  Residual deep neural networks (ResNets) are mathematically described as
interacting particle systems. In the case of infinitely many layers the ResNet
leads to a system of coupled system of ordinary differential equations known as
neural differential equations. For large scale input data we derive a
mean--field limit and show well--posedness of the resulting description.
Further, we analyze the existence of solutions to the training process by using
both a controllability and an optimal control point of view. Numerical
investigations based on the solution of a formal optimality system illustrate
the theoretical findings.
