# A Physics-Informed Deep Learning Paradigm for Traffic State and Fundamental Diagram Estimation

arXiv: https://arxiv.org/abs/2106.03142 · HTML: https://mlsanity.com/paper/2106.03142/

## Key facts

- Cited by 37 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 26 references that are in the map.
- First posted on arXiv 6 Jun 2021.

## Builds on (most cited first, 8 of 8)

- [Hidden Physics Models: Machine Learning of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1708.00588/) (2017)
- [A Physics-Informed Deep Learning Paradigm for Car-Following Models](https://mlsanity.com/paper/2012.13376/) (2020)
- [Macroscopic Traffic Flow Modeling with Physics Regularized Gaussian Process: A New Insight into Machine Learning Applications](https://mlsanity.com/paper/2002.02374/) (2020)
- [A Survey on Autonomous Vehicle Control in the Era of Mixed-Autonomy: From Physics-Based to AI-Guided Driving Policy Learning](https://mlsanity.com/paper/2007.05156/) (2020)
- [Low-Rank Hankel Tensor Completion for Traffic Speed Estimation](https://mlsanity.com/paper/2105.11335/) (2021)
- [Physics-informed Learning for Identification and State Reconstruction of Traffic Density](https://mlsanity.com/paper/2103.13852/) (2021)
- [Learning-based Traffic State Reconstruction using Probe Vehicles](https://mlsanity.com/paper/2011.05031/) (2020)
- [Learning-based State Reconstruction for a Scalar Hyperbolic PDE under noisy Lagrangian Sensing](https://mlsanity.com/paper/2011.09871/) (2020)

## Built on by (most cited first, 30 of 37)

- [On the Limitations of Physics-informed Deep Learning: Illustrations Using First Order Hyperbolic Conservation Law-based Traffic Flow Models](https://mlsanity.com/paper/2302.12337/) (2023)
- [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)
- [Hierarchical Consensus-Based Multi-Agent Reinforcement Learning for Multi-Robot Cooperation Tasks](https://mlsanity.com/paper/2407.08164/) (2024)
- [TrafficFlowGAN: Physics-informed Flow based Generative Adversarial Network for Uncertainty Quantification](https://mlsanity.com/paper/2206.09319/) (2022)
- [Physics-Informed Deep Learning For Traffic State Estimation: A Survey and the Outlook](https://mlsanity.com/paper/2303.02063/) (2023)
- [ESP: Exploiting Symmetry Prior for Multi-Agent Reinforcement Learning](https://mlsanity.com/paper/2307.16186/) (2023)
- [Short-term traffic prediction using physics-aware neural networks](https://mlsanity.com/paper/2109.10253/) (2021)
- [Higher-order Spatio-temporal Physics-incorporated Graph Neural Network for Multivariate Time Series Imputation](https://mlsanity.com/paper/2405.10995/) (2024)
- [Quantifying Uncertainty In Traffic State Estimation Using Generative Adversarial Networks](https://mlsanity.com/paper/2206.09349/) (2022)
- [Traffic expertise meets residual RL: Knowledge-informed model-based residual reinforcement learning for CAV trajectory control](https://mlsanity.com/paper/2408.17380/) (2024)
- [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)
- [IDM-Follower: A Model-Informed Deep Learning Method for Long-Sequence Car-Following Trajectory Prediction](https://mlsanity.com/paper/2210.10965/) (2022)
- [Generalized adaptive smoothing based neural network architecture for traffic state estimation](https://mlsanity.com/paper/2301.03439/) (2023)
- [Physics-informed Machine Learning for Calibrating Macroscopic Traffic Flow Models](https://mlsanity.com/paper/2307.06267/) (2023)
- [Refining time-space traffic diagrams: A simple multiple linear regression model](https://mlsanity.com/paper/2204.04457/) (2022)
- [Training Physics-Informed Neural Networks via Multi-Task Optimization for Traffic Density Prediction](https://mlsanity.com/paper/2307.03920/) (2023)
- [Privacy-Preserving Data Fusion for Traffic State Estimation: A Vertical Federated Learning Approach](https://mlsanity.com/paper/2401.11836/) (2024)
- [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 Survey of Physics-Informed AI for Complex Urban Systems](https://mlsanity.com/paper/2506.13777/) (2025)
- [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)
- [Incorporating Nonlocal Traffic Flow Model in Physics-informed Neural Networks](https://mlsanity.com/paper/2308.11818/) (2023)
- [Learning "Look-Ahead" Nonlocal Traffic Dynamics in a Ring Road](https://mlsanity.com/paper/2312.02770/) (2023)
- [Graph Neural Network Surrogates to leverage Mechanistic Expert Knowledge towards Reliable and Immediate Pandemic Response](https://mlsanity.com/paper/2411.06500/) (2024)
- [Learning to Learn the Macroscopic Fundamental Diagram using Physics-Informed and meta Machine Learning techniques](https://mlsanity.com/paper/2508.14137/) (2025)
- [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)
- [Reconstructing Physics-Informed Machine Learning for Traffic Flow Modeling: a Multi-Gradient Descent and Pareto Learning Approach](https://mlsanity.com/paper/2505.13241/) (2025)

## Abstract (arXiv)

Traffic state estimation (TSE) bifurcates into two categories, model-driven and data-driven (e.g., machine learning, ML), while each suffers from either deficient physics or small data. To mitigate these limitations, recent studies introduced a hybrid paradigm, physics-informed deep learning (PIDL), which contains both model-driven and data-driven components. This paper contributes an improved version, called physics-informed deep learning with a fundamental diagram learner (PIDL+FDL), which integrates ML terms into the model-driven component to learn a functional form of a fundamental diagram (FD), i.e., a mapping from traffic density to flow or velocity. The proposed PIDL+FDL has the advantages of performing the TSE learning, model parameter identification, and FD estimation simultaneously. We demonstrate the use of PIDL+FDL to solve popular first-order and second-order traffic flow models and reconstruct the FD relation as well as model parameters that are outside the FD terms. We then evaluate the PIDL+FDL-based TSE using the Next Generation SIMulation (NGSIM) dataset. The experimental results show the superiority of the PIDL+FDL in terms of improved estimation accuracy and data efficiency over advanced baseline TSE methods, and additionally, the capacity to properly learn the unknown underlying FD relation.