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. The Lorenz system and Chua's oscillator (both numerically simulated and experimental systems) are used to check the effectiveness of our method. We demonstrate 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. 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 a cheap surrogate model for predicting the dynamics of systems where complete observations are unavailable.

使用局部状态观察研究动力系统的问题在于其适用于许多现实世界的系统。我们通过提出具有局部状态输入和局部或全部状态输出的回声状态网络框架来解决这个问题，结合Lorenz系统和Chua的振荡器进行验证，表明该方法的有效性。因此，我们证明了ESN作为一种自主动力系统具有短期预测数个Lyapunov时间的能力，并能有效地学习系统的动态，因此具有作为预测完全观测不可用的系统的动态的廉价代理模型的潜力。

基于 ESN 的部分观测动态系统的预测时域变异性