k-Inductive Neural Barrier Certificates for Unknown Nonlinear Dynamics

2026-05-19

AI Summary

Traditional barrier certificates are overly conservative when system dynamics are partially or fully unknown, as they require the certificate function to be non-increasing at every time step, thereby compromising the trade-off between safety and flexibility. This work proposes k-inductive neural barrier certificates (k-NBCs), which permit the certificate function to increase over up to \(k-1\) consecutive steps before decreasing. For the first time, the k-inductive barrier concept is integrated with neural networks, leveraging a data-driven formulation of Willems’ lemma to construct a CEGIS-SMT verification framework applicable to unknown nonlinear systems. The approach eliminates reliance on exact system models or restrictive function classes, successfully generating barrier certificates that simultaneously ensure formal safety guarantees and operational flexibility in three nonlinear systems with partially unknown dynamics.

Abstract

While conventional (k=1) discrete-time barrier certificate conditions impose strict safety constraints by requiring the function to be non-increasing at every step, k-inductive barrier certificates relax this by allowing a temporary increase -- up to k-1 times, each within a threshold $ε$ -- while maintaining overall safety, and improving flexibility. This paper leverages neural networks and constructs k-inductive neural barrier certificates (k-NBCs) for (partially) unknown nonlinear systems. While neural networks offer scalability in the design process, they lack formal guarantees, requiring additional approaches such as counterexample-guided inductive synthesis (CEGIS) with satisfiability modulo theories (SMT) for verification. However, the CEGIS-SMT framework requires knowledge of system dynamics, which is unavailable in practical settings. To address this, we leverage the generalization of the Willems et al.'s fundamental lemma, using a single state trajectory, to construct a data-driven representation of (partially) unknown models for SMT verification without sacrificing accuracy. Additionally, CEGIS-SMT further removes the constraint of restricting barrier certificates to specific function classes, such as sum-of-squares, enabling greater flexibility in their design. We validate our approach on three nonlinear case studies with (partially) unknown dynamics.

Problem

• k-inductive barrier certificates
• unknown nonlinear dynamics
• neural barrier certificates
• safety verification
• data-driven verification

Innovation

• k-inductive barrier certificates
• neural barrier certificates
• data-driven verification
• CEGIS-SMT
• unknown nonlinear dynamics