Research Article
Open Access
Toward a Theory of Granular Digital Twins: Mathematical Foundations for Adaptive Digital Twin Systems
1
Lincoln University College, Petaling Jaya 47301, Selangor, Malaysia
* Corresponding Author:
Jinghan Li,
[email protected]
Volume 2, Issue 4
2026
Received: 10 August 2026
Revised: 29 September 2026
Accepted: 11 October 2026
Published: 19 October 2026
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CC BY 4.0 License
Article Information
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.
Graphical Abstract
Keywords
digital twin
granular computing
information granules
granular twin algebra
lattice theory
fixed-point theory
adaptive systems
Data Availability Statement
No data were generated or analyzed in this study.
Funding
This work was supported without any funding.
Conflicts of Interest
The author declares no conflicts of interest.
AI Use Statement
The author declares that no generative AI was used in the preparation of this manuscript.
Ethical Approval and Consent to Participate
Not applicable. This study is purely theoretical and did not involve human participants, animals, personal data, or identifiable individual information.
References
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APA Style
Li, J. (2026). Toward a Theory of Granular Digital Twins: Mathematical Foundations for Adaptive Digital Twin Systems. ICCK Journal of Applied Mathematics, 2(4), 302-314. https://doi.org/10.62762/JAM.2026.667211
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TY - JOUR AU - Li, Jinghan PY - 2026 DA - 2026/10/19 TI - Toward a Theory of Granular Digital Twins: Mathematical Foundations for Adaptive Digital Twin Systems JO - ICCK Journal of Applied Mathematics T2 - ICCK Journal of Applied Mathematics JF - ICCK Journal of Applied Mathematics VL - 2 IS - 4 SP - 302 EP - 314 DO - 10.62762/JAM.2026.667211 UR - https://www.icck.org/article/abs/JAM.2026.667211 KW - digital twin KW - granular computing KW - information granules KW - granular twin algebra KW - lattice theory KW - fixed-point theory KW - adaptive systems AB - 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. SN - 3068-5656 PB - Institute of Central Computation and Knowledge LA - English ER -
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@article{Li2026Toward,
author = {Jinghan Li},
title = {Toward a Theory of Granular Digital Twins: Mathematical Foundations for Adaptive Digital Twin Systems},
journal = {ICCK Journal of Applied Mathematics},
year = {2026},
volume = {2},
number = {4},
pages = {302-314},
doi = {10.62762/JAM.2026.667211},
url = {https://www.icck.org/article/abs/JAM.2026.667211},
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.},
keywords = {digital twin, granular computing, information granules, granular twin algebra, lattice theory, fixed-point theory, adaptive systems},
issn = {3068-5656},
publisher = {Institute of Central Computation and Knowledge}
}
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