# Minimal Deterministic Echo State Networks Outperform Random Reservoirs in Learning Chaotic Dynamics

arXiv: https://arxiv.org/abs/2507.06050 · HTML: https://mlsanity.com/paper/2507.06050/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 30 of its 90 references that are in the map.
- First posted on arXiv 8 Jul 2025.

## Builds on (most cited first, 30 of 30)

- [Next Generation Reservoir Computing](https://mlsanity.com/paper/2106.07688/) (2021)
- [Chaos as an interpretable benchmark for forecasting and data-driven modelling](https://mlsanity.com/paper/2110.05266/) (2021)
- [Echo state networks are universal](https://mlsanity.com/paper/1806.00797/) (2018)
- [Forecasting Chaotic Systems with Very Low Connectivity Reservoir Computers](https://mlsanity.com/paper/1910.00659/) (2019)
- [Reservoir computing approaches for representation and classification of multivariate time series](https://mlsanity.com/paper/1803.07870/) (2018)
- [Model-free prediction of spatiotemporal dynamical systems with recurrent neural networks: Role of network spectral radius](https://mlsanity.com/paper/1910.04426/) (2019)
- [Using Machine Learning to Anticipate Tipping Points and Extrapolate to Post-Tipping Dynamics of Non-Stationary Dynamical Systems](https://mlsanity.com/paper/2207.00521/) (2022)
- [A Systematic Exploration of Reservoir Computing for Forecasting Complex Spatiotemporal Dynamics](https://mlsanity.com/paper/2201.08910/) (2022)
- [Zero-shot forecasting of chaotic systems](https://mlsanity.com/paper/2409.15771/) (2024)
- [Robust Optimization and Validation of Echo State Networks for learning chaotic dynamics](https://mlsanity.com/paper/2103.03174/) (2021)
- [Generative learning for nonlinear dynamics](https://mlsanity.com/paper/2311.04128/) (2023)
- [Echo State Networks trained by Tikhonov least squares are L2(μ) approximators of ergodic dynamical systems](https://mlsanity.com/paper/2005.06967/) (2020)
- [Model-free inference of unseen attractors: Reconstructing phase space features from a single noisy trajectory using reservoir computing](https://mlsanity.com/paper/2108.04074/) (2021)
- [True Zero-Shot Inference of Dynamical Systems Preserving Long-Term Statistics](https://mlsanity.com/paper/2505.13192/) (2025)
- [Learning strange attractors with reservoir systems](https://mlsanity.com/paper/2108.05024/) (2021)
- [Panda: A pretrained forecast model for chaotic dynamics](https://mlsanity.com/paper/2505.13755/) (2025)
- [Determination of the edge of criticality in echo state networks through Fisher information maximization](https://mlsanity.com/paper/1603.03685/) (2016)
- [Learning unseen coexisting attractors](https://mlsanity.com/paper/2207.14133/) (2022)
- [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)
- [ReservoirComputing.jl: An Efficient and Modular Library for Reservoir Computing Models](https://mlsanity.com/paper/2204.05117/) (2022)
- [Simple Cycle Reservoirs are Universal](https://mlsanity.com/paper/2308.10793/) (2023)
- [Optimal Recurrent Network Topologies for Dynamical Systems Reconstruction](https://mlsanity.com/paper/2406.04934/) (2024)
- [Chaotic attractor reconstruction using small reservoirs -- the influence of topology](https://mlsanity.com/paper/2402.16888/) (2024)
- [Learning Continuous Chaotic Attractors with a Reservoir Computer](https://mlsanity.com/paper/2110.08631/) (2021)
- [Machine Learning for Predicting Chaotic Systems](https://mlsanity.com/paper/2407.20158/) (2024)
- [Composing Modeling and Simulation with Machine Learning in Julia](https://mlsanity.com/paper/2105.05946/) (2021)
- [Universality of Real Minimal Complexity Reservoir](https://mlsanity.com/paper/2408.08071/) (2024)
- [Linear Simple Cycle Reservoirs at the edge of stability perform Fourier decomposition of the input driving signals](https://mlsanity.com/paper/2412.00295/) (2024)
- [Deterministic Reservoir Computing for Chaotic Time Series Prediction](https://mlsanity.com/paper/2501.15615/) (2025)
- [Learning Hamiltonian dynamics by reservoir computer](https://mlsanity.com/paper/2104.14474/) (2021)

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

- [Storage and selection of multiple chaotic attractors in minimal reservoir computers](https://mlsanity.com/paper/2603.15155/) (2026)
- [Rethinking Reservoir Pruning: A Dynamical Perspective for Echo State Networks](https://mlsanity.com/paper/2608.04593/) (2026)

## Abstract (arXiv)

Machine learning (ML) is widely used to model chaotic systems. Among ML approaches, echo state networks (ESNs) have received considerable attention due to their simple construction and fast training. However, ESN performance is highly sensitive to hyperparameter choices and to its random initialization. In this work, we demonstrate that ESNs constructed using deterministic rules and simple topologies (MESNs) outperform standard ESNs in the task of chaotic attractor reconstruction. We use a dataset of more than 90 chaotic systems to benchmark 10 different minimal deterministic reservoir initializations. We find that MESNs obtain up to a 41% reduction in error compared to standard ESNs. Furthermore, we show that the MESNs are more robust, exhibiting less inter-run variation, and have the ability to reuse hyperparameters across different systems. Our results illustrate how structured simplicity in ESN design can outperform stochastic complexity in learning chaotic dynamics.