Complex and nonlinear dynamical systems often involve parameters that change with time, accurate tracking of which is essential to tasks such as state estimation, prediction, and control. Existing machine-learning methods require full state observation of the underlying system and tacitly assume adiabatic changes in the parameter. Formulating an inverse problem and exploiting reservoir computing, we develop a model-free and fully data-driven framework to accurately track time-varying parameters from partial state observation in real time. In particular, with training data from a subset of the dynamical variables of the system for a small number of known parameter values, the framework is able to accurately predict the parameter variations in time. Low- and high-dimensional, Markovian and non-Markovian nonlinear dynamical systems are used to demonstrate the power of the machine-learning based parameter-tracking framework. Pertinent issues affecting the tracking performance are addressed.

翻译：复杂非线性动力系统中的参数常随时间变化，精确追踪这些参数对于状态估计、预测和控制等任务至关重要。现有机器学习方法需要完整观测基础系统状态，并隐含假设参数呈绝热变化。通过构建反问题并利用储层计算，我们发展了一种无模型、完全数据驱动的框架，能够基于部分状态观测实时精确追踪时变参数。具体而言，该框架仅需利用系统动力学子集变量在少量已知参数值下的训练数据，即可精确预测参数随时间的变化。我们采用低维/高维、马尔可夫/非马尔可夫型非线性动力系统验证了该机器学习参数追踪框架的有效性，并讨论了影响追踪性能的相关关键问题。