Monotone Neural Barrier Certificates

This report presents a neurosymbolic framework for safety verification and control synthesis in high-dimensional monotone dynamical systems without relying on 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 network 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.