Neural reachability now errs on the safe side: ConDR pushes the learned value below the true one and gives the lowest closed-loop collision rates
Synopsis
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.
Interpretation
Neural reachability can now err in only one direction: the learned value function is pushed below the true one, so the learned unsafe set contains the true set and the risk of calling an unsafe state safe is systematically reduced. DeepReach and its ExactBC variant penalize positive and negative HJI residuals equally; a positive residual puts the learned value above the true one and misclassifies unsafe states as safe, while a negative residual only enlarges the unsafe set. On Air3D and Air6D, ConDR satisfies the subsolution property at 96% and 99% of states, whereas the baselines show the wrong sign roughly half the time; ConDR has zero false negatives on four of five Air3D seeds, with one seed as an outlier.
The mechanism writes a safety margin into the training target: the network output is constructed to stay at or below the obstacle-branch value, and a one-sided loss pushes the differential-branch residual below a negative margin. A one-sided loss alone only penalizes residuals above the margin and drifts toward the degenerate solution of declaring every state unsafe, so a complementarity term requires at least one branch to be tight, approaching the largest subsolution of the margin-shifted variational inequality. In the loss ablation, removing the complementarity term makes the model converge to classifying everything as unsafe and drops IoU to 0.086; adding the obstacle-branch construction raises subsolution satisfaction from 66% to 91% and halves collisions.
A trained network only satisfies the residual constraint approximately, so a residual bound from finite samples shifts the learned level set to give an outer set that contains the true reachable tube. For the 9D and 13D systems without ground truth, prior methods could already bound training error, but those bounds inflate the level set by the worst violation and leave the training objective that produced the violation unchanged; here the same kind of bound is applied to the variational-inequality residual and paired with subsolution training to give the shifted tube. On the 9D and 13D systems, ConDR violates the subsolution property on 1% to 3% of sampled states against 25% to 52% for DeepReach and MADR, and its shifted tube volume is the smallest on every system.
Perspective
The demonstrated scope is four benchmark systems: Air3D, Air6D, a 9D multi-vehicle system and a 13D quadrotor, all with a common time horizon, with closed-loop experiments using the learned value function as a least-restrictive safety filter; the 9D and 13D systems have no ground truth, so their outer sets rest on a finite-sample residual bound and are probabilistic.
The theorems concern the exact solution of the margin-shifted variational inequality, while the trained network satisfies the residual only approximately and the shift bound comes from finite samples, not a global certificate; once model mismatch exceeds the safety margin, collision rates degrade and IoU falls roughly linearly with the mismatch.