发表机构
Sichuan University; Friedrich–Alexander–Universität Erlangen–Nürnberg; Fundación Deusto; Universidad Autónoma de Madrid（四川大学; 埃尔朗根-纽伦堡大学; 德乌斯托基金会; 马德里自治大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究证明一类线性自治流可在任意正时刻精确实现有限点插值，并应用于浅层ReLU网络，同时揭示连续半流无法一致逼近所有连续映射的障碍。
AI 中文摘要
在至少二维的空间中，一类局部Lipschitz向量场能够实现任意给定正时刻下，不同输入与不同目标之间的所有有限对应关系，前提是该向量场是线性的，其成员生成全局流，并且在有界集上一致逼近所有具有有界支撑的光滑向量场。输入集和目标集可以重叠。证明过程构造了一个光滑的自治参考流和有限个局部修正场，然后利用一致稳定性和Brouwer度在逼近类内获得精确端点。该论证仅使用向量场的一致逼近，不需要控制其导数。应用于浅层ReLU向量场时，该结果实现了精确的有限插值，无需时间依赖系数或额外状态变量，同时给出了宽度和归一化系数强度的界。相反，没有连续的自治半流能够交换并压缩两个不相交的球。因此，其时间映射无法在紧集上一致逼近所有连续映射。这些结果区分了有限点上的精确插值与邻域的一致控制。
英文摘要
In dimension at least two, a class of locally Lipschitz vector fields realizes every finite correspondence between distinct inputs and distinct targets at any prescribed positive time, provided that it is linear, its members generate global flows, and it approximates every smooth vector field of bounded support uniformly on bounded sets. The input and target sets may overlap. The proof constructs a smooth autonomous reference flow and finitely many localized correction fields, then uses uniform stability and Brouwer degree to obtain exact endpoints within the approximating class. The argument uses only uniform approximation of the vector fields; it does not require control of their derivatives. Applied to shallow ReLU vector fields, the result gives exact finite interpolation without time-dependent coefficients or additional state variables, together with bounds on width and normalized coefficient strength. In contrast, no continuous autonomous semiflow can exchange and compress two disjoint balls. Its time maps therefore fail to approximate all continuous maps uniformly on compact sets. The results distinguish exact interpolation at finitely many points from uniform control of neighborhoods.
