Set-Based Training of Neural Barrier Certificates for Safety Verification of Dynamical Systems

2026-05-04

AI Summary

This work addresses the inefficiency in synthesizing neural barrier certificates for dynamical system safety verification and the disconnect between training and formal verification. To bridge this gap, the authors propose a set-based training approach that encodes the formal verification conditions required for valid barrier certificates into a differentiable set loss function, wherein zero loss directly corresponds to a provably valid certificate. This formulation tightly couples neural network training with formal verification, thereby eliminating the need for traditional iterative refinement loops. Experimental results demonstrate that the proposed method scales effectively to high-dimensional and complex nonlinear dynamical systems while maintaining strong verification guarantees.

Abstract

Barrier certificates are scalar functions over the state space of dynamical systems that separate all unsafe states from all reachable states. The existence of a barrier certificate formally verifies the safety of the dynamical system. Recent approaches synthesize barrier certificates by iteratively training a neural network. In each iteration, the candidate is formally verified - if successful, the barrier certificate is found. Instead, we propose a set-based training approach that tightly integrates verification into training via a set-based loss function that soundly encodes all barrier certificate properties. A loss of zero formally proves the validity of the barrier certificate, collapsing the iterative training and verification into a single training procedure. Our experiments demonstrate that our set-based training approach scales well with the system dimension and naturally handles complex nonlinear dynamics.

Problem

• barrier certificates
• safety verification
• dynamical systems
• neural networks
• set-based training

Innovation

• set-based training
• neural barrier certificates
• safety verification
• formal verification
• dynamical systems