Reachability-Robust Neural Safety Loss

Integrate this at the safety-loss stage of a trajectory-predicting world model, model-predictive policy, or RL actor. Let the neural policy output action u_t=\pi_ heta(x_t), and let a stochastic dynamics model produce K rollouts by independently sampling ensemble members, dropout masks, or disturbance vectors w_t^{(k)}. For each rollout, recursively compute x_{t+1}^{(k)}=f_\phi(x_t^{(k)},u_t^{(k)},w_t^{(k)}) and construct the specification state \delta_t^{(k)}, such as relative position and velocity. Evaluate r_t^{(k)}=a^ op\delta_t^{(k)}-b for every horizon step. Compute the soft upper and lower extrema with the log-sum-exp formulas, add \mathcal{L}_{\mathrm{reach}} to the task loss, and backpropagate through the rollouts. Use temperature au=0.2 initially and anneal to 0.02; clip gradients if long-horizon rollout gradients become unstable. At inference, also compute exact sample min and max for diagnostics. The paper supplies the min/max-over-reachable-set principle; the engineer estimates the reachable tube empirically from the stochastic rollout generator. First test on a 2-D point-mass collision-avoidance benchmark with a 2-layer MLP policy, comparing ordinary mean-trajectory collision loss against this loss at equal rollout count and equal training steps. Measure collision rate under held-out disturbances, goal-reaching success, empirical coverage of the predicted safety interval, conservatism, and training FLOPs. The method succeeds if it reduces collision rate at matched task reward without more than 2x training cost.