The Convolutional Neural Network (CNN) is one of the most prominent neural network architectures in deep learning. Despite its widespread adoption, our understanding of its universal approximation properties has been limited due to its intricate nature. CNNs inherently function as tensor-to-tensor mappings, preserving the spatial structure of input data. However, limited research has explored the universal approximation properties of fully convolutional neural networks as arbitrary continuous tensor-to-tensor functions. In this study, we demonstrate that CNNs, when utilizing zero padding, can approximate arbitrary continuous functions in cases where both the input and output values exhibit the same spatial shape. Additionally, we determine the minimum depth of the neural network required for approximation and substantiate its optimality. We also verify that deep, narrow CNNs possess the UAP as tensor-to-tensor functions. The results encompass a wide range of activation functions, and our research covers CNNs of all dimensions.

翻译：卷积神经网络（CNN）是深度学习中最重要的神经网络架构之一。尽管其被广泛采用，但由于其复杂本质，我们对其通用逼近性质的理解仍十分有限。CNN本质上作为张量到张量的映射，保留输入数据的空间结构。然而，目前关于全卷积神经网络作为任意连续张量到张量函数的通用逼近性质的研究仍较为有限。在本研究中，我们证明了在输入和输出值具有相同空间形状的情况下，采用零填充的CNN能够逼近任意连续函数。此外，我们确定了逼近所需的最小神经网络深度，并论证了其最优性。我们还验证了深层窄CNN作为张量到张量函数具有UAP。研究结果涵盖了广泛的激活函数，且我们的工作覆盖了所有维度的CNN。