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, our models exhibit robustness to noise in training data and remain effective even when prior coupling knowledge is imperfect. This approach has the potential to enhance other machine learning methods.

翻译：机器学习方法在学习混沌动力系统方面展现出潜力，能够实现无模型的短期预测和吸引子重构。然而，当应用于大规模时空混沌系统时，纯数据驱动的机器学习方法往往效率低下，因为它们需要庞大的学习模型规模和海量的训练数据才能达到可接受的性能。为应对这一挑战，我们将目标系统的空间耦合结构作为归纳偏置融入网络设计中。具体而言，我们引入了物理引导的聚类回声状态网络，利用回声状态网络的高效性作为基础模型。在基准混沌系统上的实验结果表明，我们这种物理信息方法在学习目标混沌系统方面优于现有的回声状态网络模型。此外，我们的模型对训练数据中的噪声表现出鲁棒性，即使在先验耦合知识不完善的情况下仍能保持有效性。该方法具有增强其他机器学习方法的潜力。