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.