Mathematical Opportunities in Digital Twins (MATH-DT)

TL;DR

The paper outlines the mathematical opportunities and challenges for Digital Twins (DTs), arguing that DTs require foundational advances that start from specific assets and operate in lifelong, data-driven loops, in contrast to traditional equation-first models. It highlights the need for multi-scale, multi-physics modeling, robust uncertainty quantification, model updating, and risk-aware decision-making, illustrated by neuromorphic imaging, structural health identification, and medical digital twins. The discussions span theory, computation, software, and workforce development, calling for benchmark problems, interdisciplinary institutes, and cross-agency funding to build an integrated mathematical ecosystem. The work emphasizes that progress in DTs hinges on coupling physics-based models with data-driven methodologies, rigorous verification and validation, and scalable software, with broad implications for engineering, medicine, and urban systems.

Abstract

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.

Figures (6)

• Figure 1: An actual bridge (physical system) is shown on the left and the digital twin (DT) of this bridge is shown on the right. From the sensors on the real bridge, a dynamic dataset is collected. The relevant dataset is identified (possibly using AI/ML techniques) and fed into the DT. The DT involves multiple components, depending on the purpose. The goal here is to identify a weak beam marked in blue color while accounting for the sensor data and the underlying physics (elasticity equations). Then the DT informs an engineer to make decisions about the actual bridge to make appropriate fixes. These decisions should account for uncertainty and must be risk-averse. This entire problem can be cast as a minimization problem with tracking the displacement or strain measurements subject to elasticity equations as constraints. To create a risk-averse framework, one could additionally consider uncertainty in loads and minimize risk measures, such as Conditional Value at Risk (CVaR). The impact of CVaR is illustrated in section \ref{['s:bridge']} on a crane.
• Figure 2: The panel shows the distribution of MATH-DT attendees.
• Figure 3: Some of the attendees of MATH-DT Workshop
• Figure 4: Figure shows a Digital Twin loop. In the top panel (left), a neuromorphic camera located on the international space station (ISS) captures event data over a time period. This data is shown in panel on the right. One then solves an inverse problem to obtain the image reconstruction. The latter helps, for instance, adjust the camera position to track objects of interest more accurately and this process continues.
• Figure 5: Top row: Target displacement $\bm u$ (left) and strength factor $z$ (right). The color bars at the bottom right corners, in each panel, correspond to the actual displacements and strength factors, respectively. Bottom row: Solution obtained at the 200-th optimization iteration. Panels (c) and (d), respectively display optimization results. The color bar on the top left in panel (c) displays the magnitude of the difference between target and actual displacements at the measuring points (in m). The DT is able to identify weakness in the structure.