Toward a Theory of Granular Digital Twins: Mathematical Foundations for Adaptive Digital Twin Systems

Source abstract

Digital Twin (DT) technology models and synchronizes physical systems through virtual counterparts. Existing mathematical formulations of DTs concentrate on physical-virtual synchronization and say little about how representations of the same system at different levels of detail should be related. This paper builds a mathematical foundation for Granular Digital Twins (GDTs) by combining Granular Computing with DT theory. We introduce information granules, the Granular Representation Mapping, the Granular Twin Space, and the Granular Digital Twin, and then construct a Granular Twin Algebra whose aggregation, refinement, and adaptive transformation operators act on an ordered family of granular representations. The structure is analyzed through lattice properties, Granular Twin Homomorphisms, adaptive granular evolution, and granular equilibrium. When the representation family is a complete lattice and the adaptive transformation is order-preserving, the Knaster-Tarski theorem guarantees that the set of granular equilibria is non-empty and is itself a complete lattice; each equilibrium is invariant under repeated adaptive transformation. These results are existence and invariance results only; uniqueness, convergence, and dynamical stability are not claimed. The framework moves DT theory from state-oriented synchronization toward adaptive granular representation and gives a formal basis for organizing hierarchical information in DT systems.