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

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.

Learning-based solutions to nonlinear hyperbolic PDEs: Empirical insights on generalization errors

We study learning weak solutions to nonlinear hyperbolic partial differential equations (H-PDE), which have been difficult to learn due to discontinuities in their solutions. We use a physics-informed variant of the Fourier Neural Operator ($\pi$-FNO) to learn the weak solutions. We empirically quantify the generalization/out-of-sample error of the $\pi$-FNO solver as a function of input complexity, i.e., the distributions of initial and boundary conditions. Our testing results show that $\pi$-FNO generalizes well to unseen initial and boundary conditions. We find that the generalization error grows linearly with input complexity. Further, adding a physics-informed regularizer improved the prediction of discontinuities in the solution. We use the Lighthill-Witham-Richards (LWR) traffic flow model as a guiding example to illustrate the results.

Generalized adaptive smoothing based neural network architecture for traffic state estimation

The adaptive smoothing method (ASM) is a standard data-driven technique used in traffic state estimation. The ASM has free parameters which, in practice, are chosen to be some generally acceptable values based on intuition. However, we note that the heuristically chosen values often result in un-physical predictions by the ASM. In this work, we propose a neural network based on the ASM which tunes those parameters automatically by learning from sparse data from road sensors. We refer to it as the adaptive smoothing neural network (ASNN). We also propose a modified ASNN (MASNN), which makes it a strong learner by using ensemble averaging. The ASNN and MASNN are trained and tested two real-world datasets. Our experiments reveal that the ASNN and the MASNN outperform the conventional ASM.