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.

翻译：本报告总结了2023年12月11日至13日在乔治梅森大学举办的“数字孪生中的数学机遇”（MATH-DT）研讨会的讨论内容。报告揭示了数字孪生（DTs）需要与传统方法不同的基础数学进展。传统的生物学、物理学、工程学或医学模型始于通用物理定律（如方程），通常是现实世界的简化。而DT则始于代表现实世界的特定生态系统、对象或个体（如个性化医疗），需要多尺度、多物理建模与耦合。因此，这两种过程始于仿真与建模流程的对立端点，需要不同的可靠性标准和不确定性评估。此外，与现有方法不同，DT辅助人类对物理系统做出决策，该物理系统（通过传感器）将数据反馈至DT，并在物理系统的整个生命周期内运行。尽管部分基础数学研究可在无特定应用背景下开展，但DT的研究必须始终考虑具体应用场景。例如，桥梁建模、生物系统（患者）建模或社会技术系统（城市）建模之间存在显著差异。模型涵盖工程领域的微分方程（确定性/不确定性）、生物学的随机模型（包括基于主体的模型），这些属于多尺度混合模型或不确定性下的（多目标）大规模优化问题。不存在普适的模型或方法。例如，用于预测的卡尔曼滤波器可能在工程领域有效，但在生物医学领域可能失效。零星的系统性研究表明，AI/ML方法可能在简单工程系统中失效，而在生物医学问题中表现良好。报告最终列出了“数学机遇与挑战”清单。