Controllability of continuous networks and a kernel-based learning approximation

Michael Herty; Chiara Segala; Giuseppe Visconti

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arXiv · 2403.08690

Controllability of continuous networks and a kernel-based learning approximation

Michael Herty·Chiara Segala·Giuseppe Visconti

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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.

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Michael Herty, Chiara Segala, Giuseppe Visconti. 2024-03-13. Controllability of continuous networks and a kernel-based learning approximation. https://arxiv.org/abs/2403.08690

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