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 first-order macroscopic traffic flow models with applications in estimating traffic densities for urban arterials. In this framework, an operator is trained to map heterogeneous and sparse traffic input data to the complete macroscopic traffic density in a supervised learning setting. We chose a physics-informed Fourier neural operator (-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 rarefaction solutions of certain macroscopic traffic flow models. 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 vehicle queues and traffic signal cycles, and it can predict density dynamics for heterogeneous vehicle queue distributions and multiple traffic signal cycles 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.

中文翻译：

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用于学习宏观交通流模型解决方案的傅立叶神经算子：在正向和逆向问题中的应用

深度学习方法正在成为解决交通流中的正向和逆向问题的流行计算工具。在本文中，我们研究了一种神经算子框架，用于学习一阶宏观交通流模型的解决方案，并应用于估计城市主干道的交通密度。在此框架中，操作员经过训练，可以在监督学习环境中将异构且稀疏的流量输入数据映射到完整的宏观流量密度。我们选择了基于物理的傅里叶神经算子（-FNO）作为算子，其中基于离散守恒定律的额外物理损失在训练过程中规范了问题，以改善冲击预测。我们还建议使用随机分段恒定输入数据生成的训练数据来系统地捕获某些宏观交通流模型的冲击和稀疏解。通过使用 LWR 交通流模型的实验，我们发现在预测环形路网和城市信号道路的密度动态方面具有极高的准确性。我们还发现，可以使用简单的交通密度动态（例如，由车辆队列和交通信号周期组成）来训练操作员，并且它可以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的密度动态。为了正确选择模型架构和训练数据，外推误差随着输入复杂性呈亚线性增长。添加物理正则化器有助于学习长期交通密度动态，特别是对于周期性边界数据的问题。