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 studying 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：基于部分状态观测的动力学系统研究因其在众多真实世界系统中的应用而具有重要意义。我们通过研究部分状态输入、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Lorenz系统与Chua氏振荡器（包含数值仿真与实验系统）上的应用验证了该方法的有效性。研究表明，作为自主动力学系统的ESN能够实现长达数个李雅普诺夫时间的短期预测。然而，预测时域因初始条件不同而呈现高度可变性——我们通过预测时域的分布特征对此进行了深入探究。进一步地，通过采用多种统计指标将ESN预测的长期动力学特性与数值仿真或实验动力学进行对比，观察到相似结果，证明即使使用含噪声的数值或实验数据集进行训练，ESN也能有效学习系统动力学特性。因此，我们证实了ESN作为低成本代理模型，在模拟无法获得完整观测的系统动力学方面具有巨大潜力。