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

TL;DR

This paper introduces a physics-informed deep learning framework that models stochastic traffic flow as a distribution, enabling probabilistic traffic state estimation and risk analysis, unlike traditional deterministic models.

Contribution

It develops a distributional physics constraint derived from stochastic traffic dynamics, leading to a differentiable probability flow ODE and a generative model for traffic densities.

Findings

The model captures stochastic traffic density distributions.

It enables computation of credible intervals and congestion risks.

The framework supports distributional traffic-state estimation.

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.