摘要：We introduce a new parameterization strategy that can be used to design algorithms simulating geometric flows on Wasserstein manifold, the probability density space equipped with optimal transport metric. The framework leverages the theory of optimal transport and the techniques like the push-forward operators and neural networks, leading to a system of ODEs for the parameters of neural networks. The resulting methods are mesh-less, basis-less, sample-based schemes that scale well to higher dimensional problems. The strategy works for Wasserstein gradient flows such as Fokker-Planck equation, and Wasserstein Hamiltonian flow like Schrodinger equation. Theoretical error bounds measured in Wasserstein metric is established. This presentation is based on joint work with Yijie Jin (Math, GT), Shu Liu (UCLA), Has Wu (Wells Fargo), Xiaojing Ye (Georgia State), and Hongyuan Zha (CUHK-SZ).

报告人简介：Haomin Zhou is a professor in the School of Mathematics at Georgia Institute of Technology. He received his B.S. in pure mathematics from Peking University, M.Phil in applied mathematics from the Chinese University of Hong Kong, and Ph.D. in applied mathematics from University of California, Los Angeles in 1991, 1996 and 2000 respectively. He spent 3 years in California Institute of Technology as a postdoctoral scholar and von Karman instructor, before he joined Georgia Institute of Technology as an assistant professor in 2003. His research interests are on numerical analysis and scientific computing, specialized in PDE and wavelet techniques in image processing, numerical methods for stochastic differential equations, and discrete optimal transport. He is a recipient of the NSF CAREER AWARD in applied and computational mathematics in 2007, and Feng Kang prize in scientific computing in 2019.