  In this paper, we consider the problem of training neural network (NN)
controllers for nonlinear dynamical systems that are guaranteed to satisfy
safety and liveness (e.g., reach-avoid) properties. Our approach is to combine
model-based design methodologies for dynamical systems with data-driven
approaches to achieve this target. We confine our attention to NNs with
Rectifier Linear Unit (ReLU) nonlinearity which are known to represent
Continuous Piece-Wise Affine (CPWA) functions. Given a mathematical model of
the dynamical system, we compute a finite-state abstract model that captures
the closed-loop behavior under all possible CPWA controllers. Using this
finite-state abstract model, our framework identifies a family of CPWA
functions guaranteed to satisfy the safety requirements. We augment the
learning algorithm with a NN weight projection operator during training that
enforces the resulting NN to represent a CPWA function from the provably safe
family of CPWA functions. Moreover, the proposed framework uses the
finite-state abstract model to identify candidate CPWA functions that may
satisfy the liveness properties. Using such candidate CPWA functions, the
proposed framework biases the NN training to achieve the liveness
specification. We show the efficacy of the proposed framework both in
simulation and on an actual robotic vehicle.
