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

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.

The stochastic fundamental diagram (SFD) provides a probabilistic description of the relationship between traffic density and flow or speed, enabling uncertainty-aware traffic modeling. However, existing stochastic models frequently struggle to accommodate rigorous physical constraints while retaining sufficient flexibility to capture complex nonlinear patterns. To address this, we propose a novel semiparametric SFD modeling framework by leveraging specially designed functional forms. These functions intrinsically satisfy physical constraints defined on the moments of the conditional flow distribution given traffic density while incorporating neural-network-based structures to capture complex empirical patterns. We derive a system of moment-matching equations to convert physical constraints into the parameterization of the conditional distribution, proving that a unique solution exists for the location-scale family of distributions, thereby guaranteeing model well-posedness. Furthermore, we demonstrate that the framework can be extended to non-location-scale distributions, including those requiring additional boundary constraints. Empirical evaluations on a real-world dataset reveal that our approach consistently outperforms representative baselines, delivering superior probabilistic accuracy and robust uncertainty quantification, particularly in congested regimes. Overall, the proposed framework provides a theoretically grounded and flexible foundation for stochastic traffic flow modeling.

Traffic forecasting is a challenging spatio-temporal modeling task and a critical component of urban transportation management. Current studies mainly focus on deterministic predictions, with limited considerations on the uncertainty and stochasticity in traffic dynamics. Therefore, this paper proposes an elegant yet universal approach that transforms existing models into probabilistic predictors by replacing only the final output layer with a novel Gaussian Mixture Model (GMM) layer. The modified model requires no changes to the training pipeline and can be trained using only the Negative Log-Likelihood (NLL) loss, without any auxiliary or regularization terms. Experiments on multiple traffic datasets show that our approach generalizes from classic to modern model architectures while preserving deterministic performance. Furthermore, we propose a systematic evaluation procedure based on cumulative distributions and confidence intervals, and demonstrate that our approach is considerably more accurate and informative than unimodal or deterministic baselines. Finally, a more detailed study on a real-world dense urban traffic network is presented to examine the impact of data quality on uncertainty quantification and to show the robustness of our approach under imperfect data conditions. Code available at https://github.com/Weijiang-Xiong/OpenSkyTraffic