京师数学前沿论坛 第四十四讲
京师数学前沿论坛
报告题目(Title)：Mathematical Explanations of Neural Networks and Transformers
报告人(Speaker)：台雪成教授 （Chief Scientist, Norwegian Research Centre）
地点(Place)：后主楼1124
时间(Time)：2026年9月30日，16:00-17:00
报告摘要
Neural networks such as encoder-decoder architectures, U-Net, and Transformers have achieved remarkable success in image processing and sequence modeling, yet a comprehensive mathematical understanding of their structures remains limited. In this talk, we present a unified operator-theoretic framework for interpreting these architectures through the lens of control theory, multigrid methods, and continuous modeling. We show that popular encoder-decoder networks, including U-Net, can be derived as time-discretized solutions to control problems via operator splitting and multigrid decomposition. Specifically, we introduce PottsMGNet, a network derived from the two-phase Potts model, and demonstrate how it provides a unified interpretation of a broad class of encoder-decoder architectures. We further extend this perspective to Transformers by modeling self-attention as a nonlocal integral operator within a continuous integro-differential framework and interpreting normalization as a time-dependent constraint. These insights not only provide a rigorous theoretical foundation for key neural architectures but also open new avenues for principled architecture design, robustness analysis, and interpretability across vision and language tasks.
主讲人简介
台雪成，挪威研究中心（Norwegian Research Centre）首席科学家。曾任挪威卑尔根大学教授、香港浸会大学讲座教授及系主任，2009年获第8届“冯康”科学计算奖，2026年当选美国工业与应用数学学会会士（SIAM Fellow）。台教授的研究领域主要包括数值偏微分方程（PDE）、优化、计算机视觉、图像处理与数据分析等，已发表论文200余篇，Google Scholar引用超过14000次。曾担任多个国际会议的大会主席，并多次受邀作大会报告；现任SIAM Journal on Numerical Analysis、SIAM Journal on Imaging Sciences、Journal of Mathematical Imaging and Vision、Inverse Problems and Imaging、International Journal of Numerical Analysis and Modeling等国际期刊的编委。
