  Estimating the region of attraction (ROA) of general nonlinear autonomous
systems remains a challenging problem and requires a case-by-case analysis.
Leveraging the universal approximation property of neural networks, in this
paper, we propose a counterexample-guided method to estimate the ROA of general
nonlinear dynamical systems provided that they can be approximated by piecewise
linear neural networks and that the approximation error can be bounded.
Specifically, our method searches for robust Lyapunov functions using
counterexamples, i.e., the states at which the Lyapunov conditions fail. We
generate the counterexamples using Mixed-Integer Quadratic Programming. Our
method is guaranteed to find a robust Lyapunov function in the parameterized
function class, if exists, after collecting a finite number of counterexamples.
We illustrate our method through numerical examples.
