A new two-stage training framework is proposed for synthesizing neural controllers and Lyapunov function for continuous-time systems.The framework utilizes a Zubov-inspired region of attraction characterization to estimate stability boundaries, reducing conservatism in training.State-of-the-art neural network verifier is extended for automatic bound propagation and a novel verification scheme to avoid expensive bisection.Experimental results show significant improvement in region of attractions and faster verification compared to traditional methods.