INTERPRETABLE AND EFFICIENT PATH PLANNING PROBLEMS IN THE HAMILTON JACOBI FORMULATION

Abstract

With the recent increasing integration of artificial intelligence (AI) tools into everyday life, it is more important now than ever to set precedents for interpretable results from the models and technology we use, especially for decision-critical problems such as autonomous vehicle navigation. While approaches like deep learning are popular and powerful for vehicle control methods, they still lack interpretability. To address these shortcomings, we propose novel algorithms for multi-agent path planning problems, addressing critical issues of computational efficiency and interpretability. Classical numerical methods for these Partial Differential Equations (PDEs) fall short in these high- dimensional state spaces due to the "curse of dimensionality". To overcome this, we leverage advanced numerical techniques based on generalized Hopf-Lax type formulas, offering scalable, grid-free solutions for complex scenarios. We specifically offer interpretable and scalable solutions for three distinct scenarios, all in the Hamilton-Jacobi formulation: (1) multi-agent path planning with obstacle avoidance, (2) using the linear bottleneck assignment problem (LBAP) for multi-agent path planning, and (3) multi-agent path planning emphasizing geometric pattern formation amidst obstacles. Comprehensive numerical simulations verify the algorithms' effectiveness across many scenarios, ranging from simple obstacle avoidance to dynamically evolving environments. These results demonstrate the potential that PDE-based approaches have to bring interpretable and computationally feasible solutions to path planning problems in multi-agent systems where machine-learning-based techniques may fall short.