[Paper Review] Mathematical Opportunities in Digital Twins (MATH-DT)

Harbir Antil|arXiv (Cornell University)|Feb 15, 2024

Engineering Education and Technology4 citations

TL;DR

This paper identifies foundational mathematical challenges and opportunities in digital twins (DTs), emphasizing the need for novel advances in optimization, uncertainty quantification, multi-scale modeling, and scientific machine learning. It proposes a collaborative institute to unify academia, industry, and national labs, with benchmarking and interdisciplinary training as key enablers for DT development across engineering, medicine, and socio-technical 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.

Motivation & Objective

• Address the lack of foundational mathematical frameworks tailored specifically for digital twins, distinct from traditional modeling approaches.
• Overcome the limitations of existing methods—like Kalman filters and ad hoc AI/ML—by developing mathematically rigorous, application-specific solutions.
• Establish interdisciplinary collaboration between mathematicians, domain experts, and industry to co-develop DTs for real-world systems such as bridges, biological systems, and cities.
• Create standardized benchmark problems and data sets to enable reproducible research and model validation across diverse DT applications.
• Foster long-term research infrastructure through a proposed NSF-funded Digital Twin Collaborative Institute, modeled after mathematical sciences institutes.

Proposed method

• Propose a new collaborative institute to serve as a central platform for mathematical research on digital twins, integrating academia, national labs, and industry.
• Introduce a framework for DTs that starts from personalized or system-specific realities rather than generic physical laws, requiring multi-scale, multi-physics, and hybrid modeling.
• Develop mathematical principles for model updating and maintenance under uncertainty, including data assimilation and online change detection.
• Apply advanced techniques such as adjoint-based optimization, quadrature in random variables, and risk measures like CVaR to handle high-dimensional, nonsmooth, and uncertain problems.
• Utilize randomized reduced basis methods with overlapping time subintervals to improve accuracy in advection-diffusion problems with sharp discontinuities.
• Accelerate Physics-Informed Neural Networks (PINNs) using optimized Python packages for PyTorch and TensorFlow, enabling faster implementation and simulation.

Experimental results

Research questions

• RQ1 What mathematical advances are required to support digital twins that differ fundamentally from traditional simulation pipelines?
• RQ2 How can risk-aware optimization, such as minimizing conditional-value-at-risk (CVaR), be effectively applied to structural integrity problems under uncertainty?
• RQ3 What role do randomized reduced basis methods play in improving accuracy and efficiency for advection-diffusion problems with discontinuous source terms?
• RQ4 How can physics-informed neural networks (PINNs) be accelerated through optimized software toolkits in PyTorch and TensorFlow?
• RQ5 What benchmark problems and data sets are needed to standardize and validate digital twin models across diverse domains like civil engineering, medicine, and climate systems?

Key findings

• Digital twins require fundamentally different mathematical foundations than traditional modeling, particularly in handling uncertainty, multi-scale dynamics, and real-time decision support.
• CVaR-based optimization successfully minimizes rare but high-impact failure events in structural models, outperforming standard expectation-based methods in high-dimensional, nonconvex settings.
• Randomized reduced basis methods with overlapping time subintervals significantly improve solution accuracy for advection-diffusion problems with sharp discontinuities compared to single-basis approaches.
• Accelerated PINN implementations via custom Python packages reduce development time and computational overhead, enabling faster prototyping in scientific machine learning.
• Benchmark problems such as sepsis modeling, bridge fatigue, and particle-resolved cloud chamber simulations are critical for validating and advancing DT research.
• A proposed Digital Twin Collaborative Institute would serve as a central hub for interdisciplinary research, training, and resource sharing, modeled after successful mathematics institutes.