Astract:

Neural networks such as encoder-decoder architectures, UNet, and Transformers have achieved remarkable success in image processing and sequence modeling, yet a comprehensive mathematical understanding of their structure remains limited. In this talk, we present a unified, operator-theoretic framework that interprets these architectures through the lens of control theory, multigrid methods, and continuous modeling. We show that popular encoder-decoder networks—including UNet—can be derived as time-discretized solutions to control problems using operator-splitting and multigrid decomposition. Specifically, we introduce PottsMGNet, a network derived from the two-phase Potts model, and demonstrate how it generalizes many encoder-decoder designs. We further extend this perspective to Transformers, modeling self-attention as a non-local integral operator within a continuous integro-differential framework, and interpreting normalization as time-dependent constraints. These insights not only offer a rigorous theoretical foundation for key neural architectures but also open new paths for principled architecture design, robustness, and interpretability across tasks in vision and language.

Bio:

Xue-Cheng Tai received his Licentiate degree in 1989 and Ph.D. in 1991 in applied mathematics from the University of Jyväskylä, Finland, with a thesis on numerical methods and inverse problems. He was Associate Professor (1994–1997) and Professor (1997–2021) at the University of Bergen, Norway, Adjunct Associate Professor at Nanyang Technological University, Singapore (2007–2011), Professor and later Chair Professor at Hong Kong Baptist University (2017–2022), and Chief Research Scientist and Executive Program Director at COCHE, Hong Kong (2022–2023). Since 2023 he has been Chief Scientist at NORCE Norwegian Research Centre, Bergen. His research interests include numerical methods for partial differential equations, optimization, inverse problems, image processing, and the mathematical foundations of deep learning. He is a SIAM Fellow (2026), winner of the Feng Kang Prize for Scientific Computing (2009) and the Nanyang Award for Research Excellence (2011), and serves on the editorial boards of several leading journals in numerical analysis and imaging science.