# Safety verification

For monotone dynamical systems with unknown transition maps, safety verification previously relied on dense sampling or Lipschitz bounds, with sample counts growing exponentially in dimension. Max-Monotone Networks build monotonicity into the network structure, reducing global safety conditions to checks over finitely many boundary cells, so a zero training loss yields a formally valid barrier certificate. The method certifies a 9,000-state Kuramoto oscillator network and a 1,000-state freeway traffic network, where two baselines fail to converge or exceed memory limits.

For monotone dynamical systems with unknown transition maps, safety verification previously relied on dense sampling or Lipschitz bounds, with sample counts growing exponentially in dimension. Max-Monotone Networks build monotonicity into the network structure, reducing global safety conditions to checks over finitely many boundary cells, so a zero training loss yields a formally valid barrier certificate. The method certifies a 9,000-state Kuramoto oscillator network and a 1,000-state freeway traffic network, where two baselines fail to converge or exceed memory limits.