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

Wuping Xin

eess.SYMar 10, 2026

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

Authors: Wuping Xin

Organizations: Caliper Corporation

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.

Figures & tables

	Fan–Du SLWR CACIE 2023; Trans. B 2024	Shi–Di PIDL AAAI 2021	This paper
Stochastic model	Random-parameter PDE	None (det. ARZ)	Itô SPDE
Distributional eq.	None	N/A	Fokker–Planck
Physics in learning	Forward solver	Det. PDE residual	PF-ODE residual
Output	Mean & std	Point estimate	Full $p_{\theta}(\hat{\rho};x,t)$
UQ mechanism	Parametric	Post-hoc only	Intrinsic (score)
Dynamic noise	No	No	Yes ( $\mathrm{d}W_{t}$ )

Table 1: Positioning of the proposed framework relative to the closest prior works.

Domain and probability space
Symbol	Description
$\mathcal{D}=[0,L]$	Highway segment
$Q_{T}=\mathcal{D}\times(0,T]$	Spatio-temporal domain
$(\Omega,\mathcal{F},\{\mathcal{F}_{t}\},\mathbb{P})$	Filtered probability space
$W^{k}_{t}$ $(k=1,\dots,K)$	Independent standard Brownian motions
Traffic variables and flux

Table 2: Summary of notation.

Panel of Figure 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.

Component	PIDL	This paper
Network input	$(x,t)$	$(\hat{\rho},x,t)$
Network output	$(\hat{\rho},\hat{u})$ point values	$s_{\theta}$ (score, scalar)
Physics evaluator	ARZ residual $(\hat{f}_{1},\hat{f}_{2})$	FPE residual $\mathcal{R}_{\theta,\phi}$
Data loss	$\sum\|\hat{\rho}-\rho^{\text{obs}}\|^{2}$	DSM: $\sum\|s_{\theta}+\epsilon/\sigma\|^{2}$
Physics loss	$\sum\|\hat{f}_{1}\|^{2}+\|\hat{f}_{2}\|^{2}$	$\sum\|\mathcal{R}_{\theta,\phi}\|^{2}$
Collocation domain	$(x,t)$ , 2D	$(\hat{\rho},x,t)$ , 3D

Table 3: Component-level comparison between PIDL and the proposed framework.

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Learning Traffic Flow Dynamics with Stochastic Physics-Informed Neural Cellular Automata

Federica Bragone, Matthieu Barreau

Department of Decision and Control Systems, KTH Royal Institute of Technology, Stockholm, Sweden

Neural Cellular Automata Cellular Automata Intelligent Transportation Systems

Open paper AskPDF arXiv

Traffic flow modeling is essential for understanding and predicting the collective dynamics of vehicles on road networks. Cellular automata provide a simple, interpretable yet powerful framework for representing these dynamics via local interaction rules, while retaining the ability to reproduce complex macroscopic traffic phenomena. However, learning local transition rules from data while preserving physically meaningful constraints remains challenging, particularly for stochastic models. In this work, we propose a physics-informed neural cellular automaton (PI-NCA) for data-driven traffic flow modeling. Building on the standard neural cellular automaton (NCA), we design a neural architecture that is physically consistent with the road topology and guarantees conservation of the total number of vehicles, thereby constraining the learned transition rules to physically admissible dynamics. We further extend this framework to stochastic dynamics by parameterizing probabilistic transition rules while preserving the same physics-informed constraints. We evaluate the proposed models on multiple traffic scenarios generated by the well-established Nagel-Schreckenberg and Kerner-Klenov-Wolf cellular automata. The results demonstrate that the PI-NCA successfully learns the dynamics of both traffic models and consistently outperforms a standard NCA, while the stochastic extension captures probabilistic transition rules without compromising the imposed physical constraints.

Abstract

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