Forward Trajectory Steering for Hamilton-Jacobi Reachability Analysis

2026-08-11

AI Summary

High-dimensional Hamilton-Jacobi (HJ) reachability analysis remains impractical due to the computational complexity of solving the associated HJ-I variational inequality PDE. Existing physics-informed neural network (PINN) approaches are sensitive to sampling and often rely on multi-stage training or model predictive control (MPC) supervision. This work proposes STEER2REACH (S2R), a lightweight PINN framework that constructs an adaptive collocation sampling distribution by steering forward trajectories with optimal controls derived from the current value function, augmented with perturbation signals and injected random exploration noise. Requiring only minor modifications to standard PINN training, S2R eliminates the need for auxiliary supervision or multi-stage procedures. Across multiple benchmarks, S2R matches or exceeds state-of-the-art MPC-guided methods in safety-critical metrics while significantly reducing relative L² error.

Abstract

Hamilton-Jacobi (HJ) reachability provides a mathematically rigorous framework for safe control of dynamical systems, but its practical application is bottlenecked by the computational complexity of solving Hamilton-Jacobi-Isaacs variational inequality PDEs in high dimensions. Physics-informed neural networks (PINNs) have recently emerged as a promising alternative to classical mesh-based solvers, yet their performance is highly sensitive to the choice of collocation sampling. In order to learn accurate safety value functions, existing PINNs-based HJ reachability solvers must rely on complex training pipelines and auxiliary supervision. In this work, we propose STEER2REACH (S2R), a PINNs-based HJ reachability solver that requires minimal modification on top of standard PINNs training. S2R's key contribution is a lightweight, low-overhead adaptive collocation sampling distribution constructed by steering forward trajectories using a combination of the optimal control and disturbance signals induced by the current value function, with injected stochastic exploration noise. We demonstrate that despite its simplicity, S2R achieves competitive--and in some cases improved--performance on safety metrics while reducing relative L2 error across a range of reachability benchmarks compared with SoTA MPC-guided HJ reachability solvers, all without requiring multi-stage training or MPC-based supervision.

Problem

• Hamilton-Jacobi reachability
• high-dimensional PDEs
• physics-informed neural networks
• collocation sampling
• safety value function

Innovation

• adaptive collocation sampling
• physics-informed neural networks
• Hamilton-Jacobi reachability
• forward trajectory steering
• safety value function