  Deep learning methods are emerging as popular computational tools for solving
forward and inverse problems in traffic flow. In this paper, we study a neural
operator framework for learning solutions to nonlinear hyperbolic partial
differential equations with applications in macroscopic traffic flow models. In
this framework, an operator is trained to map heterogeneous and sparse traffic
input data to the complete macroscopic traffic state in a supervised learning
setting. We chose a physics-informed Fourier neural operator ($\pi$-FNO) as the
operator, where an additional physics loss based on a discrete conservation law
regularizes the problem during training to improve the shock predictions. We
also propose to use training data generated from random piecewise constant
input data to systematically capture the shock and rarefied solutions. From
experiments using the LWR traffic flow model, we found superior accuracy in
predicting the density dynamics of a ring-road network and urban signalized
road. We also found that the operator can be trained using simple traffic
density dynamics, e.g., consisting of $2-3$ vehicle queues and $1-2$ traffic
signal cycles, and it can predict density dynamics for heterogeneous vehicle
queue distributions and multiple traffic signal cycles $(\geq 2)$ with an
acceptable error. The extrapolation error grew sub-linearly with input
complexity for a proper choice of the model architecture and training data.
Adding a physics regularizer aided in learning long-term traffic density
dynamics, especially for problems with periodic boundary data.
