  This paper deals with a special type of Lyapunov functions, namely the
solution of Zubov's equation. Such a function can be used to characterize the
domain of attraction for systems of ordinary differential equations. We derive
and prove an integral form solution to Zubov's equation. For numerical
computation, we develop two data-driven methods. One is based on the
integration of an augmented system of differential equations; and the other one
is based on deep learning. The former is effective for systems with a
relatively low state space dimension and the latter is developed for high
dimensional problems. The deep learning method is applied to a New England
10-generator power system model. We prove that a neural network approximation
exists for the Lyapunov function of power systems such that the approximation
error is a cubic polynomial of the number of generators. The error convergence
rate as a function of n, the number of neurons, is proved.
