  Residual neural networks are state-of-the-art deep learning models. Their
continuous-depth analog, neural ordinary differential equations (ODEs), are
also widely used. Despite their success, the link between the discrete and
continuous models still lacks a solid mathematical foundation. In this article,
we take a step in this direction by establishing an implicit regularization of
deep residual networks towards neural ODEs, for nonlinear networks trained with
gradient flow. We prove that if the network is initialized as a discretization
of a neural ODE, then such a discretization holds throughout training. Our
results are valid for a finite training time, and also as the training time
tends to infinity provided that the network satisfies a Polyak-Lojasiewicz
condition. Importantly, this condition holds for a family of residual networks
where the residuals are two-layer perceptrons with an overparameterization in
width that is only linear, and implies the convergence of gradient flow to a
global minimum. Numerical experiments illustrate our results.
