We investigate approaches to train neural networks for controlling vehicles to follow a fixed reference trajectory robustly, while respecting limits on their velocities and accelerations. Here robustness means that if a vehicle starts inside a fixed region around the reference trajectory, it remains within this region while moving along the reference from an initial set to a target set. We consider the combination of two ideas in this paper: (a) demonstrations of the correct control obtained from a model-predictive controller (MPC) and (b) falsification approaches that actively search for violations of the property, given a current candidate. Thus, our approach creates an initial training set using the MPC loop and builds a first candidate neural network controller. This controller is repeatedly analyzed using falsification that searches for counterexample trajectories, and the resulting counterexamples are used to create new training examples. This process proceeds iteratively until the falsifier no longer succeeds within a given computational budget. We propose falsification approaches using a combination of random sampling and gradient descent to systematically search for violations. We evaluate our combined approach on a variety of benchmarks that involve controlling dynamical models of cars and quadrotor aircraft.