  While Bernal stacked bilayer graphene bears two distinct atom types in its
lattice, there exists no analytical framework addressing the number of atomic
environments that emerge in twisted bilayer graphene superlattices. In this
work, we have computationally analyzed 120 different twisted bilayer
superlattices using descriptor functions to study the emergent local
environments. Our study reveals that the number of atoms with unique local
environments depend on the superlattice size linearly. Moreover, this linear
dependence manifests itself on two distinct lines and this automatically
suggests a new classification scheme based on the local environments. As a
possible application, the use of local environments in the investigation of
vibrational properties is discussed with respect to the existing literature.
Molecular dynamics simulations are performed to calculate the phonon density of
states of the 120 structures as well as the local phonon density of states of
their individual atoms. The similarity of the contributions of local density of
states coming from atoms with the same local environment is demonstrated. Local
density of states of the atoms with unique local environments of an arbitrary
selection of the structures is then used to train a machine learning model.
This model is used to predict the phonon spectra of twisted bilayer structures.
Performance of the trained model is discussed thoroughly via different
selection of training and test sets, and it is shown that the model proves
effective in predicting the vibrational properties of any given twisted bilayer
structure. The possible applications of the generic method presented which
reaches far beyond twisted bilayer graphene is also discussed.
