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Deep learning methods are emerging as popular computational tools for solvingforward and inverse problems in traffic flow. In this paper, we study a neuraloperator framework for learning solutions to nonlinear hyperbolic partialdifferential equations with applications in macroscopic traffic flow models. Inthis framework, an operator is trained to map heterogeneous and sparse trafficinput data to the complete macroscopic traffic state in a supervised learningsetting. We chose a physics-informed Fourier neural operator ($\pi$-FNO) as theoperator, where an additional physics loss based on a discrete conservation lawregularizes the problem during training to improve the shock predictions. Wealso propose to use training data generated from random piecewise constantinput data to systematically capture the shock and rarefied solutions. Fromexperiments using the LWR traffic flow model, we found superior accuracy inpredicting the density dynamics of a ring-road network and urban signalizedroad. We also found that the operator can be trained using simple trafficdensity dynamics, e.g., consisting of $2-3$ vehicle queues and $1-2$ trafficsignal cycles, and it can predict density dynamics for heterogeneous vehiclequeue distributions and multiple traffic signal cycles $(\geq 2)$ with anacceptable error. The extrapolation error grew sub-linearly with inputcomplexity for a proper choice of the model architecture and training data.Adding a physics regularizer aided in learning long-term traffic densitydynamics, especially for problems with periodic boundary data.

Fourier neural operator for learning solutions to macroscopic traffic flow models: Application to the forward and inverse problems