报告题目 (Title)：无网格方法在求解流形和模上的偏微分方程中的应用（Applications of Mesh-free Methods in Solving PDEs on Manifolds and Modeling of Missing Dynamics）
报告人 (Speaker)： 蒋诗晓 (上海科技大学)
报告时间 (Time)：2024年4月24日（周三） 10:00
报告地点 (Place)：校本部D206
邀请人(Inviter)：秦晓雪
主办部门：永利数学系
报告摘要：In this talk, we review several mesh-free methods, including generalized moving least-squares and kernel-based approach, and discuss their applications in two problems, one being solving PDEs on Riemannian manifolds and the other recovery of missing dynamics. For the first topic, we consider the Generalized Finite Difference Method (GFDM) on unknown compact submanifolds, identified by randomly sampled data, for solving PDEs with convergence guarantees. We illustrate the approach by approximating the Laplace-Beltrami operator, where a stable approximation is achieved by a combination of Generalized Moving Least-Squares algorithm and a novel linear programming. For the second topic about missing dynamics problem, we propose a framework that reformulates the prediction problem as a supervised learning problem to approximate a map that takes the memories of the resolved and identifiable unresolved variables to the missing components in the resolved dynamics. Supporting numerical results on instructive nonlinear dynamics, including the two-layer Lorenz system, the truncated Burger-Hopf equation, the 57-mode barotropic stress model, and the Kuramoto-Sivashinsky (KS) equation.
