Overcoming the curse of dimensionality for approximating Lyapunov functions with deep neural networks under a small-gain condition

Abstract

We propose a deep neural network architecture for storing approximate Lyapunov functions of systems of ordinary differential equations. Under a small-gain condition on the system, 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.

In this version a couple of typos and inaccuracies were fixed. In the previous version version a missing "-" sign in eq. (2) was added (thanks to Sergey Dashkovskiy for spotting this) and three typos were corrected (thanks to Manuel Schaller for reporting them)