Towards KANReach: Assessing the Feasibility of KANs for Solving Hamilton-Jacobi Reachability

Abstract

Hamilton-Jacobi (HJ) reachability is commonly used to determine Safe Flight Envelopes (SFEs), yet it is often limited by the "curse of dimensionality". While Deep Neural Network (DNN)-based solvers like DeepReach mitigate scaling issues, they remain computationally and memory-intensive.

This paper introduces KANReach, a novel solver that leverages Kolmogorov-Arnold Networks (KANs) with B-spline activation functions for grid-free reachable set estimation. The architecture is evaluated using three case studies: a first-order regulator, a double integrator, and a 3D Dubins car. Benchmarking against DeepReach reveals that at comparable model sizes, KANReach achieves superior accuracy via Value Function Regression (VFR). However, it currently faces difficulties in Physics-Informed Learning (PIL) that limit its training efficiency relative to traditional DNNs. Additionally, this paper proposes Absolute Maximum Error Bounding (AMEB), a technique that exploits the unique convex-hull property of B-splines to derive formal safety guarantees that verify the computed SFE.