Convergence Guarantees for Neural Network-Based Hamilton–Jacobi Reachability
Keywords: Nonlinear PDEs, physics-informed neural networks (PINNs), deep Galerkin method (DGM), optimal control, Hamilton-Jacobi reachability
TL;DR: We provide a novel uniform convergence guarantee demonstrating that a popular neural network-based algorithm can accurately solve first-order, nonlinear PDEs arising in optimal control theory.
Abstract: We provide a novel uniform convergence guarantee for DeepReach, a deep learning-based method for solving Hamilton–Jacobi–Isaacs (HJI) equations arising in 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 examining the impact that our technical modifications of the algorithm have on its empirical performance.
Submission Number: 4
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