With the increasing adoption of Deep Learning (DL) for critical tasks, such as autonomous driving, the evaluation of the quality of systems that rely on DL has become crucial. Once trained, DL systems produce an output for any arbitrary numeric vector provided as input, regardless of whether it is within or outside the validity domain of the system under test. Hence, the quality of such systems is determined by the intersection between their validity domain and the regions where their outputs exhibit a misbehaviour.

In this paper, we introduce the notion of frontier of behaviours, i.e., the inputs at which the DL system starts to misbehave. If the frontier of misbehaviours is outside the validity domain of the system, the quality check is passed. Otherwise, the inputs at the intersection represent quality deficiencies of the system. We developed DeepJanus, a search-based tool that generates frontier inputs for DL systems. The experimental results obtained for the lane keeping component of a self-driving car show that the frontier of a well trained system contains almost exclusively unrealistic roads that violate the best practices of civil engineering, while the frontier of a poorly trained one includes many valid inputs that point to serious deficiencies of the system.