k-Inductive Neural Barrier Certificates for Unknown Nonlinear Dynamics

Problem
This paper addresses the limitations of conventional discrete-time barrier certificates, specifically the k=1 case, which imposes stringent safety constraints by requiring non-increasing behavior at every time step. The authors propose a novel framework for k-inductive barrier certificates (k-NBCs) that allows for temporary increases in the barrier function, up to k-1 times, within a specified threshold ε. This work is particularly relevant for systems with unknown nonlinear dynamics, where traditional methods struggle to provide formal guarantees. The paper is a preprint and has not yet undergone peer review.

Method
The core technical contribution is the development of k-inductive neural barrier certificates (k-NBCs) that utilize neural networks to model (partially) unknown nonlinear systems. The authors employ a counterexample-guided inductive synthesis (CEGIS) approach combined with satisfiability modulo theories (SMT) for verification. A key innovation is the application of Willems et al.’s fundamental lemma, which allows the construction of a data-driven representation of system dynamics using a single state trajectory. This representation facilitates SMT verification without requiring complete knowledge of the system dynamics. The framework also relaxes the constraints on the types of functions used for barrier certificates, moving beyond traditional sum-of-squares constraints, thus enhancing design flexibility.

Results
The authors validate their k-NBC approach through three nonlinear case studies involving (partially) unknown dynamics. While specific numerical results are not detailed in the abstract, the paper reports significant improvements in flexibility and safety guarantees compared to traditional barrier certificate methods. The effectiveness of the k-NBCs is demonstrated through comparative analysis against baseline methods, although exact performance metrics (e.g., success rates, computational efficiency) are not explicitly provided in the abstract.

Limitations
The authors acknowledge that the reliance on a single state trajectory for system representation may limit the robustness of the approach in highly dynamic environments. Additionally, the CEGIS-SMT framework, while flexible, may still face challenges in scalability and computational efficiency, particularly for high-dimensional systems. The paper does not address potential issues related to the generalization of neural networks in safety-critical applications, nor does it explore the implications of model inaccuracies on the verification process.

Why it matters
This work has significant implications for the design of safety-critical systems, particularly in robotics and autonomous vehicles, where understanding and ensuring safety in the presence of unknown dynamics is crucial. The introduction of k-inductive barrier certificates enhances the flexibility of safety verification methods, potentially leading to more robust and adaptable control strategies. By leveraging neural networks for dynamic modeling and verification, this research paves the way for future advancements in safe reinforcement learning and adaptive control systems.

Authors: Ben Wooding, Hongchao Zhang, Taylor T. Johnson, Abolfazl Lavaei
Source: arXiv:2605.20108
URL: https://arxiv.org/abs/2605.20108v1