In the first part of this talk we will see the interplay between Linear-Control Systems and Residual Neural Networks, observed for the first in 2017.   We will see how one can reconstruct a diffeomorphism  as an ensemble optimal control problem. The discretization of this control problem leads to a ResNet.  
In the second part of the talk we adapt the technique of the diffeomorphisms reconstruction to approach the problem of normalizing flows by using some optimal transport.