# Macroscopic Traffic Flow Modeling with Physics Regularized Gaussian Process: A New Insight into Machine Learning Applications

arXiv: https://arxiv.org/abs/2002.02374 · HTML: https://mlsanity.com/paper/2002.02374/

## Key facts

- Cited by 25 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 6 Feb 2020.

## Built on by (most cited first, 25 of 25)

- [A Physics-Informed Deep Learning Paradigm for Traffic State and Fundamental Diagram Estimation](https://mlsanity.com/paper/2106.03142/) (2021)
- [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)
- [Incorporating Kinematic Wave Theory into a Deep Learning Method for High-Resolution Traffic Speed Estimation](https://mlsanity.com/paper/2102.02906/) (2021)
- [Short-term traffic prediction using physics-aware neural networks](https://mlsanity.com/paper/2109.10253/) (2021)
- [Quantifying Uncertainty In Traffic State Estimation Using Generative Adversarial Networks](https://mlsanity.com/paper/2206.09349/) (2022)
- [Traffic State Estimation from Vehicle Trajectories with Anisotropic Gaussian Processes](https://mlsanity.com/paper/2303.02311/) (2023)
- [Network-wide Freeway Traffic Estimation Using Sparse Sensor Data: A Dirichlet Graph Auto-Encoder Approach](https://mlsanity.com/paper/2503.15845/) (2025)
- [PAGP: A physics-assisted Gaussian process framework with active learning for forward and inverse problems of partial differential equations](https://mlsanity.com/paper/2204.02583/) (2022)
- [IDM-Follower: A Model-Informed Deep Learning Method for Long-Sequence Car-Following Trajectory Prediction](https://mlsanity.com/paper/2210.10965/) (2022)
- [Physics-informed Machine Learning for Calibrating Macroscopic Traffic Flow Models](https://mlsanity.com/paper/2307.06267/) (2023)
- [Perimeter Control with Heterogeneous Metering Rates for Cordon Signals: A Physics-Regularized Multi-Agent Reinforcement Learning Approach](https://mlsanity.com/paper/2308.12985/) (2023)
- [Efficient and Robust Freeway Traffic Speed Estimation under Oblique Grid using Vehicle Trajectory Data](https://mlsanity.com/paper/2411.05842/) (2024)
- [MoGERNN: An Inductive Traffic Predictor for Unobserved Locations](https://mlsanity.com/paper/2501.12281/) (2025)
- [Potential failures of physics-informed machine learning in traffic flow modeling: theoretical and experimental analysis](https://mlsanity.com/paper/2505.11491/) (2025)
- [A Hybrid Physics Machine Learning Approach for Macroscopic Traffic State Estimation](https://mlsanity.com/paper/2202.01888/) (2022)
- [A hybrid data driven-physics constrained Gaussian process regression framework with deep kernel for uncertainty quantification](https://mlsanity.com/paper/2205.06494/) (2022)
- [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)
- [Learning to Learn the Macroscopic Fundamental Diagram using Physics-Informed and meta Machine Learning techniques](https://mlsanity.com/paper/2508.14137/) (2025)
- [PMA-Diffusion: A Physics-guided Mask-Aware Diffusion Framework for TSE from Sparse Observations](https://mlsanity.com/paper/2512.06183/) (2025)
- [Physics-Informed Deep Learning for Traffic State Estimation](https://mlsanity.com/paper/2101.06580/) (2021)
- [Estimating Traffic Speeds using Probe Data: A Deep Neural Network Approach](https://mlsanity.com/paper/2104.09686/) (2021)
- [Closed-Loop Neural Operator-Based Observer of Traffic Density](https://mlsanity.com/paper/2504.04873/) (2025)
- [Reconstructing Physics-Informed Machine Learning for Traffic Flow Modeling: a Multi-Gradient Descent and Pareto Learning Approach](https://mlsanity.com/paper/2505.13241/) (2025)
- [Theory Foundation of Physics-Enhanced Residual Learning](https://mlsanity.com/paper/2509.00348/) (2025)

## Abstract (arXiv)

Despite the wide implementation of machine learning (ML) techniques in traffic flow modeling recently, those data-driven approaches often fall short of accuracy in the cases with a small or noisy dataset. To address this issue, this study presents a new modeling framework, named physics regularized machine learning (PRML), to encode classical traffic flow models (referred as physical models) into the ML architecture and to regularize the ML training process. More specifically, a stochastic physics regularized Gaussian process (PRGP) model is developed and a Bayesian inference algorithm is used to estimate the mean and kernel of the PRGP. A physical regularizer based on macroscopic traffic flow models is also developed to augment the estimation via a shadow GP and an enhanced latent force model is used to encode physical knowledge into stochastic processes. Based on the posterior regularization inference framework, an efficient stochastic optimization algorithm is also developed to maximize the evidence lowerbound of the system likelihood. To prove the effectiveness of the proposed model, this paper conducts empirical studies on a real-world dataset which is collected from a stretch of I-15 freeway, Utah. Results show the new PRGP model can outperform the previous compatible methods, such as calibrated pure physical models and pure machine learning methods, in estimation precision and input robustness.