Traffic Flow Theory

Car-following (CF) models is a fundamental component of microscopic traffic simulation and underpin the analysis of collective traffic dynamics, while its modeling becomes more challenging due to diverse human driving behaviors and the increasing presence of automated vehicle. Analytical CF models offer interpretable representations of longitudinal vehicle interactions, but their prescribed functional structures may limit the representation of heterogeneous driving behavior. This study proposes an agent-centered, physics-aware framework for the empirical discovery of CF laws from multi-source trajectory data. The framework iteratively combines agent-guided hypothesis generation, informed by traffic flow knowledge and state-dependent reconstruction errors, with evolutionary symbolic search and physics-aware multi-objective evaluation. Using NGSIM US-101, highD, and exiD, the framework discovers the Adaptive Equilibrium Response Model (AERM), which couples state-dependent relative-speed response with spacing-dependent equilibrium-speed restoration. AERM admits analytical equilibrium speed-spacing and flow-density relationships; its stability conditions are derived for the zero-delay formulation, and platoon simulations examine disturbance propagation. Evaluated against seven classical CF models on Shanghai naturalistic driving data, NGSIM I-80, and Waymo trajectories, AERM attains the lowest spacing and speed reconstruction errors under trajectory-specific calibration, reducing spacing RMSE by 9.11%, 7.59%, and 21.73% relative to the strongest baselines, and remains competitive under dataset-level shared parameters. These results demonstrate the fitting flexibility and analytical tractability of the discovered CF law, providing an interpretable basis for microscopic traffic simulation.

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.

Existing models of mixed-autonomy weaving ramps characterize how altruistic connected and automated vehicles (CAVs) can improve traffic efficiency at the population level, but provide limited insight into how such behavior emerges from decentralized vehicle interactions or how it is affected by finite populations, heterogeneous preferences, and imperfect information. We develop an agent-based model of a macroscopic weaving-ramp framework in which individual vehicles adapt their lane choices using an evolutionary game-theoretic update rule and altruism-based objectives providing a microscopic interpretation of the original Wardrop model. We prove convergence of the decentralized dynamics to the unique equilibrium predicted by the macroscopic theory. Beyond reproducing aggregate equilibrium behavior, the framework enables the study of deployment-level questions that cannot be addressed by static analysis. Simulation results demonstrate close agreement with the macroscopic predictions while revealing how convergence rates, adaptation to changing traffic conditions, heterogeneous altruism levels among CAVs, and imperfect state information influence system performance and the distribution of altruistic burden across vehicles. These results provide a bridge between equilibrium traffic theory and decentralized mixed-autonomy deployment.

Successive weaving segments (SWSs) on urban expressways are bottlenecks prone to recurrent congestion and collisions, requiring fine-grained active traffic management (ATM). Existing approaches struggle to balance the adaptive performance of data-driven optimization with the resilience and transferability of model-based control. We propose a hybrid framework to coordinate lane-level variable speed limits (VSLs) and ramp metering across SWSs. First, we reconstruct L-METANET, a lane-level macroscopic traffic flow model that captures free and forced lane changes. Second, we combine XGBoost-SHAP with a random-parameters binary logit (RPBL) model to derive analytical equations for merging and diverging collision risks and formulate system cost and reward functions. Third, we develop MPC-STMAPPO, a hierarchical controller integrating model predictive control (MPC) and multi-agent reinforcement learning (MARL). Its upper MPC layer uses L-METANET for long-horizon rolling optimization and generates baseline commands; its lower spatiotemporal MAPPO (ST-MAPPO) layer, enhanced with Mamba cells and graph attention, produces residual actions for short-horizon adjustment. Real-world experiments on the 18-km Eastern Expressway in Changchun, China, show that L-METANET accurately reproduces lane-changing-induced flow redistribution and capacity drops, with state evolution aligned with ground truth. XGBoost-SHAP-RPBL achieves AUCs above 0.80 in most tasks, outperforming conventional logit models. MPC-STMAPPO converges faster and performs better across multiple metrics than MPC- and MARL-based baselines. Under randomly fluctuating demand, it also significantly outperforms pure MARL in generalization, demonstrating strong potential for industrial deployment.

Large language models (LLMs) are increasingly adopted as closed-loop controllers in physical multi-agent systems, yet their emergent collective dynamics remain incompletely characterised. We deploy 22 LLM agents as direct, real-time target-speed controllers (per 0.5 s cycle, with IDM as collision-avoidance clamp) on a 230 m ring road under the Sugiyama 2008 paradigm, reproducing human-like stop-and-go waves. Six matched controls spanning stochasticity (white noise, OU noise, temperature), population variance, and dynamical instability (delay, OV model) are systematically excluded. The surviving phenomenon, termed Sustained Heterogeneity (SH), is the persistent, approximately temperature-insensitive (approx. 8 percent across a 6x T sweep), per-cycle divergence in LLM-chosen target-speed adjustments, propagating through a three-stage cascade of drift, gap erosion, and nonlinear braking. Across four traffic densities, the critical LLM penetration fraction p_c decreases monotonically from no transition at density 43.5 veh/km to p_c approx 0.23 at density 95.7 veh/km, consistent with an initiation-threshold model governed by trigger distance, stochasticity, and fleet size. Chain-of-thought analysis of 39,600 decisions across three seeds shows agents engage in multi-factor safety reasoning, yet systematic divergence persists, implying stability must be enforced at the dynamics layer. This is the first study to identify a previously uncharacterised collective mechanism in LLM-controlled traffic and map a density-dependent phase boundary p_c(rho).

Driver behavior is heterogeneous, context-dependent, and changes over time, and these properties shape the traffic phenomena we observe. Most models, however, fix in advance which behavioral variables interact and how. Behavior outside that form is absorbed as noise, while models flexible enough to capture it tend to lose interpretability. We introduce a quantum-inspired representation of driver behavior that combines properties usually treated separately or in part: it is continuous, probabilistic, context-dependent, history-dependent, and represents interactions among behavioral variables as learned from data. Each driver is encoded as an evolving density matrix, providing a unified representation of behavioral uncertainty, temporal evolution, and context-dependent behavioral variation. Trained without supervision on the I-24 MOTION dataset, the framework recovers three interpretable driving profiles representing three regimes: free flow, transition, and congestion. The profiles capture the behavioral range of the data and the smooth transitions drivers make between regimes as conditions change. The same representation also reproduces known macroscopic phenomena, aligning with the fundamental diagram and reproducing hysteresis loops. We also show how the representation supports practical use: it supplies context-dependent parameters to classical car-following models, and gives an autonomous vehicle a live behavioral read of the surrounding drivers with a short-horizon forecast of their motion. The framework points toward models of traffic that are interpretable and trustworthy by construction. We release an open-source toolkit on GitHub (https://github.com/mselayan/quantum-driver-representation) spanning data processing, training, inference, and analysis.

Disordered traffic flow is characterized by weak or non-existent lane discipline in the presence of strong vehicle heterogeneity and continuous lateral interactions, challenging traditional lane-based modeling assumptions. This study presents an empirical study of macroscopic and microscopic aspects of disordered traffic using high-resolution UAV trajectory data collected on an urban arterial. A two-dimensional extension of Edie's framework is applied to quantify aggregate traffic variables and produce a two-dimensional fundamental diagram, revealing that traffic states cannot be adequately represented using one-dimensional formulations and highlighting the persistent role of lateral redistribution. The propagation of congestion is estimated directly from the spatiotemporal speed fields, demonstrating the emergence of coherent stop-and-go waves and showing a similar dynamics as conventional lane-based flow, in spite of the heterogeneous vehicle interactions. At the microscopic level, steady-state follower-leader identification is used to examine desired time gaps and minimum lateral spacing, vehicle dimension distributions, and kinematic characteristics, revealing pronounced inter-class heterogeneity that explains disordered traffic behavior. The study provides an empirical framework linking vehicle-level interactions and aggregate traffic dynamics and establishes a data-driven basis for the calibration and validation of traffic models for disordered mixed traffic systems.

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.

Accurate trajectory prediction in autonomous driving hinges on modeling dynamic and context-dependent interactions among traffic agents. However, most existing approaches are purely data-driven and lack structural priors, which limits their generalization under distribution shifts. In this work, interaction modeling is revisited through the structure and dynamics of traffic networks, and SWIFT (Small-World Interaction Framework for Trajectory prediction) is proposed as a unified framework that integrates small-world networks with traffic flow theory. SWIFT introduces structural inductive biases via a Small-World Interaction Network that captures both local and global dependencies, and a Flow Regime Encoder that adapts the interaction structure to scene-level traffic states. Interaction reasoning is further enhanced through a multi-relational graph module that explicitly encodes direct and higher-order agent relationships. Extensive experiments on three real-world datasets, nuScenes, MoCAD, and NGSIM, show that SWIFT consistently outperforms strong baselines in prediction accuracy across diverse traffic regimes. Beyond accuracy, SWIFT exhibits improved generalization to unseen locations and regimes, robustness under noisy observations, and strong performance with limited training data, supporting the effectiveness of its structure-aware design.

Mesoscopic traffic flow models combines the merits of both macroscopic and microscopic models by capturing individual vehicle behavior in great detail and remaining the computational efficiency. At the time of this study, the mesoscopic model proposed by Eissfeldt (2004) is used in Simulation of Urban MObility (SUMO). The movement of vehicles is governed by dynamic headways between edges. However, the model does not fully comply with the principle of the Lighthill-Whitham-Richards (LWR) model. Several problems are identified, including the incomplete consideration of queue dynamics and the limited implementation of backward traveling spaces. Two case study scenarios demonstrate that the problems lead to unrealistic onset and recovery pattern of congestion. The magnitude of congestion is generally underestimated with this model. To address these drawbacks, a proper mesoscopic discrete-time implementation of link transmission model, which follows the LWR principle, is proposed. By explicitly incorporating backward traveling spaces to capture queue spillback phenomena, the proposed model provides a more precise representation of congestion dynamics. The link density outputs are consistent with the kinematic wave theory and the microscopic traffic simulation in SUMO, thus verifying its theoretical accuracy.

Active traffic management (ATM) is frequently hindered by traditional macroscopic models and rigid empirical thresholds that fail to capture metastable phase precursors, resulting in delayed, reactive interventions. To address this, we propose SpinFlow, a physics-informed spin-field framework unifying Kerner's three-phase theory with statistical physics for continuous macroscopic traffic phase inference. Inspired by the Heisenberg model, SpinFlow parametrizes spatially varying phase weights via a latent spin vector and a competitive-equilibrium mapping, allowing synchronized flow to emerge naturally. A physics-regularized Expectation-Maximization algorithm inverts this latent structure from high-resolution trajectories, jointly optimizing the spin field while softly enforcing mass conservation and spatial smoothness. We introduce the Phase Equilibrium Degree (PED) to quantify structural alignment and topologically localize phase-transition points. Across four real-world trajectory datasets, SpinFlow achieves $R_{q}^{2}$ up to 0.940, PED drops of 94.9-100%, and interpretable phase maps that outperform three heterogeneous baselines on forward accuracy, physics consistency, and bottleneck localization. SpinFlow pinpoints congestion nucleation without prior network topology, yielding a data-driven, physics-consistent trigger for ATM.

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.