# Incorporating Coupling Knowledge into Echo State Networks for Learning Spatiotemporally Chaotic Dynamics

arXiv: https://arxiv.org/abs/2504.01532 · HTML: https://mlsanity.com/paper/2504.01532/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 44 references that are in the map.
- First posted on arXiv 2 Apr 2025.

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

- [Learning Phrase Representations using RNN Encoder-Decoder for Statistical Machine Translation](https://mlsanity.com/paper/1406.1078/) (2014)
- [Towards Physics-informed Deep Learning for Turbulent Flow Prediction](https://mlsanity.com/paper/1911.08655/) (2019)
- [Backpropagation Algorithms and Reservoir Computing in Recurrent Neural Networks for the Forecasting of Complex Spatiotemporal Dynamics](https://mlsanity.com/paper/1910.05266/) (2019)
- [Physical reservoir computing -- An introductory perspective](https://mlsanity.com/paper/2005.00992/) (2020)
- [Generalized Teacher Forcing for Learning Chaotic Dynamics](https://mlsanity.com/paper/2306.04406/) (2023)
- [Hybrid Forecasting of Chaotic Processes: Using Machine Learning in Conjunction with a Knowledge-Based Model](https://mlsanity.com/paper/1803.04779/) (2018)
- [Machine Learning Link Inference of Noisy Delay-coupled Networks with Opto-Electronic Experimental Tests](https://mlsanity.com/paper/2010.15289/) (2020)
- [Embedding bifurcations into pneumatic artificial muscle](https://mlsanity.com/paper/2305.03994/) (2023)
- [Learn one size to infer all: Exploiting translational symmetries in delay-dynamical and spatio-temporal systems using scalable neural networks](https://mlsanity.com/paper/2111.03706/) (2021)

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

- [Causal Local States: Scalable Simultaneous Causal Network Inference and Forecasting for Dynamical Systems](https://mlsanity.com/paper/2608.17452/) (2026)

## Abstract (arXiv)

Machine learning methods have shown promise in learning chaotic dynamical systems, enabling model-free short-term prediction and attractor reconstruction. However, when applied to large-scale, spatiotemporally chaotic systems, purely data-driven machine learning methods often suffer from inefficiencies, as they require a large learning model size and a massive amount of training data to achieve acceptable performance. To address this challenge, we incorporate the spatial coupling structure of the target system as an inductive bias in the network design. Specifically, we introduce physics-guided clustered echo state networks, leveraging the efficiency of the echo state networks as a base model. Experimental results on benchmark chaotic systems demonstrate that our physics-informed method outperforms existing echo state network models in learning the target chaotic systems. Additionally, we numerically demonstrate that leveraging coupling knowledge into ESN models can enhance their robustness to variations of training and target system conditions. We further show that our proposed model remains effective even when the coupling knowledge is imperfect or extracted directly from time series data. We believe this approach has the potential to enhance other machine-learning methods.