under explosive development due to its exceptional efficacy and high performance in reconstruction or/and prediction of complex physical systems. However, the mechanism triggering such effective applications of RC is still unclear, awaiting deep and systematic exploration. Here, combining the delayed embedding theory with the generalized embedding theory, we rigorously prove that RC is essentially a high dimensional embedding of the original input nonlinear dynamical system. Thus, using this embedding property, we unify into a universal framework the standard RC and the time-delayed RC where we novelly introduce time delays only into the network's output layer, and we further find a trade-off relation between the time delays and the number of neurons in RC. Based on this finding, we significantly reduce the network size of RC for reconstructing and predicting some representative physical systems, and, more surprisingly, only using a single neuron reservoir with time delays is sometimes sufficient for achieving those tasks.