Short Course Title：Towards a Mathematical Foundation of Deep Learning: From Phenomena to Theory

Lecturer：Yaoyu Zhang  

Associate Professor  

Shanghai Jiao Tong University

Course Schedule：

September 13-15, 2026，

9:00-11:00 each morning

Venue：Room 202, Zhengxin Building, Qianwei Campus, Jilin University

Abstract

Establishing a mathematical foundation for deep learning is a significant and challenging endeavor in applied mathematics. Recent theoretical advancements are transforming deep learning from a black box into a more transparent and understandable framework. This three-lecture short course offers an in-depth exploration of these developments, emphasizing a promising phenomenological approach. It is designed for those seeking an intuitive understanding of how neural networks learn from data, as well as an appreciation of their theoretical underpinnings. We begin by introducing deep learning and its intriguing mysteries, setting the stage for deeper theoretical investigation. Key phenomena include the frequency principle (or spectral bias), whereby neural networks tend to prioritize learning low-frequency components of data, and the condensation phenomenon, in which neurons within a layer cluster together, effectively reducing the number of independent neurons. We will examine these phenomena, their theoretical understanding, and their implications for the capabilities and limitations of deep learning models. Overall, this short course provides a gateway to the vibrant field of deep learning theory and invites fresh perspectives on its.

Lecture Outline

Lecture 1 - September 13: Mysteries of Deep Learning

Lecture 2 - September 14: Frequency Principle / Spectral Bias

Lecture 3 - September 15: Condensation Phenomenon

Lecturer Biography

Yaoyu Zhang is an Associate Professor at Shanghai Jiao Tong University. In 2012, he received a B.S. in Applied Physics from Zhiyuan College, Shanghai Jiao Tong University. He earned his Ph.D. in Mathematics from the School of Mathematical Sciences at Shanghai Jiao Tong University in 2016. From 2016 to 2020, he conducted postdoctoral research at New York University Abu Dhabi, the Courant Institute of Mathematical Sciences, and the Institute for Advanced Study in Princeton. He joined Shanghai Jiao Tong University in 2020. His research focuses on the theoretical foundations of deep learning. His representative contributions include the frequency principle and the condensation phenomenon in deep learning. He is also a coauthor of the book An Introduction to Deep Learning Phenomena: From the Perceptron to Large Language Models.

Contact: zhyy.sjtu@sjtu.edu.cn