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 a 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：本文以Korobov函数为基准，研究二维深度卷积神经网络（CNN）的逼近能力。我们聚焦于包含多通道卷积层（采用零填充和ReLU激活函数）及后续全连接层的二维CNN。提出了一种完全构造性的方法来构建逼近Korobov函数的二维CNN，并对所构造网络的复杂度进行了严格分析。结果表明，在连续权值选择模型下，二维CNN能够达到近乎最优的逼近率，显著缓解了维度灾难问题。该工作为二维CNN提供了坚实的理论基础，并展示了其在函数逼近中更广泛应用的潜力。