# k-Inductive Neural Barrier Certificates for Unknown Nonlinear Dynamics

arXiv: https://arxiv.org/abs/2605.20108 · HTML: https://mlsanity.com/paper/2605.20108/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 17 of its 39 references that are in the map.
- First posted on arXiv 19 May 2026.

## Builds on (most cited first, 17 of 17)

- [Neurosymbolic AI: The 3rd Wave](https://mlsanity.com/paper/2012.05876/) (2020)
- [Safe Control with Learned Certificates: A Survey of Neural Lyapunov, Barrier, and Contraction methods](https://mlsanity.com/paper/2202.11762/) (2022)
- [Learning Safe Multi-Agent Control with Decentralized Neural Barrier Certificates](https://mlsanity.com/paper/2101.05436/) (2021)
- [Formulas for Data-driven Control: Stabilization, Optimality and Robustness](https://mlsanity.com/paper/1903.06842/) (2019)
- [Safety Certification for Stochastic Systems via Neural Barrier Functions](https://mlsanity.com/paper/2206.01463/) (2022)
- [Fossil 2.0: Formal Certificate Synthesis for the Verification and Control of Dynamical Models](https://mlsanity.com/paper/2311.09793/) (2023)
- [Data-driven verification and synthesis of stochastic systems via barrier certificates](https://mlsanity.com/paper/2111.10330/) (2021)
- [Data-Driven Abstraction-Based Control Synthesis](https://mlsanity.com/paper/2206.08069/) (2022)
- [SEEV: Synthesis with Efficient Exact Verification for ReLU Neural Barrier Functions](https://mlsanity.com/paper/2410.20326/) (2024)
- [Data-driven stabilization of nonlinear polynomial systems with noisy data](https://mlsanity.com/paper/2011.07833/) (2020)
- [Learning Controllers from Data via Approximate Nonlinearity Cancellation](https://mlsanity.com/paper/2201.10232/) (2022)
- [PRoTECT: Parallelized Construction of Safety Barrier Certificates for Nonlinear Polynomial Systems](https://mlsanity.com/paper/2404.14804/) (2024)
- [Fundamental Lemma for Data-Driven Analysis of Linear Parameter-Varying Systems](https://mlsanity.com/paper/2103.16171/) (2021)
- [Learning k-Inductive Control Barrier Certificates for Unknown Nonlinear Dynamics Beyond Polynomials](https://mlsanity.com/paper/2412.07232/) (2024)
- [TRUST: Stability and Safety Controller Synthesis for Unknown Dynamical Models Using a Single Trajectory](https://mlsanity.com/paper/2503.08081/) (2025)
- [From a Single Trajectory to Safety Controller Synthesis of Discrete-Time Nonlinear Polynomial Systems](https://mlsanity.com/paper/2409.10026/) (2024)
- [Certifying Lyapunov Stability of Black-Box Nonlinear Systems via Counterexample Guided Synthesis (Extended Version)](https://mlsanity.com/paper/2503.00431/) (2025)

## Abstract (arXiv)

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.