Agentic discovery of interpretable car-following laws from multi-source trajectory data

Jiamei Xu; Jie Sun; Keshuang Tang; Jian Sun

cs.ROOct 20, 2026

Agentic discovery of interpretable car-following laws from multi-source trajectory data

Authors: Jiamei Xu, Jie Sun, Keshuang Tang, Jian Sun

Organizations: College of Transportation Engineering & Key Laboratory of Road and Traffic Engineering of the Ministry of Education, Tongji University, Shanghai, 201804, China

Abstract

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.

Figures & tables

Research direction	Representative studies	Main approach
I. Analytical Car-Following Models
Classical CF modeling	GHR, Gipps, Newell, OVM, FVDM, IDM	Explicit mathematical formulations based on predefined behavioral assumptions and traffic flow theory
Trajectory-based calibration	Kesting and Treiber (2008)	Numerical optimization of model parameters within prescribed analytical structures
II. Data-Driven and Symbolic Discovery of CF Models
Data-driven CF modeling	Locally weighted regression, LSTM	Learning nonlinear vehicle-response relationships directly from observed trajectories
Neural-symbolic discovery	SciNet-CFM	Combining neural representations and symbolic regression to discover interpretable CF relationships

Table 1: Comparison of representative approaches to car-following modeling and empirical equation discovery.

Fig. 1: Overall framework of the proposed agent-centered, physics-aware car-following law discovery method.

Dataset	Traffic environment	Role
NGSIM US-101	Freeway traffic	Discovery
highD	Highway traffic	Discovery
exiD	Highway entry and exit areas	Discovery
Shanghai NDS	Naturalistic driving	External
NGSIM I-80	Freeway traffic	External
Waymo	Autonomous-vehicle-recorded traffic	External

Table 2: Datasets used for empirical CF-law discovery and evaluation.

Component	Configuration
Sampling interval	$0.1$ s
Subtree crossover probability	$0.4$
Coefficient optimization	Bounded multistart Nelder–Mead
Acceleration clipping bounds	$[-8,4]$ m/s 2
Minimum safeguarded spacing	$0.5$ m
Directional-test tolerance	$0.05$ m/s 2

Table 3: Experimental configuration for empirical CF-law discovery.

Parameter	Configuration
Encoding	Real-valued
Population size	64
Maximum generations	10,000
Elite individuals	4
Selection	Tournament (size 3)
Crossover probability	0.9

Table 4: GA configuration for external CF-model calibration.

Parameter	Unit	Physical interpretation
$k$	–	Relative-speed response intensity
$\lambda$	s -1	Equilibrium-speed restoration rate
$v_{0}$	m/s	Desired free-flow speed
$s_{0}$	m	Standstill equilibrium spacing
$T$	s	Desired time-headway scale

Table 5: Parameters and physical interpretations of AERM.

Configuration	Discovered CF equation
Full Framework (AERM)	$\displaystyle\begin{aligned} a={}&\lambda\left[\frac{v_{0}(s-s_{0})}{s-s_{0}+v_{0}T}-v\right]+k\frac{v}{s}\left(1+\frac{s_{0}}{s}\right)\Delta v\end{aligned}$
w/o Agent-Guided Pathway	$\displaystyle a=k_{1}\frac{\Delta v}{s}+k_{2}s-\frac{k_{3}}{s}-k_{4}v+k_{5}$
w/o Evolutionary Pathway	$\displaystyle\begin{aligned} a={}&\left(\frac{k_{1}}{s}-k_{2}v\right)\Delta v+\lambda\left[v_{0}\tanh\left(\beta(s-s_{0})\right)-v\right]\end{aligned}$
w/o Physical Constraints	$\displaystyle\begin{aligned} a={}&\lambda(v_{0}-v)+\frac{\Delta v-\Delta v_{0}}{s\left[c_{1}\|\Delta v\|(c_{2}\|v\|)^{m}\right]^{n}}\end{aligned}$

Table 6: Analytical CF equations discovered under different ablation configurations.

Configuration	Gap RMSE (m) $\downarrow$	Speed RMSE (m/s) $\downarrow$	Collision rate (%) $\downarrow$
Full Framework	1.2683	0.8454	0.0407
w/o Agent-Guided Pathway	4.4877	1.0800	0.0407
w/o Evolutionary Pathway	4.5314	1.0812	0.0407
w/o Physical Constraints	8.7521	1.4986	0.0407

Table 7: Ablation results of the proposed CF-law discovery framework on NGSIM I-80.

Panel of Figure 9 — Fig. 2: Model-implied fundamental diagrams of AERM: speed–density relationship (left) and flow–density relationship (right).

Panel of Figure 10 — Fig. 3: Spatiotemporal speed evolution under two AERM parameter configurations. (Left) Congestion dissipation: $k=1.05$ , $\lambda=0.03$ s -1 . (Right) Congestion persistence: $k=0.25$ , $\lambda=0.01$ s -1 . Shared parameters are $v_{0}=40$ m/s, $s_{0}=1.042$ m, and $T=0.509$ s.

	Shanghai NDS		NGSIM I-80		Waymo
Model	Gap	Speed	Gap	Speed	Gap	Speed
GHR	10.0970	0.9948	5.6453	1.2050	5.5636	0.8506
Helly	9.3016	1.0749	5.3515	1.2242	5.9596	0.9303
Gipps	9.6389	1.0731	5.1731	1.1270	5.1695	0.8484
Newell	11.0970	1.7086	6.1859	1.3207	7.7446	1.4549
OVM	12.7097	1.9593	6.3365	1.4418	22.1557	2.4185

Table 8: Trajectory reconstruction performance with dataset-level common parameters.

	Shanghai NDS		NGSIM I-80		Waymo
Model	Gap	Speed	Gap	Speed	Gap	Speed
GHR	4.4980	0.7551	2.8763	1.0125	2.0739	0.5697
Helly	2.0142	0.5962	1.5750	0.8954	0.9933	0.4322
Gipps	1.7206	0.5289	1.3725	0.8658	0.6683	0.3551
Newell	2.0394	0.6002	1.7771	0.9102	1.0450	0.4665
OVM	2.1060	0.6162	1.8906	0.9504	1.3486	0.5488

Table 9: Trajectory reconstruction performance with trajectory-specific parameters.

Fig. 4: Comparison of observed and reconstructed CF trajectories using AERM and IDM on Shanghai NDS.

Fig. 5: Comparison of observed and reconstructed CF trajectories using AERM and IDM on NGSIM I-80.

Fig. 6: Comparison of observed and reconstructed CF trajectories using AERM and IDM on Waymo.

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Car-following behavior is fundamental to traffic flow theory, yet traditional models often fail to capture the stochasticity of naturalistic driving. This paper proposes an empirical probabilistic sampling approach to car-following modeling that bypasses conventional parametric assumptions. Under this approach, we introduce the Markov Chain Car-Following (MC-CF) model, which represents state transitions as a Markov process and predicts behavior by randomly sampling accelerations from empirical distributions within discretized state bins. Evaluation on the Waymo Open Motion Dataset (WOMD) demonstrates that MC-CF variants significantly outperform all physics-based baselines (IDM, Gipps, FVDM, and SIDM) across both one-step and open-loop trajectory prediction metrics, and remain competitive with modern data-driven baselines including neural network and Gaussian mixture model approaches. Zero-shot generalization on the Naturalistic Phoenix (PHX) dataset further confirms cross-domain transferability. Finally, microscopic ring road simulations validate the framework's scalability: by incrementally integrating unconstrained free-flow trajectories and high-speed freeway data (TGSIM) alongside a conservative inference strategy, the model substantially reduces collisions across most tested scenarios and successfully reproduces naturalistic and stochastic shockwave propagation, though crashes persist under severe shockwave conditions. Overall, the proposed MC-CF model provides a robust and scalable foundation for simulating population-level stochastic traffic behavior that requires no behavioral parameter calibration, making it well-suited for the data-rich future of intelligent transportation.

Abstract

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