报 告 人: 段玉萍  北京师范大学

报告题目: Starter-Iterator Neural Operator for High-Accuracy Simulation of Forward and Inverse Problems

报告摘要: Operator learning is an emerging interdisciplinary field that combines machine learning with scientific computing paradigms. We present a novel Starter-Iterator inspired Neural Operator (SINO). This framework reconstructs traditional iterative methods' initialization strategies and iteration formats through neural networks. Extensive experiments in dynamic systems like the Navier-Stokes equations and acoustic wave equations, as well as practical applications like super-resolution imaging and weather forecasting, demonstrate that our SINO exhibits exceptional numerical accuracy, generalization capability, and robustness.

报 告 人: 何志坚  华南理工大学

报告题目: Density estimation via periodic scaled Korobov kernel method with exponential decay condition

报告摘要: We propose the periodic scaled Korobov kernel (PSKK) method for nonparametric density estimation on $\mathbb{R}^d$. By first wrapping the target density into a periodic version through modulo operation and subsequently applying kernel ridge regression in scaled Korobov spaces, we extend the kernel approach proposed by Kazashi and Nobile (SIAM J. Numer. Anal., 2023) and eliminate its requirement for inherent periodicity of the density function. This key modification enables effective estimation of densities defined on unbounded domains. We establish rigorous mean integrated squared error (MISE) bounds, proving that for densities with smoothness of order $\alpha$ and exponential decay, our PSKK method achieves the $\mathcal{O}(M^{-1/(1+1/(2\alpha)+\epsilon)})$ MISE convergence rate with an arbitrarily small $\epsilon>0$. While matching the convergence rate of the previous kernel approach, our approach applies to a broader class of non-periodic distributions. Numerical experiments confirm the theoretical results and demonstrate a significant improvement over traditional kernel density estimation in large-sample regimes.

报 告 人: 毛志平  宁波东方理工大学

报告题目: 高效且误差可控的张量Galerkin神经网络算法求解高维方程

报告摘要: 在科学工程计算中，传统数值方法求解微分方程受到维数灾难的影响。近年来，基于神经网络的深度学习算法（比如物理信息神经网络、Deep Ritz 算法、弱对抗神经网络等）在求解高维以及复杂区域问题上展现了强大的能力与巨大的潜力。然而，受非凸优化以及积分误差的影响，这些深度学习算法在求解微分方程时表现出数值精度差、计算效率低等问题。为此，我们发展了一类高效且误差可控的张量Gakerkin神经网络算法，该算法基于贪婪算法思想，通过自适应的增加基函数以逐步提高逼近解的数值精度。通过数值例子，我们验证了算法的有效性。

报 告 人: 潘泽心  浙江大学

报告题目: $L_2$-approximation using median lattice algorithms

报告摘要: This talk presents a novel $L_2$-approximation algorithm for functions in weighted Korobov spaces, eliminating the need for prior knowledge of the smoothness parameter $\alpha$ and coordinate weights. By leveraging recent advances in median quasi-Monte Carlo methods, we construct approximations with nearly optimal convergence rates $O(n^{-\alpha})$, thereby surpassing the $O(n^{-\alpha/2})$ rates of classical lattice-based methods. Furthermore, a data-driven screening mechanism automatically identifies dominant Fourier coefficients and circumvents the curse of dimensionality in high-dimensional weighted spaces.

报 告 人: 汪波  湖南师范大学

报告题目: Wideband fast multipole method in half-plane

报告摘要: In this talk, we present a wideband fast multipole method (WBFMM) for half-plane problems subject to an impedance boundary condition. A novel far-field expansion theory is developed for the reaction component of the half-plane Green’s function, based on which a WBFMM is constructed within the framework of the free-space WBFMM. High-frequency scattering problems in the half-plane are then solved using a boundary integral approach accelerated by the proposed WBFMM.

报 告 人: 王冀鲁  哈尔滨工业大学 (深圳)

报告题目: Sharp interface modeling and simulations of two-phase ferrofluid flows

报告摘要: We propose a novel sharp interface model to describe the behavior of two-phase ferrofluid flows with unmatched densities. The model couples the Navier–Stokes equations for incompressible fluid motion with an advection-reaction equation for the magnetization field, incorporating precise jump conditions at the interface. Utilizing the techniques by Barrett, Garcke, and Nurnberg (BGN), we establish a mathematical relationship between the parameterization of the interface and its mean curvature, enabling an accurate description of the interface geometry and capturing the dynamics at the sharp interface explicitly. To solve the model, we develop a fully discrete backward Euler arbitrary–Lagrangian–Eulerian (ALE) finite element method, enhanced with a specialized mesh velocity governed by a harmonic equation to maintain mesh quality throughout the simulation. Extensive numerical examples are presented to verify the validity of the proposed model, illustrate the accuracy of the numerical scheme, and simulate the benchmark “Rosensweig instability” in both two and three dimensions.

报 告 人: 文再文  北京大学  

报告题目: Exploring the Learning-based Optimization Algorithms

报告摘要: This talk will explore new paradigms for integrating data, models, algorithms, and theories in mathematical optimization. Firstly, we try to understand acceleration methods through ordinary differential equations (ODEs). Under convergence and stability conditions, we formulate a learning optimization problem that minimizes stopping time. This involves transforming the rapid convergence observed in continuous-time models into discrete-time iterative methods based on data. Next, we introduce a Monte Carlo strategy optimization algorithm for solving integer programming problems. This approach constructs probabilistic models to learn parameterized strategy distributions from data, enabling the sampling of integer solutions. Lastly, we discuss the vision of advancing automated theorem proving through formalization assisted by artificial intelligence.

报 告 人: 夏勇  北京航空航天大学  

报告题目: 主特征值计算的优化新视角：差模型与新算法

报告摘要: 计算矩阵的最大特征值是一个基础性问题。基于主特征值问题的差形式，将经典的幂法等价为步长1/2的梯度下降法，证明了步长小于1的梯度下降法都能概率1地收敛到主特征向量（全局最优解）。进一步，基于独特的“过河拆桥”式设计，提出了我们称之为“裂-聚法”的迭代算法。裂-聚法无需借助谱先验知识，即可实现最大程度的加速，并且仅依靠矩阵-向量乘法运算即可完成计算。此外，我们对裂-聚法的收敛性质展开了深入探讨。在生成数据集和真实数据集上开展的广泛数值实验结果表明：裂-聚法相较于幂法实现了超过10倍的加速效果。

报 告 人: 杨云斐  中山大学  

报告题目: Approximating and learning smooth functions by ReLU neural networks

报告摘要: In this talk, we will discuss some recent progresses on the approximation and learning theory of ReLU neural networks. We can divide the approximation theory of neural networks into two parts according to the methods. In the first method, we approximate smooth functions by piecewise polynomials and then construct neural networks to approximate these piecewise polynomials. Using constructive approximation, one can derive optimal approximation rates in terms of the width and depth. In the second method, we consider the variation space of shallow neural networks (also called Barron space), and use random approximation method to derive approximation bounds in this space. By studying the relation between the variation space and the smooth function spaces, we can characterize the approximation error of shallow neural networks by the width and certain norm of the weights. As an application, we will discuss how these approximation results can be used in nonparametric regression problems. In particular, we will show that least squares estimations based on deep or shallow neural networks can achieve minimax optimal rates of convergence for learning smooth function classes.

报 告 人: 游俊韬  星空游戏(StarSky Sports)官方网站  

报告题目: Efficient Structured Data Recovery in Phase Retrieval and Matrix Completion

报告摘要: In this talk, we delve into two fundamental problems in data science: phase retrieval and matrix recovery. Our focus is on addressing major challenges frequently encountered in data analysis, including nonlinear measurements, large-scale or high-dimensional data, missing entries, and data corruption. To overcome these difficulties, we develop provably efficient nonconvex optimization algorithms that exploit low-dimensional data structures such as sparsity, low rank, and Hankel structure.

报 告 人: 郑伟英  中国科学院数学与系统科学研究院  

报告题目: A perfectly matched layer method for the wave scattering problem by a step-like surface

报告摘要: This talk is concerned with the convergence theory of perfectly matched layer (PML) method for wave scattering problems in a half plane bounded by a step-like surface. When a plane wave impinges upon the surface, the scattered waves compose of an outgoing radiative field and two known parts. The first part consists of two parallel reflected plane waves of different phases, which propagate in two different subregions separated by a half-line parallel to the wave direction. The second part stands for an outgoing corner-scattering field which is discontinuous and represented by a double-layer potential. A piecewise circular PML is defined by introducing two types of complex coordinates transformations in the two subregions, respectively. A PML variational problem is proposed to approximate the scattered waves. The exponential convergence of the PML solution is established by two results based on the technique of Cagniard-de Hoop (CDH) transform. First, we show that the discontinuous corner-scattering field decays exponentially in the PML. Second, we show that the transparent boundary condition (TBC) defined by the PML is an exponentially small perturbation of the original TBC defined by the radiation condition. Numerical examples validate the theory and demonstrate the effectiveness of the proposed PML.

报 告 人: 周栋焯  上海交通大学  

报告题目: 从树突到网络：解码神经系统的结构、动力学与功能

报告摘要: 大脑神经元网络是一个复杂的物理系统，其结构、动力学与功能紧密耦合，而理解它们之间的相互关系一直是神经科学领域的重要挑战。本报告从多尺度视角出发，将单神经元到网络层面的研究联系起来。首先介绍我们针对具有树突结构的神经元系统发展的一种渐近框架，将复杂的树突电缆模型约化为高效的点神经元模型，在保留树突计算特性的同时也提供了一种新的脑启发的人工神经元模型。其次讨论神经元网络的结构连接与因果连接的映射关系，结合实验上常用的四种因果性度量方法，建立它们之间的关系，并且提出实际中可行的推测神经元网络结构的计算方法。最后探讨神经元网络动力学中的均衡态，揭示兴奋性与抑制性输入如何在异质性网络连接下实现不规则放电，并进一步讨论网络均衡态在高效编码中的作用。本报告强调通过发展建模、分析和模拟等计算和应用数学研究方法，能够帮忙我们从机制上理解和解决神经科学中的一些重要科学问题。

报 告 人: 周圣高  上海交通大学  

报告题目: Multi-Physics Modeling and Computation for Electrochemical Devices

报告摘要: This talk focuses on the multi-physics modeling and computation for electrochemical devices, such as supercapacitors and solid-state batteries. For supercapacitors, accurate characterization of entropy plays a pivotal role in capturing reversible and irreversible heating during charging/discharging cycles. First, we will present a non-isothermal electrokinetic model and corresponding entropy increasing numerical methods for the prediction of temperature oscillations observed in experiments. Numerical analysis is performed to theoretically establish the structure-preserving properties that are preserved by the numerical methods. For solid-state batteries, we present a real 2D (R2D) galvanostatic model that encodes physicochemical heterogeneity in a full battery system under realistic working conditions. Simulation results are validated against experimental data with detailed discussions.