李代数与网络动力学：精确的宏观约化（有限系统）
Lie Meets Network Dynamics: Exact Macroscopic Reductions (Finite Systems)
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中文总结 AI 辅助
该研究利用李 - 谢弗斯理论为网络动力系统精确降维建立统一框架，针对平均场李 - 谢弗斯结构的系统，可从\(nd\)维降至\(md\)维，还能发现新约化方法，并通过多种方程集合进行了理论说明。
中文摘要 AI 辅助
我们利用李 - 谢弗斯理论建立了网络动力系统精确降维的统一框架。对于具有“平均场李 - 谢弗斯结构”的网络动力系统，我们证明具有局部维度\(d\)的\(n\)个节点的网络可从\(nd\)维精确降至维度为\(md\)的固定宏观系统，其中\(m\)是节点动力学所需基本解的数量。李代数结构产生的叠加原理使平均场耦合能根据宏观变量明确表示，得到一个与网络大小无关的“封闭”自洽系统。这种约化将高维网络流坍缩到由\(\gamma = d(n - m)\)个独立运动常数参数化的不变流形上。我们的框架严格解释了已知约化并提供了发现新约化的系统方法。我们用里卡蒂方程组（包括库拉托莫模型和θ神经元模型）、拟线性常微分方程和广义伯努利方程的集合来说明该理论，明确推导每种情况下的宏观流和守恒量。
英文摘要
We establish a unified framework for exact dimensional reductions in network dynamical systems using Lie-Scheffers theory. For network dynamical systems with \emph{mean-field Lie-Scheffers structure}, we prove that networks of $n$ nodes with local dimension $d$ can be exactly reduced from $ n d $ dimensions to a fixed macroscopic system of dimension $ m d $, where $m$ is the number of fundamental solutions required by the nodal dynamics. Crucially, the superposition principle resulting from the Lie-algebraic structure allows the mean-field coupling to be expressed explicitly in terms of the macroscopic variables, yielding a \emph{closed} self-consistent system independent of network size. This reduction collapses the high-dimensional network flow onto invariant manifolds parameterized by $ γ= d(n-m) $ independent constants of motion. Our framework rigorously explains known reductions and provides a \emph{systematic method to discover new ones}. We illustrate the theory with ensembles of Riccati equations (encompassing the Kuramoto model and Theta neuron model), quasi-linear ODEs, and generalized Bernoulli equations, explicitly deriving the macroscopic flows and conserved quantities for each case.
