  Perimeter Control (PC) strategies have been proposed to address urban road
network control in oversaturated situations by regulating the transfer flow of
the Protected Network (PN) based on the Macroscopic Fundamental Diagram (MFD).
The uniform metering rate for cordon signals in most existing studies overlooks
the variance of local traffic states at the intersection level, which may cause
severe local traffic congestion and degradation of the network stability. PC
strategies with heterogeneous metering rates for cordon signals allow precise
control for the perimeter but the complexity of the problem increases
exponentially with the scale of the PN. This paper leverages a Multi-Agent
Reinforcement Learning (MARL)-based traffic signal control framework to
decompose this PC problem, which considers heterogeneous metering rates for
cordon signals, into multi-agent cooperation tasks. Each agent controls an
individual signal located in the cordon, decreasing the dimension of action
space for the controller compared to centralized methods. A physics
regularization approach for the MARL framework is proposed to ensure the
distributed cordon signal controllers are aware of the global network state by
encoding MFD-based knowledge into the action-value functions of the local
agents. The proposed PC strategy is operated as a two-stage system, with a
feedback PC strategy detecting the overall traffic state within the PN and then
distributing local instructions to cordon signals controllers in the MARL
framework via the physics regularization. Through numerical tests with
different demand patterns in a microscopic traffic environment, the proposed PC
strategy shows promising robustness and transferability. It outperforms
state-of-the-art feedback PC strategies in increasing network throughput,
decreasing distributed delay for gate links, and reducing carbon emissions.
