The universal approximation property (UAP) of neural networks is a fundamental characteristic of deep learning. It is widely recognized that a composition of linear functions and non-linear functions, such as the rectified linear unit (ReLU) activation function, can approximate continuous functions on compact domains. In this paper, we extend this efficacy to the scenario of dynamical systems with controls. We prove that the control family $\mathcal{F}_1 = \mathcal{F}_0 \cup \{ \text{ReLU}(\cdot)\} $ is enough to generate flow maps that can uniformly approximate diffeomorphisms of $\mathbb{R}^d$ on any compact domain, where $\mathcal{F}_0 = \{x \mapsto Ax+b: A\in \mathbb{R}^{d\times d}, b \in \mathbb{R}^d\}$ is the set of linear maps and the dimension $d\ge2$. Since $\mathcal{F}_1$ contains only one nonlinear function and $\mathcal{F}_0$ does not hold the UAP, we call $\mathcal{F}_1$ a minimal control family for UAP. Based on this, some sufficient conditions, such as the affine invariance, on the control family are established and discussed. Our result reveals an underlying connection between the approximation power of neural networks and control systems.

翻译：神经网络的通用逼近性质（UAP）是深度学习的基本特征。人们普遍认为，线性函数与非线性函数（如修正线性单元（ReLU）激活函数）的组合能够逼近紧致域上的连续函数。本文将这一有效性扩展至带控制的动力系统场景。我们证明控制族 $\mathcal{F}_1 = \mathcal{F}_0 \cup \{ \text{ReLU}(\cdot)\} $ 足以生成能在任意紧致域上一致逼近 $\mathbb{R}^d$ 微分同胚的流映射，其中 $\mathcal{F}_0 = \{x \mapsto Ax+b: A\in \mathbb{R}^{d\times d}, b \in \mathbb{R}^d\}$ 是线性映射集合，且维度 $d\ge2$。由于 $\mathcal{F}_1$ 仅包含一个非线性函数，而 $\mathcal{F}_0$ 不具备UAP，我们将 $\mathcal{F}_1$ 称为用于UAP的最小控制族。在此基础上，我们建立并讨论了控制族的一些充分条件（如仿射不变性）。我们的结果揭示了神经网络逼近能力与控制系统之间的内在联系。