Barrier Certificates for Unknown Systems with Latent States and Polynomial Dynamics using Bayesian Inference

Certifying safety in dynamical systems is crucial, but barrier certificates - widely used to verify that system trajectories remain within a safe region - typically require explicit system models. When dynamics are unknown, data-driven methods can be used instead, yet obtaining a valid certificate requires rigorous uncertainty quantification. For this purpose, existing methods usually rely on full-state measurements, limiting their applicability. This paper proposes a novel approach for synthesizing barrier certificates for unknown systems with latent states and polynomial dynamics. A Bayesian framework is employed, where a prior in state-space representation is updated using input-output data via a targeted marginal Metropolis-Hastings sampler. The resulting samples are used to construct a candidate barrier certificate through a sum-of-squares program. It is shown that if the candidate satisfies the required conditions on a test set of additional samples, it is also valid for the true, unknown system with high probability. The approach and its probabilistic guarantees are illustrated through a numerical simulation.

@article{lefringhausen2025_2504.01807,
  title={ Barrier Certificates for Unknown Systems with Latent States and Polynomial Dynamics using Bayesian Inference },
  author={ Robert Lefringhausen and Sami Leon Noel Aziz Hanna and Elias August and Sandra Hirche },
  journal={arXiv preprint arXiv:2504.01807},
  year={ 2025 }
}