Hamilton-Jacobi (HJ) reachability provides a principled framework for synthesizing safety certificates and robust controllers for safety-critical robotic systems. However, applying reachability analysis to high-dimensional nonlinear systems remains challenging: classical grid-based solvers suffer from the curse of dimensionality, continuous-time neural solvers require accurate spatial value gradients, and reinforcement-learning-based approaches often suffer from weak boundary anchoring and non-stationary adversarial policy optimization. We propose a discrete-time neural reachability framework for control-disturbance-affine systems that learns backward reachable tubes (BRTs) and backward reach-avoid tubes (BRATs) through Bellman-Isaacs value propagation. Our key idea is to combine equation-driven self-supervision with structured policy learning: rather than computing explicit PDE-gradients, we exploit the bang-bang structure of optimal safety interventions to construct approximate teacher actions from gradient-free value probes, converting adversarial actor learning into supervised policy learning. To stabilize long-horizon value propagation, we leverage the learned actor to train the value function backward from the terminal boundary using a windowed temporal curriculum, where each window is used as the boundary condition for the next window. Across benchmark problems up to 80 dimensions, our method learns accurate reachability value functions while improving stability over existing learning-based solvers. We further demonstrate observation-space scalability on F1-tenth racing with over 16,000-dimensional egocentric inputs. The learned safety filter generalizes zero-shot to unseen tracks and transfers to a physical RC car, achieving real-time robust collision avoidance.

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