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.

翻译：深度学习方法正成为解决交通流中正问题与反问题的流行计算工具。本文研究了一种神经算子框架，用于学习非线性双曲型偏微分方程的解，并应用于宏观交通流模型。在该框架中，通过监督学习方式训练一个算子，将异构且稀疏的交通输入数据映射至完整的宏观交通状态。我们选择了基于物理信息的傅里叶神经算子（$\pi$-FNO）作为该算子，其中基于离散守恒律的额外物理损失在训练过程中对问题进行正则化，以改善激波预测。同时，我们提出使用由随机分段常数输入数据生成的训练数据，系统性地捕捉激波和稀疏解。通过LWR交通流模型的实验，我们发现该方法在预测环状道路网络和城市信号化道路的密度动态方面具有优越的准确性。我们还发现，该算子可利用简单的交通密度动态（例如包含$2-3$个车辆队列和$1-2$个交通信号周期）进行训练，并能以可接受的误差预测异构车辆队列分布和多个交通信号周期（$\geq 2$）下的密度动态。对于适当的模型架构和训练数据选择，外推误差随输入复杂度呈亚线性增长。添加物理正则化有助于学习长期交通密度动态，尤其适用于具有周期性边界条件的问题。