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

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.