Study of dynamical systems using partial state observation is an important problem due to its applicability to many real-world systems. We address the problem by proposing an echo state network (ESN) framework with partial state input with partial or full state output. Application to the Lorenz system and Chua's oscillator (both numerically simulated and experimental systems) demonstrate the effectiveness of our method. We show that the ESN, as an autonomous dynamical system, is capable of making short-term predictions up to a few Lyapunov times. However, the prediction horizon has high variability depending on the initial condition - an aspect that we explore in detail using the distribution of the prediction horizon. Further, using a variety of statistical metrics to compare the long-term dynamics of the ESN predictions with numerically simulated or experimental dynamics and observed similar results, we show that the ESN can effectively learn the system's dynamics even when trained with noisy numerical or experimental datasets. Thus, we demonstrate the potential of ESNs to serve as cheap surrogate models for simulating the dynamics of systems where complete observations are unavailable.

翻译：利用部分状态观测研究动力系统是一个重要问题，因其适用于众多实际系统。我们通过提出一种具有部分状态输入和部分或全状态输出的回声状态网络（ESN）框架来解决该问题。在洛伦兹系统和蔡氏振荡器（包括数值模拟与实验系统）上的应用验证了该方法的有效性。我们证明，作为自主动力系统的ESN能够实现长达若干李雅普诺夫时间的短期预测。然而，预测范围具有高度变异性——这一特性取决于初始条件，我们通过预测范围分布对其进行了详细探究。进一步，采用多种统计指标将ESN预测的长期动力学与数值模拟或实验动力学进行比较，并观察到相似结果，这表明即使使用含噪数值或实验数据集训练，ESN也能有效学习系统动力学。因此，我们展示了ESN作为廉价替代模型模拟因完整观测不可用而受限的系统动力学的潜力。