Deep Learning via Dynamical Systems: An Approximation Perspective

We build on the dynamical systems approach to deep learning, where deep residual networks are idealized as continuous-time dynamical systems. Although theoretical foundations have been developed on the optimization side through mean-field optimal control theory, the function approximation properties of such models remain largely unexplored, especially when the dynamical systems are controlled by functions of low complexity. In this paper, we establish some basic results on the approximation capabilities of deep learning models in the form of dynamical systems. In particular, we derive general sufficient conditions for universal approximation of functions in $L^p$ using flow maps of dynamical systems, and we also deduce some results on their approximation rates for specific cases. Overall, these results reveal that composition function approximation through flow maps present a new paradigm in approximation theory and contributes to building a useful mathematical framework to investigate deep learning.