Compositionally Verifiable Vector Neural Lyapunov Functions for Stability Analysis of Interconnected Nonlinear Systems

TL;DR

This paper introduces a compositional approach to verify neural Lyapunov functions for high-dimensional interconnected nonlinear systems, overcoming limitations of existing methods and demonstrating improved performance over traditional polynomial approaches.

Contribution

It presents a novel compositional verification method leveraging system structure to certify neural Lyapunov functions beyond current SMT solver capabilities.

Findings

Successfully verified neural Lyapunov functions for high-dimensional systems.

Demonstrated performance advantages over SOS polynomial Lyapunov functions.

Validated approach using neural Lyapunov functions trained via Zubov's PDE.

Abstract

While there has been increasing interest in using neural networks to compute Lyapunov functions, verifying that these functions satisfy the Lyapunov conditions and certifying stability regions remain challenging due to the curse of dimensionality. In this paper, we demonstrate that by leveraging the compositional structure of interconnected nonlinear systems, it is possible to verify neural Lyapunov functions for high-dimensional systems beyond the capabilities of current satisfiability modulo theories (SMT) solvers using a monolithic approach. Our numerical examples employ neural Lyapunov functions trained by solving Zubov's partial differential equation (PDE), which characterizes the domain of attraction for individual subsystems. These examples show a performance advantage over sums-of-squares (SOS) polynomial Lyapunov functions derived from semidefinite programming.