  We provide a systematic investigation of using physics-informed neural
networks to compute Lyapunov functions. We encode Lyapunov conditions as a
partial differential equation (PDE) and use this for training neural network
Lyapunov functions. We analyze the analytical properties of the solutions to
the Lyapunov and Zubov PDEs. In particular, we show that employing the Zubov
equation in training neural Lyapunov functions can lead to approximate regions
of attraction close to the true domain of attraction. We also examine
approximation errors and the convergence of neural approximations to the unique
solution of Zubov's equation. We then provide sufficient conditions for the
learned neural Lyapunov functions that can be readily verified by
satisfiability modulo theories (SMT) solvers, enabling formal verification of
both local stability analysis and region-of-attraction estimates in the large.
Through a number of nonlinear examples, ranging from low to high dimensions, we
demonstrate that the proposed framework can outperform traditional
sums-of-squares (SOS) Lyapunov functions obtained using semidefinite
programming (SDP).
