Multifaceted Mathematics for Predictive Digital Twins

The overarching goal of the M2dt Center is to catalyze research and education on multifaceted mathematical foundations of predictive digital twins for complex systems.

A digital twin is a set of virtual information constructs that mimics the structure, context, and behavior of a natural, engineered, or social system (or system-of-systems), is dynamically updated with data from its physical twin, has a predictive capability, and informs decisions that realize value. The bidirectional interaction between the virtual and the physical is central to the digital twin.

Data streams from the physical system are assimilated into the virtual representation to reduce uncertainties and improve predictions, which in turn are used as a basis for controlling the physical system, optimizing data acquisition, and providing decision support. To support critical high-consequence decisions, the DT must be trustworthy through a combination of explainability and uncertainty quantification. Digital twins are key to unlocking the full potential of decades of investment in computational modeling – moving beyond forward simulation to underpin intelligent automation in complex systems by supporting data-driven decision making.

The tight, dynamic interplay between computational models and physical systems, the rapid time scales needed for decision making and control, the heterogeneity of data sources that must be assimilated, the coupled nature of target systems, and the need for robustness to data and model uncertainties to support high-consequence decisions – these present frontier mathematical and computational challenges for DTs and place them at the forefront of applied math and computational science research today.

M2dt addresses these challenges via an integrated and multifaceted research program that unites the complementary perspectives of physics-based computational science (PDE solvers, data assimilation, reduced-order models, optimal control, uncertainty quantification) with statistical/data sciences (machine learning, graphical models, dimensionality reduction, optimal experimental design, optimization) to advance the state of the art in these areas and at their interfaces. In particular, we are pursuing an integrated and multifaceted program of research on foundations of digital twins in:

1. new mathematical and statistical frameworks and computational algorithms, and new representations blending graphical and PDE models, to enable model-based inference from data and dynamic decision making for complex systems;
2. new dimensionality reduction methods exploiting nonlinear manifolds;
3. new data-driven, machine learning-based, structure-preserving reduced-order models and surrogates of high fidelity models that address high dimensional state and parameter spaces;
4. new algorithms orchestrating use of hierarchies of models in digital twins, given real-time and heterogeneous computing constraints on latencies, bandwidth, response times, and data availability, on heterogeneous compute platforms from edge devices to exascale systems; and
5. new decomposition algorithms to tackle the challenges of data assimilation and optimal control for complex systems.