  Providing formal safety and performance guarantees for autonomous systems is
becoming increasingly important. Hamilton-Jacobi (HJ) reachability analysis is
a popular formal verification tool for providing these guarantees, since it can
handle general nonlinear system dynamics, bounded adversarial system
disturbances, and state and input constraints. However, it involves solving a
PDE, whose computational and memory complexity scales exponentially with
respect to the state dimensionality, making its direct use on large-scale
systems intractable. A recently proposed method called DeepReach overcomes this
challenge by leveraging a sinusoidal neural PDE solver for high-dimensional
reachability problems, whose computational requirements scale with the
complexity of the underlying reachable tube rather than the state space
dimension. Unfortunately, neural networks can make errors and thus the computed
solution may not be safe, which falls short of achieving our overarching goal
to provide formal safety assurances. In this work, we propose a method to
compute an error bound for the DeepReach solution. This error bound can then be
used for reachable tube correction, resulting in a safe approximation of the
true reachable tube. We also propose a scenario-based optimization approach to
compute a probabilistic bound on this error correction for general nonlinear
dynamical systems. We demonstrate the efficacy of the proposed approach in
obtaining probabilistically safe reachable tubes for high-dimensional
rocket-landing and multi-vehicle collision-avoidance problems.
