Convergence Guarantees for Neural Network-Based Hamilton-Jacobi Reachability

TL;DR

This paper establishes a uniform convergence guarantee for DeepReach, a deep learning method for solving Hamilton-Jacobi-Isaacs equations in reachability analysis, ensuring neural network approximations converge to classical solutions under certain conditions.

Contribution

It provides the first theoretical uniform convergence guarantee for DeepReach, linking loss functional convergence to neural network approximation accuracy.

Findings

DeepReach is stable if the loss functional converges to zero.

Numerical experiments replicate original results and analyze the impact of supremum norm loss.

Empirical evidence supports the theoretical convergence guarantees.

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.