  The emergence of superconductivity in slightly-misaligned graphene bilayer
[1] and moir\'e excitons in MoSe$_2$-WSe$_2$ van der Waals (vdW)
heterostructures [2] is intimately related to the formation of a 2D
superlattice in those systems. At variance, perfect primitive lattice matching
of the constituent layers has also been reported in some vdW-heterostructures
[3-5], highlighting the richness of interfaces in the 2D world. In this work,
the determination of the nature of such interface, from first principles, is
demonstrated. To do so, an extension of the Frenkel-Kontorova (FK) model [6] is
presented, linked to first-principles calculations, and used to predict lattice
coherency for a set of 56 vdW-heterostructures. Computational predictions agree
with experiments, when available. New superlattices as well as
perfectly-matching interfaces are predicted.
