Stability analysis of dynamical systems involves both qualitative and quantitative aspects. While qualitative stability determines whether an equilibrium is stable, quantitative stability characterizes the size of the region of attraction, the set of initial conditions from which trajectories converge to the equilibrium. Constructing Lyapunov functions and estimating the region of attraction are fundamental tasks in safety-critical control design and verification. However, classical approaches such as analytical Lyapunov function construction, sum-of-squares techniques, and PDE-based methods often become computationally intractable or conservative for nonlinear or high-dimensional systems. Neural networks offer a promising alternative by parameterizing Lyapunov functions or approximating solutions to stability-related PDEs. However, neural stability-learning methods often suffer from poor training stability, sparse data coverage, and difficulties in handling disturbances. This dissertation develops enhanced neural stability-learning frameworks. An augmented learning strategy with verifier-generated samples improves Lyapunov function learning, while a physics-informed neural network framework combined with policy iteration and rollout-generated anchor data enables robust region of attraction estimation through the generalized Zubov equation. Experiments on nonlinear and disturbance-affected systems demonstrate improved accuracy, robustness, and scalability compared with existing methods.