  Neural networks have been very successful in many applications; we often,
however, lack a theoretical understanding of what the neural networks are
actually learning. This problem emerges when trying to generalise to new data
sets. The contribution of this paper is to show that, for the residual neural
network model, the deep layer limit coincides with a parameter estimation
problem for a nonlinear ordinary differential equation. In particular, whilst
it is known that the residual neural network model is a discretisation of an
ordinary differential equation, we show convergence in a variational sense.
This implies that optimal parameters converge in the deep layer limit. This is
a stronger statement than saying for a fixed parameter the residual neural
network model converges (the latter does not in general imply the former). Our
variational analysis provides a discrete-to-continuum $\Gamma$-convergence
result for the objective function of the residual neural network training step
to a variational problem constrained by a system of ordinary differential
equations; this rigorously connects the discrete setting to a continuum
problem.
