Computing Lyapunov functions using deep neural networks

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