Abstract:
Deep learning is a transformative mathematical technique with broad impact across many areas of scientific and engineering research. In this talk, we will present some recent results of efficient deep learning algorithms for addressing the curse of dimensionality (CoD) encountered in a wide range of computational engineering and science problems. We focus on three problems: [1] Solving stochastic optimal controls arising from robotic systems, production management, financial portfolio optimization and risk controls. We developed a martingale neural network, based on Varadhan’s martingale formulation of PDEs, to solve the Hamilton-Jacobi-Bellman equation in dynamic programming with a dimensionality up to 10000. [2] Learning the infinite dimensional operator, mapping medium property to scattering wave field solution, by a multiscale Fourier neural operator (MscaleFNO), which is designed to overcome the spectral bias inherent in standard deep learning architecture and can be used as a surrogate model for high frequency inverse medium problems in medical imaging and geophysical explorations. [3] Sampling high dimensional transient distribution governed by Fokker-Planck equations (FPE) for many particle interacting systems from biology and statistical physics. A deep neural pushfoward map is learnt to generate the target samples through adversarial training of an ultra-weak form of the FPE.
Shot Bio:
Prof. Wei Cai is the Clements Chair Professor in Applied Mathematics at the Department of Mathematics at Southern Methodist University. He obtained his B.S. and M.S. in Mathematics from the University of Science and Technology of China (USTC) in 1982 and 1985, respectively, and his Ph.D. in Applied Mathematics at Brown University in 1989. Before he joins SMU in the fall of 2017, he was an assistant and then associate professor at the University of California at Santa Barbara during 1995-96, and a full Professor at the University of North Carolina after 1999. He has also conducted collaborative research in Peking University, USTC, Shanghai Jiao Tong University, and Fudan University. His research interest is in the development of deterministic, stochastic, and machine learning numerical methods for studying electromagnetic, fluid, and quantum phenomena with applications in CFD, meta-materials, nano-photonics, nano-electronics, biological systems, and quantum systems. His recent work on machine learning focus on the reduction of spectral bias of deep neural network with multiscale DNNs and Feynman-kac formula based networks for high dimensional PDEs and optimal stochastic controls. He has published over 140 refereed research articles and is the author of the book "Computational Methods for Electromagnetic Phenomena: electrostatics in solvation, scattering, and electron transport" published by Cambridge University Press, 2013 and the book published by Springer “ Deterministic, Stochastic, and Deep Learning Methods for Computational Electromagnetics “(2025) and 科学出版社-纯粹数学与应用数学专著《计算电磁学: 确定性, 随机和深度学习方法》 (2025). He was awarded the Feng Kang prize in scientific computing in 2005.
