Neural operators for boundary stabilization of stop-and-go traffic

This paper introduces a novel approach to PDE boundary control design using neural operators to alleviate stop-and-go traffic instabilities. Our framework leverages neural operators to design control strategies for traffic flow systems. The traffic dynamics are described by the Aw-Rascle- Zhang (ARZ) model, which consists of second-order coupled hyperbolic partial differential equations (PDEs). The backstepping method which involves constructing and solving a backstepping control kernel is widely used for boundary control of such PDE systems, but it requires intensive depth of expertise and can be time-consuming. To overcome these challenges, we present two distinct neural operator (NO) learning schemes aimed at stabilizing the traffic PDE system. The first scheme embeds NO-approximated gain kernels within a predefined backstepping controller, while the second one directly learns a boundary control law. The Lyapunov analysis is conducted to evaluate the stability of the NO-approximated gain kernels and control law. It is proved that the NO-based closed-loop system is practical stable under certain approximation accuracy conditions. To validate the efficacy of the proposed approach, simulations are conducted to compare the performance of the two neural operator controllers with a PDE backstepping controller and a Proportional Integral (PI) controller. While the NO-approximated methods exhibit larger errors compared to the backstepping controller, they consistently outperform the PI controller, demonstrating faster computation speeds across all scenarios. This result suggests that neural operators can significantly expedite and simplify the process of obtaining boundary controllers for freeway traffic stabilization systems.