Reservoir Computing (RC) is a simple and efficient model-free framework for forecasting the behavior of nonlinear dynamical systems from data. Here, we show that there exist commonly-studied systems for which leading RC frameworks struggle to learn the dynamics unless key information about the underlying system is already known. We focus on the important problem of basin prediction -- determining which attractor a system will converge to from its initial conditions. First, we show that the predictions of standard RC models (echo state networks) depend critically on warm-up time, requiring a warm-up trajectory containing almost the entire transient in order to identify the correct attractor. Accordingly, we turn to Next-Generation Reservoir Computing (NGRC), an attractive variant of RC that requires negligible warm-up time. By incorporating the exact nonlinearities in the original equations, we show that NGRC can accurately reconstruct intricate and high-dimensional basins of attraction, even with sparse training data (e.g., a single transient trajectory). Yet, a tiny uncertainty in the exact nonlinearity can render prediction accuracy no better than chance. Our results highlight the challenges faced by data-driven methods in learning the dynamics of multistable systems and suggest potential avenues to make these approaches more robust.

翻译：水库计算（RC）是一种简单且高效的无模型框架，用于根据数据预测非线性动力系统的行为。在此，我们证明存在一些常见研究的系统，除非底层系统的关键信息已知，否则主流RC框架难以学习其动力学。我们聚焦于盆预测这一重要问题——即确定系统从初始条件出发将收敛到哪个吸引子。首先，我们证明标准RC模型（回声状态网络）的预测对热身时间高度敏感，需要包含几乎整个瞬态过程的热身轨迹才能识别正确的吸引子。因此，我们转向下一代水库计算（NGRC），这是一种有吸引力的RC变体，其热身时间可忽略不计。通过纳入原始方程中的精确非线性，我们证明即使训练数据稀疏（例如单条瞬态轨迹），NGRC也能精确重构复杂且高维的吸引盆。然而，精确非线性中的微小不确定性会导致预测准确率降至仅比随机猜测略好。我们的结果突显了数据驱动方法在学习多稳态系统动力学时面临的挑战，并为增强这些方法的鲁棒性提供了潜在方向。