This paper is devoted to studying the optimal expressive power of ReLU deep neural networks (DNNs) and its application in approximation via the Kolmogorov Superposition Theorem. We first constructively prove that any continuous piecewise linear functions on $[0,1]$, comprising $O(N^2L)$ segments, can be represented by ReLU DNNs with $L$ hidden layers and $N$ neurons per layer. Subsequently, we demonstrate that this construction is optimal regarding the parameter count of the DNNs, achieved through investigating the shattering capacity of ReLU DNNs. Moreover, by invoking the Kolmogorov Superposition Theorem, we achieve an enhanced approximation rate for ReLU DNNs of arbitrary width and depth when dealing with continuous functions in high-dimensional spaces.

该研究论文探讨了ReLU深度神经网络的最佳表达能力及其在通过Kolmogorov叠加定理进行逼近方面的应用。研究通过构建性地证明了在[0,1]上，由$O(N^2L)$个线性分段组成的连续分段线性函数可以通过具有$L$隐藏层和每层$N$个神经元的ReLU深度神经网络来表示。同时，通过研究ReLU深度神经网络的分形能力，证明了这种构建是最优的。此外，通过应用Kolmogorov叠加定理，在处理高维连续函数时，实现了具有任意宽度和深度的ReLU深度神经网络的改进逼近速度。

ReLU深度神经网络的最优表达能力及其在科尔莫戈洛夫超级位置定理中的应用