Mathematical Opportunities in Digital Twins (MATH-DT)

The report describes the discussions from the Workshop on Mathematical Opportunities in Digital Twins (MATH-DT) from December 11-13, 2023, George Mason University. It illustrates that foundational Mathematical advances are required for Digital Twins (DTs) that are different from traditional approaches. A traditional model, in biology, physics, engineering or medicine, starts with a generic physical law (e.g., equations) and is often a simplification of reality. A DT starts with a specific ecosystem, object or person (e.g., personalized care) representing reality, requiring multi -scale, -physics modeling and coupling. Thus, these processes begin at opposite ends of the simulation and modeling pipeline, requiring different reliability criteria and uncertainty assessments. Additionally, unlike existing approaches, a DT assists humans to make decisions for the physical system, which (via sensors) in turn feeds data into the DT, and operates for the life of the physical system. While some of the foundational mathematical research can be done without a specific application context, one must also keep specific applications in mind for DTs. E.g., modeling a bridge or a biological system (a patient), or a socio-technical system (a city) is very different. The models range from differential equations (deterministic/uncertain) in engineering, to stochastic in biology, including agent-based. These are multi-scale hybrid models or large scale (multi-objective) optimization problems under uncertainty. There are no universal models or approaches. For e.g., Kalman filters for forecasting might work in engineering, but can fail in biomedical domain. Ad hoc studies, with limited systematic work, have shown that AI/ML methods can fail for simple engineering systems and can work well for biomedical problems. A list of `Mathematical Opportunities and Challenges' concludes the report.