This paper explores the capability of deep neural networks to capture key characteristics of vehicle dynamics, and their ability to perform coupled longitudinal and lateral control of a vehicle. To this extent, two different artificial neural networks are trained to compute vehicle controls corresponding to a reference trajectory, using a dataset based on high-fidelity simulations of vehicle dynamics. In this study, control inputs are chosen as the steering angle of the front wheels, and the applied torque on each wheel. The performance of both models, namely a Multi-Layer Perceptron (MLP) and a Convolutional Neural Network (CNN), is evaluated based on their ability to drive the vehicle on a challenging test track, shifting between long straight lines and tight curves. A comparison to conventional decoupled controllers on the same track is also provided.