We prove a quantitative result for the approximation of functions of regularity $C^k$ (in the sense of real variables) defined on the complex cube $\Omega_n := [-1,1]^n +i[-1,1]^n\subseteq \mathbb{C}^n$ using shallow complex-valued neural networks. Precisely, we consider neural networks with a single hidden layer and $m$ neurons, i.e., networks of the form $z \mapsto \sum_{j=1}^m \sigma_j \cdot \phi\big(\rho_j^T z + b_j\big)$ and show that one can approximate every function in $C^k \left( \Omega_n; \mathbb{C}\right)$ using a function of that form with error of the order $m^{-k/(2n)}$ as $m \to \infty$, provided that the activation function $\phi: \mathbb{C} \to \mathbb{C}$ is smooth but not polyharmonic on some non-empty open set. Furthermore, we show that the selection of the weights $\sigma_j, b_j \in \mathbb{C}$ and $\rho_j \in \mathbb{C}^n$ is continuous with respect to $f$ and prove that the derived rate of approximation is optimal under this continuity assumption. We also discuss the optimality of the result for a possibly discontinuous choice of the weights.

翻译：我们证明了关于在复立方体$\Omega_n := [-1,1]^n + i[-1,1]^n \subseteq \mathbb{C}^n$上定义的、正则性为$C^k$（在实变量的意义上）的函数使用浅层复值神经网络逼近的一个定量结果。精确地，我们考虑具有单个隐藏层和$m$个神经元的神经网络，即形如$z \mapsto \sum_{j=1}^m \sigma_j \cdot \phi\big(\rho_j^T z + b_j\big)$的网络，并证明：当激活函数$\phi: \mathbb{C} \to \mathbb{C}$在某个非空开集上光滑但不为多调和函数时，当$m \to \infty$，可以用这种形式的函数以$m^{-k/(2n)}$阶误差逼近$C^k \left( \Omega_n; \mathbb{C}\right)$中的每一个函数。此外，我们展示了权重$\sigma_j, b_j \in \mathbb{C}$和$\rho_j \in \mathbb{C}^n$的选择关于$f$是连续的，并证明了在此连续性假设下，所得逼近率是最优的。我们还讨论了在权重可能非连续选择情况下结果的最优性。