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.

翻译：我们提出了一种新颖的卷积神经网络架构，称为\emph{周期性CNN}，该架构将周期性边界条件引入卷积层。我们的主要理论贡献是一个严格的逼近定理：在$d$维输入空间中，周期性CNN能够逼近依赖于$d-1$个线性变量的岭函数，而在更低维的岭设置（$d-2$个或更少变量）中此类逼近是不可能的。这一结果建立了对周期性CNN表达能力的一个精确刻画。除了理论意义外，我们的研究表明周期性CNN特别适用于数据天然具有高内在维度岭状结构的问题，例如环绕域上的图像分析、物理信息学习以及材料科学。因此，这项工作既拓展了CNN逼近理论的数学基础，也突显了一类具有出人意料且具有实际应用价值的逼近能力的架构。