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Differentiable Stochastic Traffic Dynamics: Physics-Informed Generative Modelling in Transportation

Wuping Xin

TL;DR

This work develops a framework in which the physics constraint itself is distributional and directly derived from stochastic traffic-flow dynamics, providing a basis for distributional traffic-state estimation and for stochastic fundamental-diagram analysis in a physics-informed generative setting.

Abstract

Macroscopic traffic flow is stochastic, but the physics-informed deep learning methods currently used in transportation literature embed deterministic PDEs and produce point-valued outputs; the stochasticity of the governing dynamics plays no role in the learned representation. This work develops a framework in which the physics constraint itself is distributional and directly derived from stochastic traffic-flow dynamics. Starting from an Ito-type Lighthill-Whitham-Richards model with Brownian forcing, we derive a one-point forward equation for the marginal traffic density at each spatial location. The spatial coupling induced by the conservation law appears as an explicit conditional drift term, which makes the closure requirement transparent. Based on this formulation, we derive an equivalent deterministic Probability Flow ODE that is pointwise evaluable and differentiable once a closure is specified. Incorporating this as a physics constraint, we then propose a score network with an advection-closure module, trainable by denoising score matching together with a Fokker-Planck residual loss. The resulting model targets a data-conditioned density distribution, from which point estimates, credible intervals, and congestion-risk measures can be computed. The framework provides a basis for distributional traffic-state estimation and for stochastic fundamental-diagram analysis in a physics-informed generative setting.

Differentiable Stochastic Traffic Dynamics: Physics-Informed Generative Modelling in Transportation

TL;DR

This work develops a framework in which the physics constraint itself is distributional and directly derived from stochastic traffic-flow dynamics, providing a basis for distributional traffic-state estimation and for stochastic fundamental-diagram analysis in a physics-informed generative setting.

Abstract

Macroscopic traffic flow is stochastic, but the physics-informed deep learning methods currently used in transportation literature embed deterministic PDEs and produce point-valued outputs; the stochasticity of the governing dynamics plays no role in the learned representation. This work develops a framework in which the physics constraint itself is distributional and directly derived from stochastic traffic-flow dynamics. Starting from an Ito-type Lighthill-Whitham-Richards model with Brownian forcing, we derive a one-point forward equation for the marginal traffic density at each spatial location. The spatial coupling induced by the conservation law appears as an explicit conditional drift term, which makes the closure requirement transparent. Based on this formulation, we derive an equivalent deterministic Probability Flow ODE that is pointwise evaluable and differentiable once a closure is specified. Incorporating this as a physics constraint, we then propose a score network with an advection-closure module, trainable by denoising score matching together with a Fokker-Planck residual loss. The resulting model targets a data-conditioned density distribution, from which point estimates, credible intervals, and congestion-risk measures can be computed. The framework provides a basis for distributional traffic-state estimation and for stochastic fundamental-diagram analysis in a physics-informed generative setting.
Paper Structure (67 sections, 4 theorems, 36 equations, 1 figure, 3 tables, 1 algorithm)

This paper contains 67 sections, 4 theorems, 36 equations, 1 figure, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. The disconnect between stochastic traffic physics and deep learning
  3. The core obstacle and our resolution
  4. Significance beyond a single application
  5. Distributional traffic state estimation.
  6. Stochastic fundamental diagram.
  7. Physics-informed generative modelling for traffic.
  8. Contributions
  9. Paper organisation
  10. Literature Review
  11. Stochastic macroscopic traffic flow modelling
  12. Kinetic theory and the Prigogine--Herman programme.
  13. Randomised fundamental diagrams and noise-augmented models.
  14. Stochastic models with explicit physical attribution.
  15. Random-parameter SLWR models.
  16. ...and 52 more sections

Key Result

Proposition 4.3

Suppose Assumptions ass:flux--ass:density hold and that the conditional drift coefficient $b$ in eq:b-def is measurable and locally bounded. Then, for each fixed $x\in\mathcal{D}$, the one-point marginal density $p(\hat{\rho};x,t)$ of $Y_t^x=\rho(x,t)$ satisfies with initial condition understood in the distributional sense (equivalently, one may begin from a mollified initial law $\delta_\epsilo

Figures (1)

  • Figure 1: Architectural comparison between PIDL and the proposed framework. Both share a two-component structure (estimation network + physics evaluator), but the proposed framework operates at the distributional level: the network learns a score function, the physics evaluator regularises via the Fokker--Planck equation, and the output is a data-conditioned distributional estimate.

Theorems & Definitions (27)

  • Definition 3.1: Itô-type SLWR with finite-dimensional noise
  • Remark 3.2: On the form of the stochastic forcing
  • Remark 3.3: Generality of the flux
  • Remark 3.4: Regularity of the noise coefficients
  • Remark 3.5: Practical scope of Assumption \ref{['ass:smooth']}
  • Remark 3.6: On Assumption \ref{['ass:density']}
  • Definition 4.1: Conditional drift coefficient
  • Remark 4.2: Transportation interpretation
  • Proposition 4.3: Exact one-point Fokker--Planck equation
  • proof
  • ...and 17 more