汇报标题 (Title)：无网格步骤在求解流形和模上的偏微分方程中的利用（Applications of Mesh-free Methods in Solving PDEs on Manifolds and Modeling of Missing Dynamics）
汇报人 (Speaker)： 蒋诗晓 (上？萍即笱)
汇报功夫 (Time)：2024年4月24日（周三） 10:00
汇报地址 (Place)：校本部D206
约请人(Inviter)：秦晓雪
主办部门：理学院数学系
汇报提要：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.
