  We outline the general framework of machine learning (ML) methods for
multi-scale dynamical modeling of condensed matter systems, and in particular
of strongly correlated electron models. Complex spatial temporal behaviors in
these systems often arise from the interplay between quasi-particles and the
emergent dynamical classical degrees of freedom, such as local lattice
distortions, spins, and order-parameters. Central to the proposed framework is
the ML energy model that, by successfully emulating the time-consuming
electronic structure calculation, can accurately predict a local energy based
on the classical field in the intermediate neighborhood. In order to properly
include the symmetry of the electron Hamiltonian, a crucial component of the ML
energy model is the descriptor that transforms the neighborhood configuration
into invariant feature variables, which are input to the learning model. A
general theory of the descriptor for the classical fields is formulated, and
two types of models are distinguished depending on the presence or absence of
an internal symmetry for the classical field. Several specific approaches to
the descriptor of the classical fields are presented. Our focus is on the
group-theoretical method that offers a systematic and rigorous approach to
compute invariants based on the bispectrum coefficients. We propose an
efficient implementation of the bispectrum method based on the concept of
reference irreducible representations. Finally, the implementations of the
various descriptors are demonstrated on well-known electronic lattice models.
