PDE MODELS FOR DEEP NEURAL NETWORKS:
LEARNING THEORY,
CALCULUS OF VARIATIONS AND OPTIMAL CONTROL
报告专家：Peter Markowich 教授 (KAUST)
报告时间：11月10日（周一）10：30-11：30
报告地点：国家天元数学西南中心516
报告摘要：
We propose a partial differential-integral equation (PDE) framework for deep neural networks (DNNs) and their associated learning problems by taking the continuum limits of both network width and depth. The proposed model captures the complex interactions among hidden nodes, overcoming limitations of traditional discrete and ordinary differential equation (ODE)-based models. Controllability and optimality conditions for the learning task with its associated PDE forward problem are established using variational calculus, the Pontryagin Maximum Principle, and the Hamilton-Jacobi-Bellman equation, framing the deep learning process as a PDE-constrained optimization problem. This approach facilitates the development of new network architectures and numerical methods that improve upon traditional layer-by-layer gradient descent techniques. The talk provides a mathematical foundation for connecting neural networks, PDE theory, variational analysis, and optimal control. By integrating these fields, we offer a robust framework that enhances deep learning models’ stability, efficiency, and interpretability.
专家简介：
Professor Peter A. Markowich is a Distinguished Professor at the King Abdullah University of Science and Technology (KAUST) in Saudi Arabia. Markowich's career is marked by significant contributions to the field of applied mathematics, particularly in the analysis of partial differential equations (PDEs) and their applications across various scientific domains including physics, biology, and engineering. His professional trajectory includes roles at prestigious institutions including the Mathematics Research Center at the University of Wisconsin-Madison, University of Texas at Austin, Purdue University, TU Berlin, University of Vienna and the University of Cambridge. His accolades include the Wittgenstein Award of the Austrian Science Fund, the Royal Society Wolfson Research Merit Award, the Humboldt Research Award, and the Inaugural KAUST Investigator Award. Markowich is a Fellow of several academies, including the European Academy of Sciences (EURASC), the AMS, the IMA (UK), the European Academy of Sciences and Arts, and he is a Foreign Member of the Austrian Academy of Sciences. He also gave an invited lecture at ICM in 2010.
邀请人：吕琦
