A Minimal Control Family of Dynamical Syetem for Universal Approximation

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.