# ConDR

DeepReach replaces the grid in Hamilton–Jacobi reachability with a neural value function but penalizes both sides of the residual equally, so the learned value can sit above the true one and unsafe states get labeled safe. ConDR instead trains toward a subsolution of the HJI variational inequality: the obstacle branch and terminal condition hold by construction, a one-sided loss pushes the differential residual below a safety margin, and a complementarity term blocks the degenerate all-unsafe solution. On Air3D, Air6D, a 9D multi-vehicle system and a 13D quadrotor, ConDR reports the lowest false-negative rate and the lowest closed-loop collision rates, with at most a small rise in intervention.

DeepReach replaces the grid in Hamilton–Jacobi reachability with a neural value function but penalizes both sides of the residual equally, so the learned value can sit above the true one and unsafe states get labeled safe. ConDR instead trains toward a subsolution of the HJI variational inequality: the obstacle branch and terminal condition hold by construction, a one-sided loss pushes the differential residual below a safety margin, and a complementarity term blocks the degenerate all-unsafe solution. On Air3D, Air6D, a 9D multi-vehicle system and a 13D quadrotor, ConDR reports the lowest false-negative rate and the lowest closed-loop collision rates, with at most a small rise in intervention.