In this talk, I will show how control theoretic techniques can shed light on the approximation capabilities of Deep Neural Networks (DNN) with limited width. Specifically, I will demonstrate that Residual Networks (ResNet), a type of DNN, can approximate trajectories on the set of probability measures due to the controllability properties of the continuity equation. I will then use geometric control theory to design more efficient ResNets when additional data information is available, particularly when it is known that data that lies on a lower-dimensional manifold. I will then present numerical experiments on two model manifolds – the two-dimensional sphere and the 3-dimensional orthogonal group SO(3) – which have applications in mechanical systems, such as spacecraft and satellites, to validate the theoretical results.