Data Science Seminar by Wei Chi: DeepMartNet - A Martingale based Deep Neural Network Algorithm for Eigenvalue/BVP Problems of PDEs and Optimal Stochastic Controls

Speaker: Wei Cai, professor of mathematics, Southern Methodist University

Title:  DeepMartNet - A Martingale based Deep Neural Network Algorithm for Eigenvalue/BVP Problems of PDEs and Optimal Stochastic Controls

Abstract: In this talk, we will present a deep neural network (DNN) learning algorithm for solving high dimensional Eigenvalue (EV) and boundary value problems (BVPs)  for elliptic operators and initial BVPs (IBVPs) of quasi-linear parabolic equations as well as optimal stochastic controls. The method is based on the Martingale property in the stochastic representation for the eigenvalue/BVP/IBVP problems and martingale principle for optimal stochastic controls. A loss function based on the Martingale property can be used for an efficient optimization by sampling the stochastic processes associated with the elliptic operators or value process  for stochastic controls. The Martingale property conforms naturally with the stochastic gradient descent process  for the DNN optimization. The proposed algorithm can be used for eigenvalue problems and BVPs and  IBVPs with Dirichlet, Neumann, and Robin boundaries in bounded or unbounded domains and some feedback stochastic control problems. Numerical results for BVP and EV problems in high dimensions will be presented.

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.  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 130 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.  He was awarded the Feng Kang prize in scientific computing in 2005.

 

Data Science Seminar