Learning Robust Regions of Attraction Using Rollout-Enhanced Physics-Informed Neural Networks with Policy Iteration

The region of attraction is a key metric of the robustness of systems. This paper addresses the numerical solution of the generalized Zubov's equation, which produces a special Lyapunov function characterizing the robust region of attraction for perturbed systems. To handle the highly nonlinear characteristic of the generalized Zubov's equation, we propose a physics-informed neural network framework that employs a policy iteration training scheme with rollout to approximate the viscosity solution. In addition to computing the optimal disturbance during the policy improvement process, we incorporate neural network-generated value estimates as anchor points to facilitate the training procedure to prevent singularities in both low- and high-dimensional systems. Numerical simulations validate the effectiveness of the proposed approach.