Bilal Thonnam Thodi

Bilal Thonnam Thodi, Sai Venkata Ramana Ambadipudi, Saif Eddin Jabari

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 ($π$-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.

Bilal Thonnam Thodi, Sai Venkata Ramana Ambadipudi, Saif Eddin Jabari

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 ($π$-FNO) to learn the weak solutions. We empirically quantify the generalization/out-of-sample error of the $π$-FNO solver as a function of input complexity, i.e., the distributions of initial and boundary conditions. Our testing results show that $π$-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.

Bilal Thonnam Thodi, Zaid Saeed Khan, Saif Eddin Jabari et al.

Space-time visualizations of macroscopic or microscopic traffic variables is a qualitative tool used by traffic engineers to understand and analyze different aspects of road traffic dynamics. We present a deep learning method to learn the macroscopic traffic speed dynamics from these space-time visualizations, and demonstrate its application in the framework of traffic state estimation. Compared to existing estimation approaches, our approach allows a finer estimation resolution, eliminates the dependence on the initial conditions, and is agnostic to external factors such as traffic demand, road inhomogeneities and driving behaviors. Our model respects causality in traffic dynamics, which improves the robustness of estimation. We present the high-resolution traffic speed fields estimated for several freeway sections using the data obtained from the Next Generation Simulation Program (NGSIM) and German Highway (HighD) datasets. We further demonstrate the quality and utility of the estimation by inferring vehicle trajectories from the estimated speed fields, and discuss the benefits of deep neural network models in approximating the traffic dynamics.

Bilal Thonnam Thodi, Zaid Saeed Khan, Saif Eddin Jabari et al.

We propose a kinematic wave-based Deep Convolutional Neural Network (Deep CNN) to estimate high-resolution traffic speed fields from sparse probe vehicle trajectories. We introduce two key approaches that allow us to incorporate kinematic wave theory principles to improve the robustness of existing learning-based estimation methods. First, we propose an anisotropic traffic kernel for the Deep CNN. The anisotropic kernel explicitly accounts for space-time correlations in macroscopic traffic and effectively reduces the number of trainable parameters in the Deep CNN model. Second, we propose to use simulated data for training the Deep CNN. Using a targeted simulated data for training provides an implicit way to impose desirable traffic physical features on the learning model. In the experiments, we highlight the benefits of using anisotropic kernels and evaluate the transferability of the trained model to real-world traffic using the Next Generation Simulation (NGSIM) and the German Highway Drone (HighD) datasets. The results demonstrate that anisotropic kernels significantly reduce model complexity and model over-fitting, and improve the physical correctness of the estimated speed fields. We find that model complexity scales linearly with problem size for anisotropic kernels compared to quadratic scaling for isotropic kernels. Furthermore, evaluation on real-world datasets shows acceptable performance, which establishes that simulation-based training is a viable surrogate to learning from real-world data. Finally, a comparison with standard estimation techniques shows the superior estimation accuracy of the proposed method.

Ouafa Benkraouda, Bilal Thonnam Thodi, Hwasoo Yeo et al.

We propose a statistical learning-based traffic speed estimation method that uses sparse vehicle trajectory information. Using a convolutional encoder-decoder based architecture, we show that a well trained neural network can learn spatio-temporal traffic speed dynamics from time-space diagrams. We demonstrate this for a homogeneous road section using simulated vehicle trajectories and then validate it using real-world data from NGSIM. Our results show that with probe vehicle penetration levels as low as 5\%, the proposed estimation method can provide a sound reconstruction of macroscopic traffic speeds and reproduce realistic shockwave patterns, implying applicability in a variety of traffic conditions. We further discuss the model's reconstruction mechanisms and confirm its ability to differentiate various traffic behaviors such as congested and free-flow traffic states, transition dynamics, and shockwave propagation.

Saif Eddin Jabari, Deepthi Mary Dilip, DianChao Lin et al.

This paper presents a mesoscopic traffic flow model that explicitly describes the spatio-temporal evolution of the probability distributions of vehicle trajectories. The dynamics are represented by a sequence of factor graphs, which enable learning of traffic dynamics from limited Lagrangian measurements using an efficient message passing technique. The approach ensures that estimated speeds and traffic densities are non-negative with probability one. The estimation technique is tested using vehicle trajectory datasets generated using an independent microscopic traffic simulator and is shown to efficiently reproduce traffic conditions with probe vehicle penetration levels as little as 10\%. The proposed algorithm is also compared with state-of-the-art traffic state estimation techniques developed for the same purpose and it is shown that the proposed approach can outperform the state-of-the-art techniques in terms reconstruction accuracy.