An approach to the time-accurate prediction of chaotic solutions is by learning temporal patterns from data. Echo State Networks (ESNs), which are a class of Reservoir Computing, can accurately predict the chaotic dynamics well beyond the predictability time. Existing studies, however, also showed that small changes in the hyperparameters may markedly affect the network's performance. The aim of this paper is to assess and improve the robustness of Echo State Networks for the time-accurate prediction of chaotic solutions. The goal is three-fold. First, we investigate the robustness of routinely used validation strategies. Second, we propose the Recycle Validation, and the chaotic versions of existing validation strategies, to specifically tackle the forecasting of chaotic systems. Third, we compare Bayesian optimization with the traditional Grid Search for optimal hyperparameter selection. Numerical tests are performed on two prototypical nonlinear systems that have both chaotic and quasiperiodic solutions. Both model-free and model-informed Echo State Networks are analysed. By comparing the network's robustness in learning chaotic versus quasiperiodic solutions, we highlight fundamental challenges in learning chaotic solutions. The proposed validation strategies, which are based on the dynamical systems properties of chaotic time series, are shown to outperform the state-of-the-art validation strategies. Because the strategies are principled-they are based on chaos theory such as the Lyapunov time-they can be applied to other Recurrent Neural Networks architectures with little modification. This work opens up new possibilities for the robust design and application of Echo State Networks, and Recurrent Neural Networks, to the time-accurate prediction of chaotic systems.

中文翻译：

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学习混沌动力学的回声状态网络的鲁棒优化和验证

混沌解的时间精确预测的一种方法是通过从数据中学习时间模式。回声状态网络（ESN）是油藏计算的一类，可以准确地预测混沌动力学，远远超出了可预测的时间。但是，现有研究还表明，超参数的细微变化可能会明显影响网络的性能。本文的目的是评估和改善回声状态网络的鲁棒性，以进行时间精确的混沌解预测。目标是三重。首先，我们研究了常规使用的验证策略的鲁棒性。其次，我们提出了回收验证以及现有验证策略的混沌版本，以专门解决混沌系统的预测问题。第三，我们将贝叶斯优化与传统的网格搜索进行比较，以选择最佳的超参数。在两个具有混沌和拟周期解的原型非线性系统上进行了数值测试。分析了无模型和模型通知的回声状态网络。通过比较网络在学习混沌解和拟周期解中的鲁棒性，我们强调了学习混沌解的基本挑战。所提出的基于混沌时间序列动力学系统特性的验证策略表现出优于最新的验证策略。由于这些策略是原则性的-它们基于诸如Lyapunov时间之类的混沌理论-因此无需进行任何修改就可以将其应用于其他递归神经网络体系结构。