# Incorporating Kinematic Wave Theory into a Deep Learning Method for High-Resolution Traffic Speed Estimation

arXiv: https://arxiv.org/abs/2102.02906 · HTML: https://mlsanity.com/paper/2102.02906/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 78 references that are in the map.
- First posted on arXiv 4 Feb 2021.

## Builds on (most cited first, 20 of 20)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Domain Randomization for Transferring Deep Neural Networks from Simulation to the Real World](https://mlsanity.com/paper/1703.06907/) (2017)
- [Hyperband: A Novel Bandit-Based Approach to Hyperparameter Optimization](https://mlsanity.com/paper/1603.06560/) (2016)
- [Training Deep Networks with Synthetic Data: Bridging the Reality Gap by Domain Randomization](https://mlsanity.com/paper/1804.06516/) (2018)
- [The highD Dataset: A Drone Dataset of Naturalistic Vehicle Trajectories on German Highways for Validation of Highly Automated Driving Systems](https://mlsanity.com/paper/1810.05642/) (2018)
- [Macroscopic Traffic Flow Modeling with Physics Regularized Gaussian Process: A New Insight into Machine Learning Applications](https://mlsanity.com/paper/2002.02374/) (2020)
- [Nonlinear Traffic Prediction as a Matrix Completion Problem with Ensemble Learning](https://mlsanity.com/paper/2001.02492/) (2020)
- [Learning-based Traffic State Reconstruction using Probe Vehicles](https://mlsanity.com/paper/2011.05031/) (2020)
- [Highway Traffic State Estimation with Mixed Connected and Conventional Vehicles](https://mlsanity.com/paper/1504.06879/) (2015)
- [Traffic Data Imputation using Deep Convolutional Neural Networks](https://mlsanity.com/paper/2002.04406/) (2020)
- [Learning-based State Reconstruction for a Scalar Hyperbolic PDE under noisy Lagrangian Sensing](https://mlsanity.com/paper/2011.09871/) (2020)
- [Multi-scale Perimeter Control Approach in a Connected-Vehicle Environment](https://mlsanity.com/paper/1612.05910/) (2016)
- [Traffic state estimation using stochastic Lagrangian dynamics](https://mlsanity.com/paper/1806.02692/) (2018)
- [Learning Traffic Flow Dynamics using Random Fields](https://mlsanity.com/paper/1806.08764/) (2018)
- [Position weighted backpressure intersection control for urban networks](https://mlsanity.com/paper/1810.11406/) (2018)
- [Sparse Travel Time Estimation from Streaming Data](https://mlsanity.com/paper/1804.08130/) (2018)
- [Backpressure Control with Estimated Queue Lengths for Urban Network Traffic](https://mlsanity.com/paper/2006.15549/) (2020)
- [Pay for Intersection Priority: A Free Market Mechanism for Connected Vehicles](https://mlsanity.com/paper/2001.01813/) (2020)
- [The State-of-the-Art of Coordinated Ramp Control with Mixed Traffic Conditions](https://mlsanity.com/paper/1908.00573/) (2019)
- [Short-Term Traffic Forecasting Using High-Resolution Traffic Data](https://mlsanity.com/paper/2006.12292/) (2020)

## Built on by (most cited first, 12 of 12)

- [Low-Rank Hankel Tensor Completion for Traffic Speed Estimation](https://mlsanity.com/paper/2105.11335/) (2021)
- [Fourier neural operator for learning solutions to macroscopic traffic flow models: Application to the forward and inverse problems](https://mlsanity.com/paper/2308.07051/) (2023)
- [Physics-Informed Deep Learning For Traffic State Estimation: A Survey and the Outlook](https://mlsanity.com/paper/2303.02063/) (2023)
- [Traffic State Estimation from Vehicle Trajectories with Anisotropic Gaussian Processes](https://mlsanity.com/paper/2303.02311/) (2023)
- [Urban traffic analysis and forecasting through shared Koopman eigenmodes](https://mlsanity.com/paper/2409.04728/) (2024)
- [Generalized adaptive smoothing based neural network architecture for traffic state estimation](https://mlsanity.com/paper/2301.03439/) (2023)
- [Efficient and Robust Freeway Traffic Speed Estimation under Oblique Grid using Vehicle Trajectory Data](https://mlsanity.com/paper/2411.05842/) (2024)
- [Inverting the Fundamental Diagram and Forecasting Boundary Conditions: How Machine Learning Can Improve Macroscopic Models for Traffic Flow](https://mlsanity.com/paper/2303.12740/) (2023)
- [Estimating Traffic Speeds using Probe Data: A Deep Neural Network Approach](https://mlsanity.com/paper/2104.09686/) (2021)
- [Physics-informed Machine Learning of Parameterized Fundamental Diagrams](https://mlsanity.com/paper/2208.00880/) (2022)
- [Newell's theory based feature transformations for spatio-temporal traffic prediction](https://mlsanity.com/paper/2307.05949/) (2023)
- [Machine Unlearning of Traffic State Estimation and Prediction](https://mlsanity.com/paper/2507.17984/) (2025)

## Abstract (arXiv)

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.