Dr. Pengzhan Jin（National Engineering Laboratory for Big Data Analysis and Applications）

唐贻发 研究员

Manifold Function Encoder: Identifying Different Functions Defined on Different Manifolds

1月26日（周一）15:00-16:00，南楼733

We propose the Manifold Function Encoder (MFE) for identifying different functions defined on different manifolds. Both a manifold in Euclidean space and a function defined on this manifold can be viewed as bounded linear functionals on a suitable space of continuous functions. From this perspective, we treat manifold functions as elements of the dual space. By expanding them in the dual space based on appropriate approximating sequence of bases, we obtain a corresponding method for encoding manifold functions, that is MFE. Especially, we prove that MFE achieves super-algebraic convergence based on smooth bases commonly used in spectral methods, such as Legendre polynomials and Fourier basis. We further extend MFE to handle more complex cases, including joint manifold functions of different dimensions and manifold functions with different measures. In addition, we show the approximation theory for MFE-based operator learning, in particular learning the solution mappings of PDEs defined on varying domains, together with several numerical experiments including the 2-d Poisson equation and the 3-d elasticity problem on the real-world bearing.

报告人简介：金鹏展，北京大学大数据分析与应用技术国家工程实验室助理研究员，主要研究方向为机器学习与科学计算。2016年本科毕业于中国科学技术大学数学系，2021年博士毕业于中国科学院数学与系统科学研究院，2021年至2024年为北京大学数学科学学院博雅博士后。曾两次访问美国布朗大学应用数学系。曾获中国博士后科学基金面上一等资助、国自然青年科学基金C类资助。在Math. Comp.，SIAM系列，Nature Mach. Intell.，Neural Netw.等期刊发表论文10余篇，总被引用5000余次。