周期卷积网络可精准逼近高维流形数据，理论突破传统限制。
Theory of periodic convolutional neural network
- 引入周期边界条件的卷积层，提升对高维流形结构的建模能力
- 理论上证明可逼近依赖d-1个线性变量的脊函数，而d-2维时不可能
- 适用于图像环面域、物理信息学习等高内在维度问题
我们提出一种新型卷积神经网络——周期卷积神经网络（periodic CNN），将周期边界条件引入卷积层。主要理论贡献为严格逼近定理：周期CNN可在d维输入空间中逼近依赖d−1个线性变量的脊函数，而在更低维脊结构（d−2或更少变量）下此逼近不可能实现。该结果精确刻画了周期CNN的表达能力。除理论外，研究暗示周期CNN特别适合具有高内在维度脊结构的数据，如环面域上的图像分析、物理信息学习和材料科学。本工作既拓展了CNN逼近理论的数学基础，也揭示了一类具有意外且实际相关逼近能力的架构。
原文摘要 · Abstract (English)
We introduce a novel convolutional neural network architecture, termed the \emph{periodic CNN}, which incorporates periodic boundary conditions into the convolutional layers. Our main theoretical contribution is a rigorous approximation theorem: periodic CNNs can approximate ridge functions depending on $d-1$ linear variables in a $d$-dimensional input space, while such approximation is impossible in lower-dimensional ridge settings ($d-2$ or fewer variables). This result establishes a sharp characterization of the expressive power of periodic CNNs. Beyond the theory, our findings suggest that periodic CNNs are particularly well-suited for problems where data naturally admits a ridge-like structure of high intrinsic dimension, such as image analysis on wrapped domains, physics-informed learning, and materials science. The work thus both expands the mathematical foundation of CNN approximation theory and highlights a class of architectures with surprising and practically relevant approximation capabilities.
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