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.

翻译：为自主系统提供形式化的安全与性能保证正变得日益重要。Hamilton-Jacobi (HJ) 可达性分析是一种流行的形式化验证工具，可提供此类保证，因为它能处理一般非线性系统动力学、有界对抗性系统扰动以及状态和输入约束。然而，该方法涉及求解偏微分方程 (PDE)，其计算和内存复杂度随状态维度呈指数增长，从而使得在大规模系统上直接使用变得困难。一种称为DeepReach的新方法通过利用正弦神经PDE求解器来解决高维可达性问题，其计算需求取决于底层可达管道的复杂度，而非状态空间维度。不幸的是，神经网络可能产生误差，因此计算出的解可能不安全，这未能实现我们提供形式化安全保证的首要目标。本文提出一种方法来计算DeepReach解的误差界。该误差界可用于对可达管道进行修正，从而获得真实可达管道的安全近似。我们还提出一种基于场景的优化方法，用于计算一般非线性动力学系统下该误差修正的概率界。我们通过高维火箭着陆和多车避碰问题，证明了所提方法在获得概率安全可达管道方面的有效性。