Distinguished Seminar in Computational Science and Engineering
March 20, 2025, 12-1PM
45-432 in Building 45 and Zoom Webinar
Mathematics of Digital Twins and Transfer Learning
Alexandre Tartakovsky
Professor
Department of Civil and Environmental Engineering
University of Illinois at Urbana-Champaign
Abstract:
A digital twin for physical systems governed by partial differential equations (PDEs) is a real-time virtual replica, capable of simulating system behavior under various conditions. This talk will present the mathematical foundation for developing digital twins as surrogate models for PDE systems. We require the surrogate model to support real-time inference, be differentiable with respect to control parameters, and be adaptable to new conditions with a reasonable amount of data. We employ the KL-NN method as the surrogate model, treating the PDE state and control parameters as stochastic processes, which are decomposed into mean functions and fluctuations. The fluctuations are approximated with zero-mean truncated Karhunen–Loève expansion (KLE), enabling a reduced-order representation. A parameterized mapping then relates KLE coefficients of control parameters to KLE coefficients of the state.
To adapt the KL-NN surrogate model trained under one set of conditions (source) to another set (target), we apply moment equation analysis to investigate the transferability of the KL-NN surrogate components. The analysis shows that for linear PDEs, all components of the KL-NN surrogate model can be transferred, except for the mean function, which can be learned with just one additional simulation under the target mean condition. For certain nonlinear PDE models—specifically when the variability in the random control parameters is small—the eigenfunctions can also be transferred. In these cases, the mean function and the parameters of the mapping between KLE coefficients can be retrained using “few-shot” learning. We provide examples, including linear and nonlinear time-dependent diffusion equations, demonstrating that transfer learning for the KL-NN model can be achieved with minimal solution samples under target conditions.
Bio:
Alexandre Tartakovsky is a Professor in the Department of Civil and Environmental Engineering at the University of Illinois Urbana-Champaign and a Lab Fellow at the Pacific Northwest National Laboratory. His research focuses on scientific machine learning, multiscale mathematics, uncertainty quantification, and Lagrangian particle methods. Dr. Tartakovsky received master’s degree in applied mathematics from Kazan State University in Russia and Ph.D. in Hydrology from the University of Arizona. Prior to joining PNNL, he was a postdoctoral research scientist at the Idaho National Laboratory.
Mathematics of Digital Twins and Transfer Learning
Alexandre Tartakovsky
University of Illinois at Urbana-Champaign