  We develop a versatile deep neural network architecture, called Lyapunov-Net,
to approximate Lyapunov functions of dynamical systems in high dimensions.
Lyapunov-Net guarantees positive definiteness, and thus it can be easily
trained to satisfy the negative orbital derivative condition, which only
renders a single term in the empirical risk function in practice. This
significantly reduces the number of hyper-parameters compared to existing
methods. We also provide theoretical justifications on the approximation power
of Lyapunov-Net and its complexity bounds. We demonstrate the efficiency of the
proposed method on nonlinear dynamical systems involving up to 30-dimensional
state spaces, and show that the proposed approach significantly outperforms the
state-of-the-art methods.
