# Learning strange attractors with reservoir systems

arXiv: https://arxiv.org/abs/2108.05024 · HTML: https://mlsanity.com/paper/2108.05024/

## Key facts

- Cited by 17 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 56 references that are in the map.
- First posted on arXiv 11 Aug 2021.

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

- [Next Generation Reservoir Computing](https://mlsanity.com/paper/2106.07688/) (2021)
- [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)
- [Fading memory echo state networks are universal](https://mlsanity.com/paper/2010.12047/) (2020)
- [Echo State Networks trained by Tikhonov least squares are L2(μ) approximators of ergodic dynamical systems](https://mlsanity.com/paper/2005.06967/) (2020)
- [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)
- [Learn to Synchronize, Synchronize to Learn](https://mlsanity.com/paper/2010.02860/) (2020)

## Built on by (most cited first, 17 of 17)

- [Catch-22s of reservoir computing](https://mlsanity.com/paper/2210.10211/) (2022)
- [Infinite-dimensional next-generation reservoir computing](https://mlsanity.com/paper/2412.09800/) (2024)
- [Reservoir kernels and Volterra series](https://mlsanity.com/paper/2212.14641/) (2022)
- [Infinite-dimensional reservoir computing](https://mlsanity.com/paper/2304.00490/) (2023)
- [Learning mechanical systems from real-world data using discrete forced Lagrangian dynamics](https://mlsanity.com/paper/2505.20370/) (2025)
- [Memory of recurrent networks: Do we compute it right?](https://mlsanity.com/paper/2305.01457/) (2023)
- [State-space systems as dynamic generative models](https://mlsanity.com/paper/2404.08717/) (2024)
- [On the emergence of numerical instabilities in Next Generation Reservoir Computing](https://mlsanity.com/paper/2505.00846/) (2025)
- [Stochastic dynamics learning with state-space systems](https://mlsanity.com/paper/2508.07876/) (2025)
- [Minimal Deterministic Echo State Networks Outperform Random Reservoirs in Learning Chaotic Dynamics](https://mlsanity.com/paper/2507.06050/) (2025)
- [Thermodynamic limit in learning period three](https://mlsanity.com/paper/2405.08825/) (2024)
- [Unsupervised Learning in Echo State Networks for Input Reconstruction](https://mlsanity.com/paper/2501.11409/) (2025)
- [Forecasting the Forced van der Pol Equation with Frequent Phase Shifts Using Reservoir Computing](https://mlsanity.com/paper/2404.14651/) (2024)
- [On the dimension of pullback attractors in recurrent neural networks](https://mlsanity.com/paper/2501.11357/) (2025)
- [A tensor network approach for chaotic time series prediction](https://mlsanity.com/paper/2505.17740/) (2025)
- [Data-Specific Hyper-Parameter Design: A Paradigm Shift in Reservoir Computing](https://mlsanity.com/paper/2605.25221/) (2026)
- [A KKL Observer Perspective on Reservoir Computing](https://mlsanity.com/paper/2610.04343/) (2026)

## Abstract (arXiv)

This paper shows that the celebrated Embedding Theorem of Takens is a particular case of a much more general statement according to which, randomly generated linear state-space representations of generic observations of an invertible dynamical system carry in their wake an embedding of the phase space dynamics into the chosen Euclidean state space. This embedding coincides with a natural generalized synchronization that arises in this setup and that yields a topological conjugacy between the state-space dynamics driven by the generic observations of the dynamical system and the dynamical system itself. This result provides additional tools for the representation, learning, and analysis of chaotic attractors and sheds additional light on the reservoir computing phenomenon that appears in the context of recurrent neural networks.