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

An analytical approach to real-time bus signal priority system for isolated intersections

Bus signal priority (BSP) is an active traffic management measure to reduce bus travel delay at signalized intersections and to improve the bus service reliability. In this paper, we present a real-time BSP system with a primary focus on its practical implementation. We tackle two inter-related issues of existing priority systems, namely, real-time computation and solution optimality, using an analytical approach. The core of the proposed priority system involves two signal controller actions – red truncation (RT) and green extension (GE), which determine the priority timings based on the objective of minimizing total person delay incurring at the subject intersection. We demonstrate the analytical approach by deriving closed-form expressions for optimal RT and GE for a two-phase signal using cumulative count curves. The inputs required for these priority models are based on average traffic and bus conditions limited to the current signal cycle alone. Solutions for the RT and GE models indicate that three dimensionless variables – ratio of bus-arrival time to traffic queuing time, ratio of bus passenger occupancy to other vehicles’ average passenger occupancy, and ratio of traffic demand to saturation flow ratio – govern the priority decisions. Simulation results showed significant delay reduction for buses (≈22%) with a negligible impact on other traffic users during low to medium traffic conditions and high bus frequencies (3 min headway)

Generalized Adaptive Smoothing Using Matrix Completion for Traffic State Estimation

The Adaptive Smoothing Method (ASM) is a data-driven approach for traffic state estimation. It interpolates unobserved traffic quantities by smoothing measurements along spatio-temporal directions defined by characteristic traffic wave speeds. The standard ASM consists of a superposition of two a priori estimates weighted by a heuristic weight factor. In this paper, we propose a systematic procedure to calculate the optimal weight factors. We formulate the a priori weights calculation as a constrained matrix completion problem, and efficiently solve it using the Alternating Direction Method of Multipliers (ADMM) algorithm. Our framework allows one to further improve the conventional ASM, which is limited by utilizing only one pair of congested and free flow wave speeds, by considering multiple wave speeds. Our proposed algorithm does not require any field-dependent traffic parameters, thus bypassing frequent field calibrations as required by the conventional ASM. Experiments using NGSIM data show that the proposed ADMM-based estimation incurs lower error than the ASM estimation