发表机构
Johns Hopkins University（约翰斯·霍普金斯大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出一个统一分析框架，研究多粒子 Oja 流，证明有限粒子系统收敛于平均场方程，并揭示其低维不变结构，最终在角中心高斯族中实现参数化约化描述。
AI 中文摘要
我们开发了一个分析框架，用于研究一族相互作用粒子系统，我们将其称为多粒子 Oja 流。该族系统共有的相互作用结构出现在多种应用中，包括高阶同步模型、意见动力学以及由 Transformer 架构导出的自注意力动力学，这促使我们建立一个统一的数学框架来进行分析。我们首先建立了有限粒子系统和平均场系统的适定性。我们还证明了当粒子数趋于无穷时，有限粒子动力学收敛到平均场方程。接着，我们研究了动力学背后的低维结构。特别地，我们证明了平均场解可以通过一个作用于初始分布的时间相关归一化线性变换来表示，该变换由矩阵常微分方程控制。基于这一表示，我们识别出一大类在平均场动力学下不变的参数族。当初始分布属于这些族之一时，演化在所有时间内保持在同一族内，因此可以通过相应族参数的动力学来刻画。作为一个具体例子，我们在角中心高斯族内发展了这种约化，并说明了由此产生的参数动力学如何为几个代表性系统提供平均场演化的约化描述。
英文摘要
We develop an analytical framework for studying a family of interacting particle systems that we refer to as the multi-particle Oja flow. The common interaction structure underlying this family arises in a variety of applications, including higher-order synchronization models, opinion dynamics, and self-attention dynamics derived from Transformer architectures, motivating a unified mathematical framework for their analysis. We first establish well-posedness for both the finite-particle and the mean-field systems. We also prove convergence of the finite-particle dynamics to the mean-field equation as the number of particles tends to infinity. We then investigate low-dimensional structures underlying the dynamics. In particular, we show that the mean-field solution admits a representation through a time-dependent normalized linear transformation acting on the initial distribution, with the transformation governed by a matrix ODE. Building on this representation, we identify a broad class of parametric families that are invariant under the mean-field dynamics. When the initial distribution belongs to one of these families, the evolution remains within the same family for all time and can therefore be characterized through the dynamics of the corresponding family parameters. As a concrete example, we develop this reduction within the angular central Gaussian family and illustrate how the resulting parameter dynamics provide reduced descriptions of the mean-field evolution for several representative systems.
