  Residual networks (ResNets) have displayed impressive results in pattern
recognition and, recently, have garnered considerable theoretical interest due
to a perceived link with neural ordinary differential equations (neural ODEs).
This link relies on the convergence of network weights to a smooth function as
the number of layers increases. We investigate the properties of weights
trained by stochastic gradient descent and their scaling with network depth
through detailed numerical experiments. We observe the existence of scaling
regimes markedly different from those assumed in neural ODE literature.
Depending on certain features of the network architecture, such as the
smoothness of the activation function, one may obtain an alternative ODE limit,
a stochastic differential equation or neither of these. These findings cast
doubts on the validity of the neural ODE model as an adequate asymptotic
description of deep ResNets and point to an alternative class of differential
equations as a better description of the deep network limit.
