This paper investigates approximation capabilities of two-dimensional (2D) deep convolutional neural networks (CNNs), with Korobov functions serving as a benchmark. We focus on 2D CNNs, comprising multi-channel convolutional layers with zero-padding and ReLU activations, followed by a fully connected layer. We propose a fully constructive approach for building 2D CNNs to approximate Korobov functions and provide rigorous analysis of the complexity of the constructed networks. Our results demonstrate that 2D CNNs achieve near-optimal approximation rates under the continuous weight selection model, significantly alleviating the curse of dimensionality. This work provides a solid theoretical foundation for 2D CNNs and illustrates their potential for broader applications in function approximation.

翻译：本文研究二维深度卷积神经网络（CNN）的逼近能力，以Korobov函数作为基准测试对象。我们聚焦于由多通道卷积层（含零填充和ReLU激活函数）及全连接层构成的二维CNN。针对Korobov函数的逼近问题，我们提出一种完全构造性的二维CNN构建方法，并对所构造网络的复杂度进行严格分析。研究结果表明，在连续权重选择模型下，二维CNN能够实现近乎最优的逼近速率，显著缓解维度灾难问题。本工作为二维CNN奠定了坚实的理论基础，并揭示了其在函数逼近领域更广泛的应用潜力。