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This paper presents a mathematical analysis of ODE-Net, a continuum model ofdeep neural networks (DNNs). In recent years, Machine Learning researchers haveintroduced ideas of replacing the deep structure of DNNs with ODEs as acontinuum limit. These studies regard the "learning" of ODE-Net as theminimization of a "loss" constrained by a parametric ODE. Although theexistence of a minimizer for this minimization problem needs to be assumed,only a few studies have investigated its existence analytically in detail. Inthe present paper, the existence of a minimizer is discussed based on aformulation of ODE-Net as a measure-theoretic mean-field optimal controlproblem. The existence result is proved when a neural network, which describesa vector field of ODE-Net, is linear with respect to learnable parameters. Theproof employs the measure-theoretic formulation combined with the direct methodof Calculus of Variations. Secondly, an idealized minimization problem isproposed to remove the above linearity assumption. Such a problem is inspiredby a kinetic regularization associated with the Benamou--Brenier formula anduniversal approximation theorems for neural networks. The proofs of theseexistence results use variational methods, differential equations, andmean-field optimal control theory. They will stand for a new analytic way toinvestigate the learning process of deep neural networks.

Variational formulations of ODE-Net as a mean-field optimal control problem and existence results