A Minimal Control Family of Dynamical Systems for Universal Approximation

• URL: http://arxiv.org/abs/2312.12903v2
• Date: Mon, 31 Mar 2025 01:01:12 GMT
• Title: A Minimal Control Family of Dynamical Systems for Universal Approximation
• Authors: Yifei Duan, Yongqiang Cai,
• Abstract summary: The universal approximation property (UAP) holds a fundamental position in deep learning.<n>We show that it can approximate continuous functions on compact domains.<n>Our results reveal an underlying connection between the approximation power of neural networks and control systems.
• Score: 5.217870815854702
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: The universal approximation property (UAP) holds a fundamental position in deep learning, as it provides a theoretical foundation for the expressive power of neural networks. It is widely recognized that a composition of linear and nonlinear 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 a scenario containing dynamical systems with controls. We prove that the control family $\mathcal{F}_1$ containing all affine maps and the nonlinear ReLU map is sufficient for generating flow maps that can approximate orientation-preserving (OP) diffeomorphisms on any compact domain. Since $\mathcal{F}_1$ contains only one nonlinear function and the UAP does not hold if we remove the nonlinear function, we call $\mathcal{F}_1$ a minimal control family for the UAP. On this basis, several mild sufficient conditions, such as affine invariance, are established for the control family and discussed. Our results reveal an underlying connection between the approximation power of neural networks and control systems and could provide theoretical guidance for examining the approximation power of flow-based models.

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