Post by Lekan Molu

🔐 Can Hamilton–Jacobi reachability escape the curse of dimensionality? Introducing HJ-Gauss Classical Hamilton–Jacobi (HJ) safety verification computes backward reachable tubes (BRTs) by solving nonlinear PDEs on state-space grids. Unfortunately, standard level-set methods require memory scaling as $O(M^n)$, where $M$ is the number of grid points per dimension and $n$ is the state dimension. In our latest work, we show that a viscous Hamilton–Jacobi equation admits a Cole–Hopf-inspired quasilinearization, yielding a sequence of linear heat equations through frozen-coefficient Picard iteration. Each Picard iterate admits a Gaussian heat-kernel representation via the Feynman–Kac formula, allowing both the value function and its spatial derivatives to be recovered through Monte Carlo expectations. The result is a grid-free, memory-scalable framework whose memory requirements scale linearly with state dimension and Monte Carlo sample count, replacing combinatorial grid explosion with controllable sampling error. The trade: exchange exponential state-space discretization for Monte Carlo estimation. The payoff: scalable HJ reachability with provable $O(N^{-1/2})$ concentration guarantees and local linear convergence under Picard contractivity assumptions. As a stress test, we analyzed safety geometry in flocking systems containing up to 100,000 simulated European starlings (Sturnus vulgaris), modeled as interacting 4D Dubins vehicles under a modified Vicsek flocking model with multi-predator pursuit-evasion dynamics. The resulting safety landscapes captured defensive cordon formation, topological collapse, flock fragmentation, and flash expansion as emergent geometric phenomena. We additionally benchmarked against classical level-set methods on multi-agent pursuit-evasion games up to 45 dimensions, obtaining relative $L^2$ errors between 0.03–0.20, substantially below the theoretical viscosity-regularization accuracy bound $O(\sqrt{\delta}) \approx 0.28$, with wall-clock times of 14–26 seconds per 2D slice on CPU. The broader question remains compelling: Can stochastic representations and kernel methods provide a viable path toward scalable safety verification beyond the traditional grid paradigm? 📄 Paper: https://lnkd.in/gX-VYsDt Collaborators: 🏛️ Cranfield University & Seoul National University 🌐