  We propose a deep neural network architecture and a training algorithm for
computing approximate Lyapunov functions of systems of nonlinear ordinary
differential equations. Under the assumption that the system admits a
compositional Lyapunov function, we prove that the number of neurons needed for
an approximation of a Lyapunov function with fixed accuracy grows only
polynomially in the state dimension, i.e., the proposed approach is able to
overcome the curse of dimensionality. We show that nonlinear systems satisfying
a small-gain condition admit compositional Lyapunov functions. Numerical
examples in up to ten space dimensions illustrate the performance of the
training scheme.
