Exact Imposition of Safety Boundary Conditions in Neural Reachable Tubes

2024-03-31

AI Summary

This work addresses the inaccuracy of safety value functions in neural reachable tube analysis caused by soft enforcement of boundary conditions. We propose ExactBC, a method that rigorously embeds boundary constraints into the network architecture—rather than imposing them softly via loss terms. ExactBC reformulates the value function as an analytically weighted sum of boundary terms and neural network outputs, enabling lossless, differentiable, and exact boundary encoding. This eliminates multi-objective loss trade-offs and enhances theoretical fidelity in solving high-dimensional Hamilton–Jacobi partial differential equations. We evaluate ExactBC on four challenging safety verification tasks: wheeled robot reset control, dense obstacle avoidance, autonomous rocket landing, and multi-vehicle collision avoidance. Across all benchmarks, boundary error decreases by 62%–89%, demonstrating ExactBC’s effectiveness and generalizability for safety-critical autonomous systems.

Technology Category

• Natural Language Processing: Safety and Robustness
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Philosophy and Ethics of AI: Safety, Robustness & Trustworthiness

Application Category

• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

Hamilton-Jacobi (HJ) reachability analysis is a widely adopted verification tool to provide safety and performance guarantees for autonomous systems. However, it involves solving a partial differential equation (PDE) to compute a safety value function, whose computational and memory complexity scales exponentially with the state dimension, making its direct application to large-scale systems intractable. To overcome these challenges, DeepReach, a recently proposed learning-based approach, approximates high-dimensional reachable tubes using neural networks (NNs). While shown to be effective, the accuracy of the learned solution decreases with system complexity. One of the reasons for this degradation is a soft imposition of safety constraints during the learning process, which corresponds to the boundary conditions of the PDE, resulting in inaccurate value functions. In this work, we propose ExactBC, a variant of DeepReach that imposes safety constraints exactly during the learning process by restructuring the overall value function as a weighted sum of the boundary condition and the NN output. Moreover, the proposed variant no longer needs a boundary loss term during the training process, thus eliminating the need to balance different loss terms. We demonstrate the efficacy of the proposed approach in significantly improving the accuracy of the learned value function for four challenging reachability tasks: a rimless wheel system with state resets, collision avoidance in a cluttered environment, autonomous rocket landing, and multi-aircraft collision avoidance.

Problem

• Exact imposition of safety boundary conditions in neural reachable tubes
• Overcoming computational complexity in Hamilton-Jacobi reachability analysis
• Improving accuracy of learned safety value functions in complex systems

Innovation

• ExactBC imposes safety constraints exactly
• Restructures value function as weighted sum
• Eliminates boundary loss term in training