Neural operators for adaptive control of freeway traffic

The uncertainty in human driving behaviors leads to stop-and-go traffic congestion on freeway. The freeway traffic dynamics are governed by the Aw–Rascle–Zhang (ARZ) traffic Partial Differential Equation (PDE) models with unknown relaxation time. Motivated by the adaptive traffic control problem, this paper presents a neural operator (NO) based adaptive boundary control design for the coupled 2 × 2 hyperbolic systems with uncertain spatially varying in-domain coefficients and boundary parameter. In traditional adaptive control for PDEs, solving backstepping kernel online can be computationally intensive, as it updates the estimation of coefficients at each time step. To address this challenge, we use operator learning, i.e. DeepONet, to learn the mapping from system parameters to the kernels functions. DeepONet, a class of deep neural networks designed for approximating operators, has shown strong potential for approximating PDE backstepping designs in recent studies. Unlike previous works that focus on approximating single kernel equation associated with the scalar PDE system, we extend this framework to approximate PDE kernels for a class of the first-order coupled 2 × 2 hyperbolic kernel equations. Our approach demonstrates that DeepONet is nearly two orders of magnitude faster than traditional PDE solvers for generating kernel functions, while maintaining a loss on the order of 10⁻³. In addition, we rigorously establish the system's stability via Lyapunov analysis when employing DeepONet-approximated kernels in the adaptive controller. The proposed adaptive control is compared with reinforcement learning (RL) methods. Our approach guarantees stability and does not rely on initial values, which is essential for rapidly changing traffic scenarios. This is the first time this operator learning framework has been applied to the adaptive control of the ARZ traffic model, significantly enhancing the real-time applicability of this design framework for mitigating traffic congestion.