Safety Verification of Nonlinear Systems with Bayesian Neural Network Controllers

Bayesian neural networks (BNNs) retain NN structures with a probability distribution placed over their weights. With the introduced uncertainties and redundancies, BNNs are proper choices of robust controllers for safety-critical control systems. This paper considers the problem of verifying the safety of nonlinear closed-loop systems with BNN controllers over unbounded-time horizon. In essence, we compute a safe weight set such that as long as the BNN controller is always applied with weights sampled from the safe weight set, the controlled system is guaranteed to be safe. We propose a novel two-phase method for the safe weight set computation. First, we construct a reference safe control set that constraints the control inputs, through polynomial approximation to the BNN controller followed by polynomial-optimization-based barrier certificate generation. Then, the computation of safe weight set is reduced to a range inclusion problem of the BNN on the system domain w.r.t. the safe control set, which can be solved incrementally and the set of safe weights can be extracted. Compared with the existing method based on invariant learning and mixed-integer linear programming, we could compute safe weight sets with larger radii on a series of linear benchmarks. Moreover, experiments on a series of widely used nonlinear control tasks show that our method can synthesize large safe weight sets with probability measure as high as 95% even for a large-scale system of dimension 7.