A deformation-based framework for learning solution mappings of PDEs defined on varying domains

讲座题目：A deformation-based framework for learning solution mappings of PDEs defined on varying domains
讲座时间 Datetime: 2025年4月 11日，周五，9:00-11:00
地点 Venue: 腾讯会议：307-715-753密码：202504
主持人 Host：李露 副教授
报告人 Speaker: 金鹏展 助理研究员
单位 Affiliation: 北京大学大数据分析与应用技术国家工程实验室
报告摘要 Abstract:
In this work, we establish a deformation-based framework for learning solution mappings of PDEs defined on varying domains. The union of functions defined on varying domains can be identified as a metric space according to the deformation, then the solution mapping is regarded as a continuous metric-to-metric mapping, and subsequently can be represented by another continuous metric-to-Banach mapping using two different strategies, referred to as the D2D framework and the D2E framework, respectively. We point out that such a metric-to-Banach mapping can be learned by neural networks, hence the solution mapping is accordingly learned. With this framework, a rigorous convergence analysis is built for the problem of learning solution mappings of PDEs on varying domains. As the theoretical framework holds based on several pivotal assumptions which need to be verified for a given specific problem, we study the star domains as a typical example, and other situations could be similarly verified. We finally present several numerical experiments to validate our theoretical results.

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。在Math. Comp.，SIAM系列，Nature Mach. Intell.，Neural Networks等期刊发表论文10余篇。