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Traffic Flow Reconstruction from Limited Collected Data

Nail Baloul, Amaury Hayat, Thibault Liard, Pierre Lissy

TL;DR

The paper tackles reconstructing traffic density from sparse data by using only initial and final positions of a small set of probe vehicles, bridging microscopic Follow-the-Leader dynamics and the macroscopic LWR conservation law. It proposes an ODE-constrained learning framework that implicitly enforces mass conservation through the underlying FtL-to-LWR convergence, avoiding explicit PDE constraints. A residual-network architecture models time-stepped vehicle interactions, trained to match observed probe final positions and recover a discrete density $\rho^N$ that converges to the entropy solution of the LWR model as $N\to\infty$, given growth conditions on the inter-probe counts $\alpha^N$. Theoretical convergence is complemented by numerical experiments under Greenshields velocity, showing accurate density reconstruction and agreement with Godunov solutions even with limited data, illustrating the method’s potential for sparse-sensor traffic state estimation. The approach offers a practical, low-data-demand path to scalable traffic density estimation with convergence guarantees, and future work includes validation on real-world data sets.

Abstract

We propose an efficient method for reconstructing traffic density with low penetration rate of probe vehicles. Specifically, we rely on measuring only the initial and final positions of a small number of cars which are generated using microscopic dynamical systems. We then implement a machine learning algorithm from scratch to reconstruct the approximate traffic density. This approach leverages learning techniques to improve the accuracy of density reconstruction despite constraints in available data. For the sake of consistency, we will prove that, if only using data from dynamical systems, the approximate density predicted by our learned-based model converges to a well-known macroscopic traffic flow model when the number of vehicles approaches infinity.

Traffic Flow Reconstruction from Limited Collected Data

TL;DR

The paper tackles reconstructing traffic density from sparse data by using only initial and final positions of a small set of probe vehicles, bridging microscopic Follow-the-Leader dynamics and the macroscopic LWR conservation law. It proposes an ODE-constrained learning framework that implicitly enforces mass conservation through the underlying FtL-to-LWR convergence, avoiding explicit PDE constraints. A residual-network architecture models time-stepped vehicle interactions, trained to match observed probe final positions and recover a discrete density that converges to the entropy solution of the LWR model as , given growth conditions on the inter-probe counts . Theoretical convergence is complemented by numerical experiments under Greenshields velocity, showing accurate density reconstruction and agreement with Godunov solutions even with limited data, illustrating the method’s potential for sparse-sensor traffic state estimation. The approach offers a practical, low-data-demand path to scalable traffic density estimation with convergence guarantees, and future work includes validation on real-world data sets.

Abstract

We propose an efficient method for reconstructing traffic density with low penetration rate of probe vehicles. Specifically, we rely on measuring only the initial and final positions of a small number of cars which are generated using microscopic dynamical systems. We then implement a machine learning algorithm from scratch to reconstruct the approximate traffic density. This approach leverages learning techniques to improve the accuracy of density reconstruction despite constraints in available data. For the sake of consistency, we will prove that, if only using data from dynamical systems, the approximate density predicted by our learned-based model converges to a well-known macroscopic traffic flow model when the number of vehicles approaches infinity.
Paper Structure (17 sections, 2 theorems, 18 equations, 8 figures)

This paper contains 17 sections, 2 theorems, 18 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Traffic flow models
  3. Microscopic model
  4. Macroscopic model
  5. Data-driven based model
  6. Mathematical formulation
  7. Parametrized ODE system
  8. Global existence and uniqueness
  9. Discrete density
  10. Learning-based density estimation
  11. ODE constrained optimization problem
  12. Dataset generation
  13. Learning architecture
  14. Learning procedure
  15. Convergence of the model
  16. ...and 2 more sections

Key Result

Lemma III.1

Let $\left(x_0(\cdot), \dots, x_n(\cdot)\right)$ be the solution of ODE system 3b and $v$ satisfy hypotheses (v1)-(v3). Then the discrete maximum principle holds; for all $i=0,\dots, n-1$ and for all $t\in [0,T]$, where $M\coloneqq \underset{i=0,\dots, n-1}{\max}\left(\frac{\alpha^N_i L}{N(\bar{x}_{i+1}-\bar{x}_{i})}\right)$ denotes the maximum discrete density at initial time $t=0$.

Figures (8)

  • Figure 1: Learning procedure
  • Figure 2: Comparison between predictions and data with $N=2000$
  • Figure 3: Comparison between predictions and data with $N=4000$
  • Figure 4: Comparison between the predicted density and Godunov scheme density with $N=4000$ and $T=0.2$
  • Figure 5: Reconstructed final traffic density
  • ...and 3 more figures

Theorems & Definitions (3)

  • Lemma III.1
  • Proposition V.1
  • Remark V.2