Monotone Neural Control Barrier Certificates

This work presents a neurosymbolic framework for synthesizing and verifying safety controllers in high-dimensional monotone dynamical systems using only linear sample complexity, without requiring explicit models or conservative Lipschitz bounds. The approach combines the expressiveness of neural networks with the rigor of symbolic reasoning via barrier certificates, functional analogs of inductive invariants that formally guarantee safety. Prior data-driven methods often treat dynamics as black-box models, relying on dense state-space discretization or Lipschitz overapproximations, leading to exponential sample complexity. In contrast, monotonicity -- a pervasive structural property in many real-world systems -- provides a symbolic scaffold that simplifies both learning and verification. Exploiting order preservation reduces verification to localized boundary checks, transforming a high-dimensional problem into a tractable, low-dimensional one. Barrier certificates are synthesized using monotone neural networks -- architectures with embedded monotonicity constraints -- trained via gradient-based optimization guided by barrier conditions. This enables scalable, formally sound verification directly from simulation data, bridging black-box learning and formal guarantees within a unified neurosymbolic framework. Empirical results on three large-scale benchmarks -- a 1,000-dimensional freeway traffic model, a 50-dimensional urban traffic network, and a 13,000-dimensional power grid -- demonstrate the scalability and effectiveness of the approach in real-world, safety-critical systems.