  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.
