摘要：

In this talk, we first introduce the Diffusion Maps (DM) algorithm which is one popular manifold learning method. Since the classical DM diverges near the boundary, we recently developed a Ghost Point Diffusion Maps (GPDM) method which provides a pointwisely convergent estimator even near the boundary of a manifold. We investigate various applications of DM and GPDM in uncertainty quantifications. First, we solve the elliptic and time-dependent parabolic PDEs on unknown smooth manifolds without and with boundaries that naturally arise in PDE models. Second, we apply DM and Recurrent Neural Network to model reduction problems. Last, we use DM and GPDM for parameter estimations in ODE and PDE models in Bayesian framework.

简介：

蒋诗晓，上海科技大学，数学科学研究院的副教授。2010年本科毕业于上海交通大学数学系。2017年博士毕业于上海交通大学，数学系和自然科学研究院。2017年至2020年，在宾州州立数学系，做了一年博士后两年、副研教授。主要研究方向是算子估计，流形学习，非线性色散波。多篇论文发表于Journal of Computational Physics, Research in the Mathematical Sciences, Entropy等学术期刊上。

摘要：

In this talk, we first introduce the Diffusion Maps (DM) algorithm which is one popular manifold learning method. Since the classical DM diverges near the boundary, we recently developed a Ghost Point Diffusion Maps (GPDM) method which provides a pointwisely convergent estimator even near the boundary of a manifold. We investigate various applications of DM and GPDM in uncertainty quantifications. First, we solve the elliptic and time-dependent parabolic PDEs on unknown smooth manifolds without and with boundaries that naturally arise in PDE models. Second, we apply DM and Recurrent Neural Network to model reduction problems. Last, we use DM and GPDM for parameter estimations in ODE and PDE models in Bayesian framework.

简介：

蒋诗晓，上海科技大学，数学科学研究院的副教授。2010年本科毕业于上海交通大学数学系。2017年博士毕业于上海交通大学，数学系和自然科学研究院。2017年至2020年，在宾州州立数学系，做了一年博士后两年、副研教授。主要研究方向是算子估计，流形学习，非线性色散波。多篇论文发表于Journal of Computational Physics, Research in the Mathematical Sciences, Entropy等学术期刊上。