报告人：Wang LiLian（Nanyang Technological University）

邀请人：王海永

（一）

报告题目：Kolmogorov-Arnold’s Superpositions: Constructions and Neural Networks

报告摘要：The Kolmogorov–Arnold Representation Theorem (KART, 1957), also known as Kolmogorov–Arnold’s Superposition Theorem, offers a mathematically elegant framework for expressing any high-dimensional continuous function as a superposition of one-dimensional continuous functions. This foundational result has recently gained renewed interest, particularly in the design of neural network architectures. However, a major challenge remains: the one-dimensional functions resulted from all constructions are highly non-smooth. In this talk, we present a novel approximate version of KART involving C^2 inner and outer functions and discuss its potential applications in neural network design. This is a joint work with L. Song (Lanzhou University) and J. Toscane & G. Karniadakis (Brown University).  

报告时间：2025年4月26日（星期六）8:30-10:00

报告地点：科技楼南楼702会议室

（二）

报告题目：Low-Regularity Estimates of Time-Splitting Fourier Spectral Method for Logarithmic Schrodinger Equation

报告摘要：The logarithmic Schrodinger equation (LogSE) in applications exhibits rich dynamics and possesses some unique properties that the usual Schrodinger equations may not have. However, the presence of the logarithmic nonlinear term: f(u)=u log(|u|^2) poses significant challenges in both numerical solution and error analysis, largely due to the non-differentiability. In this talk, we shall characterize the low regularity of the nonlinearity and underlying solution in suitable fractional Sobolev space and derive the first set of error estimates for the time-splitting Fourier spectral method with initial values of fractional order regularity. These new results are supported by recent regularity estimates of the LogSE in PDE community and by ample numerical evidence. This talk will be based on joint works with Xiaolong Zhang (Hunan Normal University).  

报告时间：2025年4月27日（星期日）8:00-9:00

报告地点：科技楼南楼706会议室