  Koopman spectral theory has grown in the past decade as a powerful tool for
dynamical systems analysis and control. In this paper, we show how recent
data-driven techniques for estimating Koopman-Invariant subspaces with neural
networks can be leveraged to extract Lyapunov certificates for the underlying
system. In our work, we specifically focus on systems with a limit-cycle,
beyond just an isolated equilibrium point, and use Koopman eigenfunctions to
efficiently parameterize candidate Lyapunov functions to construct
forward-invariant sets under some (unknown) attractor dynamics. Additionally,
when the dynamics are polynomial and when neural networks are replaced by
polynomials as a choice of function approximators in our approach, one can
further leverage Sum-of-Squares programs and/or nonlinear programs to yield
provably correct Lyapunov certificates. In such a polynomial case, our
Koopman-based approach for constructing Lyapunov functions uses significantly
fewer decision variables compared to directly formulating and solving a
Sum-of-Squares optimization problem.
