Computing Lyapunov Functions Using Deep Neural Networks

Abstract

We propose a deep neural network architecture and associated loss functions for 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.