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.

The RoboCup 2D Simulation League incorporates several challenging features, setting a benchmark for Artificial Intelligence (AI). In this paper we describe some of the ideas and tools around the development of our team, Gliders2012. In our description, we focus on the evaluation function as one of our central mechanisms for action selection. We also point to a new framework for watching log files in a web browser that we release for use and further development by the RoboCup community. Finally, we also summarize results of the group and final matches we played during RoboCup 2012, with Gliders2012 finishing 4th out of 19 teams.