656	RWTH Publication No: 989614        2024
TITLE	Controllability of continuous networks and a kernel-based learning approximation
AUTHORS	Michael Herty, Chiara Segala, Giuseppe Visconti
ABSTRACT	Residual deep neural networks are formulated as interacting particle systems leading to a description through neural differential equations, and, in the case of large input data, through mean-field neural networks. The mean-field description allows also the recast of the training processes as a controllability problem for the solution to the mean-field dynamics. We show theoretical results on the controllability of the linear microscopic and mean-field dynamics through the Hilbert Uniqueness Method and propose a computational approach based on kernel learning methods to solve numerically, and efficiently, the training problem. Further aspects of the structural properties of the mean-field equation will be reviewed.
KEYWORDS	Neural networks, mean-field limit, controllability, kernel methods