  One important problem in constructing the reduced dynamics of molecular
systems is the accurate modeling of the non-Markovian behavior arising from the
dynamics of unresolved variables. The main complication emerges from the lack
of scale separations, where the reduced dynamics generally exhibits pronounced
memory and non-white noise terms. We propose a data-driven approach to learn
the reduced model of multi-dimensional resolved variables that faithfully
retains the non-Markovian dynamics. Different from the common approaches based
on the direct construction of the memory function, the present approach seeks a
set of non-Markovian features that encode the history of the resolved
variables, and establishes a joint learning of the extended Markovian dynamics
in terms of both the resolved variables and these features. The training is
based on matching the evolution of the correlation functions of the extended
variables that can be directly obtained from the ones of the resolved
variables. The constructed model essentially approximates the multi-dimensional
generalized Langevin equation and ensures numerical stability without empirical
treatment. We demonstrate the effectiveness of the method by constructing the
reduced models of molecular systems in terms of both one-dimensional and
four-dimensional resolved variables.
