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）是深度学习中最突出的神经网络架构之一。尽管其被广泛采用，但由于其复杂特性，我们对它的普适逼近性质的理解仍然有限。CNN本质上作为张量到张量的映射，能够保留输入数据的空间结构。然而，将全卷积神经网络作为任意连续张量到张量函数来研究其普适逼近性质的工作仍然有限。在本研究中，我们证明了当采用零填充时，CNN能够在输入和输出值具有相同空间形状的情况下逼近任意连续函数。此外，我们确定了逼近所需的最小网络深度，并验证了其最优性。我们还证实了深层窄CNN具备作为张量到张量函数的普适逼近性质（UAP）。这些结果涵盖了广泛的激活函数，且我们的研究涉及所有维度的CNN。