Taming the Learning Dynamics of Neural Barriers Using Projected Anchor Sets

Learning neural control barrier functions (CBFs) requires shaping safety boundaries of complex high-dimensional geometry while finding safety-preserving controls along them. Direct training of neural network representations couples neural network training, worst-case state search, and control selection in a difficult min–max–min optimization problem that can destabilize learning. We propose new approaches for taming the learning dynamics of neural barriers, grounded in Nagumo’s theorem that characterizes forward invariance through boundary conditions. Our formulation exploits a common structure in safety problems: violations are observable in a low-dimensional space even when the system state is high-dimensional. Our method, projected anchor sets, uses these observations to shape a barrier in the full state space. Multistep recovery searches backtrack along failed rollouts to states where changing the actions can still avert failure. These states anchor the candidate zero level set, while their recovery trajectories supervise both barrier fitting and policy fine-tuning. The resulting loop learns from its own safety feedback, without expert safety demonstrations or a separate task reward. We connect the formulation to high-order CBFs and specify conditions under which the learned boundary and policy ensure forward invariance.