Seminars on Numerical Algorithms, Analyses, and Applications: Weak Generative Sampler to Efficiently Sample Invariant Distribution

：周翔（香港城市大学）
：2024-07-03 09:00
：海韵园教学楼406

报告人：周翔（香港城市大学）

时  间：2024年7月3日9:00

地  点：海韵园教学楼406

内容摘要:

Sampling invariant distributions from an Ito diffusion process presents a significant challenge in stochastic simulation. Traditional numerical solvers for stochastic differential equations require both a fine step size and a lengthy simulation period, resulting in both biased and correlated samples. Current deep learning-based method solves the stationary Fokker--Planck equation to determine the invariant probability density function in form of deep neural networks, but they generally do not directly address the problem of sampling from the computed density function. In this work, we introduce a framework that employs a weak generative sampler (WGS) to directly generate independent and identically distributed (iid) samples induced by a transformation map derived from the stationary Fokker-Planck equation. Our proposed loss function is based on the weak form of the Fokker--Planck equation, integrating normalizing flows to characterize the invariant distribution and facilitate sample generation from the base distribution. Our randomized test function circumvents the need for mini-max optimization in the traditional weak formulation. Distinct from conventional generative models, our method neither necessitates the computationally intensive calculation of the Jacobian determinant nor the invertibility of the transformation map. A crucial component of our framework is the adaptively chosen family of test functions in the form of Gaussian kernel functions with centres selected from the generated data samples. Experimental results on several benchmark examples demonstrate the effectiveness of our method, which offers both low computational costs and excellent capability in exploring multiple metastable states. This joint work of Z. Cai, Y. Cao, Y. Huang and X. Zhou is supported by Hong Kong RGC. 

个人简介：

Prof. Xiang Zhou received his BSc from Peking University (School of Mathematical Sciences) and PhD from Princeton University (PACM). Before joining City University in 2012, he worked as a research associate at Princeton University and Brown University. His major research focus is the study of rare event and the development of new computational methods for stochastic models and machine learning algorithms. His research works include the transitions in stochastic dynamical systems, rare-event simulation, saddle-point calculations and high dimensional problems for controls. His research results have turned into peer-reviewed papers in SIAM journals, Journal of Computational Physics, Chaos, Nonlinearity and Annals of Applied Probability, etc. He has a joint appointment at Department of Mathematics, College of Science, but recruits PhD students only via School of Data Science.

 

联系人：陈黄鑫