HJ-Gauss: A Monte-Carlo HJ Reachability Scheme

TL;DR

The paper introduces HJ-Gauss, a scalable Monte Carlo method for high-dimensional reachability analysis using linearized PDEs and Gaussian heat-kernel representations, overcoming grid-based limitations.

Contribution

It proposes a novel grid-free, memory-efficient Monte Carlo scheme for solving viscous Hamilton-Jacobi PDEs in high dimensions, enabling practical reachability analysis.

Findings

Achieves low relative L2 errors of 0.03-0.20 in pursuit-evasion games.

Scales to problems with up to 45 dimensions.

Runs in 14-26 seconds per 2D slice on CPU.

Abstract

Backward reachable tubes (BRTs), computed via viscous Hamilton-Jacobi (HJ) partial differential equations, provide principled safety certificates for learned controllers and planning algorithms in trustworthy machine learning. However, classical grid-based HJ solvers require $O(M^n)$ memory footprint for $M$ grid points per $n$ state dimension. This renders them impractical for high-dimensional systems. We address this bottleneck with a local PDE linearization that enables a frozen-coefficient sampling scheme for the viscous HJ PDE: a generalized Cole-Hopf-type transformation reduces the nonlinear HJ equation to a sequence of linear heat equations whose solutions admit Gaussian heat-kernel representations. The value function and its spatial gradient are then recovered via roll-outs of Monte Carlo expectations on Gaussian densities, yielding a storage and grid-free algorithm that scales as $N\cdot n$ for $N$ samples. This decoupling of memory from dimensionality enables reachability analysis on problems where grid-based methods are simply impossible. We prove a finite-sample concentration bound $O(N^{-1/2})$ error and conditional linear convergence for the introduced Monte-Carlo Picard iterative scheme. Numerical validation on pursuit-evasion games demonstrates relative $L^2_{\text{rel}}$ errors of $0.03 - 0.20$, with $14-26$ second wall-clock times per 2D slice on a CPU. Crucially, the method scales with validation on up to (but not limited to) $n=45$-dimensional multi-agent games.