Originally arising in the context of interacting particle systems in statistical physics, dynamical systems and differential equations on networks/graphs have permeated into a broad number of mathematical areas as well as into many applications. One central problem in the field is to find suitable approximations of the dynamics as the number of nodes/vertices tends to infinity, i.e., in the large graph limit. A cornerstone in this context are Vlasov-Fokker-Planck equations (VFPEs) describing a particle density on a mean-field level. For all-to-all coupled systems, it is quite classical to prove the rigorous approximation by VFPEs for many classes of particle systems. For dense graphs converging to graphon limits, one also knows that mean-field approximation holds for certain classes of models, e.g., for the Kuramoto model on graphs. Yet, the space of intermediate density and sparse graphs is clearly extremely relevant. Here we prove that the Kuramoto model can be be approximated in the mean-field limit by far more general graph limits than graphons. In particular, our contributions are as follows. (I) We show, how to introduce operator theory more abstractly into VFPEs by considering graphops. Graphops have recently been proposed as a unifying approach to graph limit theory, and here we show that they can be used for differential equations on graphs. (II) For the Kuramoto model on graphs we rigorously prove that there is a VFPE equation approximating it in the mean-field sense. (III) This mean-field VFPE involves a graphop, and we prove the existence, uniqueness, and continuous graphop-dependence of weak solutions. (IV) On a technical level, our results rely on designing a new suitable metric of graphop convergence and on employing Fourier analysis on compact abelian groups to approximate graphops using summability kernels.
Mean field repulsive Kuramoto models: Phase locking and spatial signs
振荡单位群体的自同步现象在神经科学中很自然地出现。然而，在某些情况下，相干态的形成是有害的。在这篇文章中，我们研究了一个排斥的平均场仓本模型，它描述了单位圆上的n个点的时间演化，这些点被转换成非相干的锁相态。最近已经表明，这样的系统可以减少到一个三维系统的常微分方程，其数学结构是密切相关的双曲几何。然后用莫比乌斯变换描述了仓本动力系统的轨道。我们表明，这种基本的动态执行统计推断，通过动态计算M-估计的散布矩阵。我们还描述了使用泰勒变换矩阵的随机初始条件下的极限锁相状态。此外，我们表明，排斥Kuramoto模型不仅动态地进行稳健的协方差矩阵估计，而且还进行数据处理：n个点的初始配置被动态转换为限制锁相状态，该状态令人惊讶地等于非参数统计的空间符号。这使得符号经验协方差矩阵等于1 2 id 2，即单位圆上均匀分布的随机向量的方差-协方差矩阵。
Equivalence of synchronization states in the Kuramoto flow: a unified framework
Scuola Internazionale Superiore di Studi Avanzati (SISSA)（高等国际研究学院）；Georgia Institute of Technology（佐治亚理工学院）；University of Illinois Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
Quantitative Clustering in Mean-Field Transformer Models
Department of Mathematics, Massachusetts Institute of Technology（数学系，麻省理工学院）；Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology（电气工程与计算机科学系，麻省理工学院）
