Mathematics of Digital Twins: A New Mathematical Paradigm for Integrating Data, Models, Decisions

MURI grant 2025–2028 | Funded by Air Force Office of Science Research

Overview

DT is a digital representation of a physical asset and its interactions with the environment, synchronized and continuously refined with data, driven by mission objectives, adapted to situational changes, with self-learning and self-improving AI capabilities, to provide real-time assessment, analysis, optimization and control, and decision-making.

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This definition differs from others in two distinct novel features. First, our DT is designed to adapt not only to environmental changes but also to situational changes and mission objective changes. Second, a self-learning AI (SLAI) component of our DT enables it to constantly ``learn'' from the asset (physical twin), the environment, and the mission objectives. Thus, our DT self-refines and improves its fidelity in the course of its life-cycle, producing better and more accurate real-time control and decision making. Our DT paradigm (shown below) emphasizes bidirectional flow of information and data between the physical asset and its DT. As the physical asset navigates the real world, all the data, including those that define the mission objectives and environment, are assimilated into its DT. This process is termed P2D (physical-to-digital). Meanwhile, the DT provides control and optimization decisions to the physical asset, a process termed D2P (digital-to-physical). As the physical asset evolves in physical time, its counterpart, the DT, evolves in virtual time. The P2D and D2P processes enable seamless interactions between the physical asset and its DT. The latter is updated by the changing data and environment via the P2D process; whereas the physical asset is adapted by the optimized decision from the DT via the D2P process.

Aims

Design a novel conceptual DT paradigm tailored to the specific needs of DoD missions: to achieve this goal, we will supplement the emphasis on information exchange between the physical and digital twins with the construction of a novel PDDP (physical-digital/digial-physical) framework; the latter includes a critical D2D (digital-to-digital) component to enable real-time control and optimization of the physical twin.
Identify and further develop of a set of enabling mathematical and numerical methods (e.g., tools from graph theory, machine learning methods, reduced-order modeling) tailored to DTs.
Demonstrate our DT framework on problems in additive manufacturing: 3D metal printing: real-time optimization of production via model simulation, data fusion, and control of laser power beam; and monitoring, prediction and control of long-term materials defects, i.e., material aging.