Title: Convergence Guarantees for Neural Network-Based Hamilton-Jacobi Reachability Show Chinese title

Title: 基于神经网络的哈密顿-雅可比可达性收敛性保证

Abstract: We provide a novel uniform convergence guarantee for DeepReach, a deep learning-based method for solving Hamilton-Jacobi-Isaacs (HJI) equations associated with reachability analysis. Specifically, we show that the DeepReach algorithm, as introduced by Bansal et al. in their eponymous paper from 2020, is stable in the sense that if the loss functional for the algorithm converges to zero, then the resulting neural network approximation converges uniformly to the classical solution of the HJI equation, assuming that a classical solution exists. We also provide numerical tests of the algorithm, replicating the experiments provided in the original DeepReach paper and empirically examining the impact that training with a supremum norm loss metric has on approximation error. Show Chinese abstract

Abstract: 我们为基于深度学习的DeepReach方法提供了新的均匀收敛保证，该方法用于求解与可达性分析相关的Hamilton-Jacobi-Isaacs（HJI）方程。 具体而言，我们证明了Bansal等人在2020年发表的同名论文中引入的DeepReach算法具有稳定性：如果算法的损失函数收敛到零，则在经典解存在的前提下，由此得到的神经网络逼近将一致收敛到HJI方程的经典解。 我们还提供了算法的数值测试，复制了原始DeepReach论文中的实验，并通过经验研究了使用上确界范数损失度量训练对逼近误差的影响。