Neural network solution of stochastic Pontryagin maximum principle for urban road network perimeter control

In contemporary perimeter control strategies, reliance on simplified accumulation-based macroscopic fundamental diagram (MFD) models with minimal scatter in macroscopic variables is common. Yet, the inherent state-dependent stochasticity driving road network evolution challenges the efficacy of such deterministic approaches, potentially undermining desired objectives when supply and demand variabilities are ignored. To fill the gap, our study integrates an analytical state-dependent stochastic dynamics model into the MFD framework, incorporating Brownian noise to capture the stochastic essence of traffic evolution. This advancement enables the modeling of stochastic gridlock processes. We formulate a stochastic optimal control problem for perimeter regulation based on this novel MFD model, targeting both delay minimization and gridlock prevention. Utilizing the stochastic Pontryagin maximum principle (SPMP), we express the optimal control conditions as a coupled forward-backward stochastic differential equation (FBSDE). To solve this FBSDE, we reformulate it as an optimization problem and develop neural network-assisted solution methodologies. Our numerical studies validate the convergence of the proposed solution method and demonstrate significant performance enhancements: not only can gridlocks be effectively mitigated by accounting for the model’s stochastic nature and associated objectives, but also the total cumulative trip completion rate improves by up to 25% when optimizing for both delay reduction and gridlock avoidance. These findings underscore the potential of integrating stochastic dynamics into perimeter control strategies for more resilient and efficient traffic management systems.