cs.LGAug 26, 2026
SaveQuantum-Inspired Modeling of Driving Behavior
Organizations: Civil & Environmental Engineering, University of Nebraska–Lincoln, Lincoln, NE, USA
Abstract
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.
Figures & tables
Figure 1: Conceptual motivation for the proposed representation. Drivers differ from one another, change over time, respond to context, and remain probabilistic. The framework represents these properties through an evolving density-matrix state in a nonlinear feature space, allowing behavioral profiles and within-profile modes to emerge from observed behavior and context, and ultimately supporting macroscopic traffic phenomena and prediction.
Figure 2: The design philosophy behind the framework. The same heterogeneous driver behavior can be represented in two ways. Imposing fixed interactions forces behavior into a predefined template, causing unmatched variation to appear as noise and preventing the emergence of the correct macroscopic structure. Learning interactions directly from the data preserves this variation and reproduces the observed traffic patterns while remaining interpretable. A behavioral representation is therefore trustworthy only if what it learns aggregates into known traffic phenomena.
Figure 3: Heterogeneity under three fitted representations on a common leader-follower simulation. (a) Gap to the leader over time for drivers d1 – d4 . (b) Hard K-means clustering conflates d2 and d3 . (c) A small neural network yields latent coordinates with no direct behavioral interpretation. (d) The density-matrix representation describes each driver as a soft mixture over three behavioral profiles with within-profile modes.
Figure 4: Two simulated drivers ( d1 and d2 ) responding to a leader as traffic density rises into congestion. (a) Speeds over time against rising density. (b) A static representation calibrates one fixed IDM per driver, placing them in rigid behavioral classes insensitive to context. (c) A fixed mixture assigns constant proportions over three reference IDM regimes, differing between drivers but not over time. (d) The density-matrix representation resolves how each driver’s composition shifts among free-form behavioral profiles (not governed by IDM formulation) over time, capturing when and how sharply each driver transitions.
Figure 5: Acceleration versus headway across three speed ranges. (a) A single global regression misses the speed interaction. (b) A Gaussian mixture with few components averages across regimes. (c) A small neural network predicts accurately but yields no interpretable profile structure. (d) TThe density-matrix representation assigns observations to three profiles; the inset shows the mixture weights varying smoothly with speed, and the intermediate range (green) is captured as its own profile.
| Concept | Meaning |
|---|---|
| Behavioral vector | A short vector of observed driving quantities at a single instant. This is the driver’s action, measured directly. |
| Context | A vector of variables describing the surrounding traffic situation. Context is what the driver is responding to and is read from the neighborhood. |
| State | A density matrix at time that summarizes everything the model has learned about the driver from past behavior and past context. It also evolves through context-driven predictions and observed behavior. |
| Profile | A density matrix representing a recurring pattern of driving behavior. For example, a profile can represent congested car-following, characterized generally by lower speeds and shorter spacing. |
| Profile mixture | A weighted combination of profiles for a driver at time , where the weights depend on the current traffic context. As context changes, the contribution of each profile changes smoothly. For example, increasing traffic density may shift weight from a free-flow profile toward a congested car-following profile. |
| Modes within a profile | Each profile can contain multiple behavioral modes, with weights indicating their relative importance. For example, a congested car-following profile may include both smooth and aggressive car-following behaviors, with different weights assigned to each mode. |
Table 1: Core concepts used throughout the framework.
| Type | Variable | Description |
|---|---|---|
| Behavioral | Speed ( ) | Magnitude of longitudinal velocity (ft/s) |
| Behavioral | Spacing ( ) | Gap to same-lane leader (ft) |
| Behavioral | Jerk ( ) | Rate of change of acceleration (ft/s 3 ) |
| Contextual | Density ( ) | Vehicles within a 150 m radius |
| Contextual | Speed entropy ( ) | Entropy of neighbors’ 1 s speed changes |
| Contextual | Acceleration entropy ( ) | Entropy of forward neighbors’ accelerations |
Table 2: Selected behavioral and contextual variables.
Figure 6: The RFF projection. The three behavioral variables (speed, spacing, jerk) form a short vector of size . The random Fourier projection maps it into a much longer feature vector of size , where each feature is one nonlinear combination of the inputs, . The expanded representation lets the model separate behaviors that look similar in the original variables.
Figure 7: Structure of a profile density matrix , shown as a truncated matrix over the RFF feature indices. Diagonal terms (blue) give the weight of each RFF feature in the profile. Off-diagonal terms (red) give the coupling between two features, which enables the profile to represent interactions among behaviors. The eigendecomposition of yields its behavioral modes (green): each bar is one mode, and its height is the corresponding eigenvalue, indicating how strongly that mode contributes to the profile.
Figure 8: Estimation pipeline of the quantum-inspired driver behavioral profiling framework. Trajectories are reduced to behavioral and contextual variables, the behavioral state is mapped through a RFF projection, and the per-driver density-matrix state is evolved by context-driven profile mixture and observation-driven update. Profiles and context coefficients are estimated by minimizing the NLL and interpreted through eigendecomposition.
Figure 9: Spatio-temporal speed field for westbound I-24 traffic. Color encodes mean speed from low (red) through moderate (yellow) to high (green).
Figure 10: Speed (a) and spacing (b) characteristics of the eligible westbound population. All four lanes follow a common congestion profile, with modest cross-lane variation in onset and recovery. Spacing is concentrated at short following distances but extend across the full observation window.
| Profile 1 | 0.999 | 0.001 | |||
| Profile 2 | 0.852 | 0.077 | 0.034 | 0.020 | 0.011 |
| Profile 3 | 0.998 | 0.002 |
Table 3: Top eigenvalues of each profile. Profile 2 carries multi-modal structure that the other two profiles lack.
| Profile 1 | Profile 2 | Profile 3 | |
|---|---|---|---|
| Profile 1 | — | ||
| Profile 2 | — | ||
| Profile 3 | — |
Table 4: Pairwise Frobenius distances between profiles. The geometric maximum for any two rank-one density matrices is .
Figure 11: Empirical behavioral signatures per profile. Each row shows of speed, spacing, and jerk over the frames dominated by that profile.
| Profile | |||
|---|---|---|---|
| Profile 1 | |||
| Profile 2 | |||
| Profile 3 |
Table 5: Pairwise variable interaction, measured by mutual information (in nats), under each profile.
Figure 12: Joint distributions of speed and jerk for the three profiles. Dashed and solid contours enclose the and density regions, respectively.
Figure 13: Learned context coefficients for each profile. Positive coefficients increase a profile’s contribution as the context variable rises; negative coefficients suppress it.
Figure 14: Empirical signatures of the three modes within Profile 2. Markers show mode means and cross-hairs indicate standard deviation for the frames associated with each mode.
Figure 15: Example Profile 2-dominated trajectory from I-24 MOTION. Speed, jerk, and spacing are shown over time, with background shading indicating the dominant mode of Profile 2 at each frame. Mode changes track the sign of jerk, which shows congested driving is separated by acceleration behavior.
Figure 16: State evolution for a driver entering congestion. The bottom panel shows the profile composition over time. As traffic conditions change, the driver’s state transitions smoothly from Profile 1 (free flow) to Profile 3 (transition) and finally to Profile 2 (congestion).
Figure 17: Fundamental diagram for the westbound I-24 MOTION dataset, with cells colored by the dominant inferred profile. Profile 1 occupies the low density free flow region, Profile 3 dominates near capacity, and Profile 2 occupies the congested branch.
Figure 18: Hysteresis on a westbound I-24 MOTION wave (lane 1). Top left: the wave on the time-space diagram. Top right: the resulting traced through a representative trajectory. Bottom: the average profile composition at each point.
Figure 19: Integration of the representation with IDM. The model’s equation is unchanged. Its target parameters are supplied by the evolving driver state and adapt to context.
Figure 20: Representation-supported car following. A single leader (purple) drives through a free-flow, stop-and-go and recovery cycle. A standard IDM follower (dashed) uses fixed parameters. A representation-supported IDM follower (red) uses the same IDM equation but reads its target parameters from the evolving state . Top: speed. Middle: position. Bottom: profile composition , shifting from Profile 1 (free flow) through Profile 3 (transition) to Profile 2 (congestion) as context changes.
Figure 21: Short-horizon ( ) forecast of human-driver speed from the trained state representation, evaluated on lane 1 of I-24 MOTION under calm, busy, and between-waves conditions. Each point is one vehicle; axes compare forecast and realized speed at the horizon, with the diagonal marking perfect prediction.
Figure 22: Profile composition along lane 1 of I-24 MOTION at three different instants. Each bar is one vehicle at its corridor position, split into profile weights . Inferred vehicles cover roughly of the lane-1 population.
Appendix figures & tables8 assets
Supplementary material from the paper’s appendix.
Appendix
| Code | Variable | Definition |
|---|---|---|
| Behavioral variables | ||
| B1 | Speed | Longitudinal speed |
| B2 | Acceleration | Longitudinal acceleration |
| B3 | Jerk | Rate of change of acceleration |
| B4 | Lateral velocity | Lateral velocity |
| B5 | Lateral acceleration | Lateral acceleration |
Table A1: Candidate variables.
| Code | Type | WCS | BCS | TE | CDC | A | G | NaN |
|---|---|---|---|---|---|---|---|---|
| B1 | beh. | 0.53 | 0.98 | 0.07 | 0.86 | 0.61 | 0.42 | 0.00 |
| C1 | ctx. | 0.80 | 0.69 | 0.19 | 0.77 | 0.61 | 0.53 | 0.00 |
| C4 | ctx. | 0.36 | 1.07 | 0.12 | 0.84 | 0.60 | 0.44 | 0.04 |
| C3 | ctx. | 0.29 | 1.10 | 0.11 | 0.87 | 0.59 | 0.42 | 0.04 |
| C2 | ctx. | 0.87 | 0.56 | 0.16 | 0.70 | 0.57 | 0.48 | 0.00 |
| C8 | ctx. | 0.96 | 0.30 | 0.45 | 0.44 | 0.54 | 0.49 | 0.04 |
Table A2: Variable ranking results based on the four evaluation criteria.
| Code | Name | Rationale |
|---|---|---|
| Behavioral | ||
| B1 | Speed | Highest-scoring behavioral variable; strong context-sensitivity and driver consistency; no missing data. |
| B7 | Spacing | Principal car-following measure; strong driver-specific signal. |
| B3 | Jerk | The only strongly time-evolving behavioral variable; retained so the state is not static. |
| Context | ||
| C1 | Omni density | Strongest context variable; balanced across criteria; no missing data. |
Table A3: Selected variables.
Figure B1 : Selection of the number of behavioral profiles . Columns correspond to models estimated with under identical training conditions. The top row shows profile eigenvalue composition, the middle row shows profile dominant mode concentration in speed–spacing space, and the bottom row shows context-activation coefficients. The per-observation NLL is reported above each column. Profile colors are not comparable across columns.
Figure B2 : Selection of the entropy regularization weight . Columns correspond to models estimated with under identical training conditions. The top row shows the profile eigenvalue spectrum , the middle row shows the profile dominant-mode locations in speed–spacing space, and the bottom row shows context-activation coefficients. The per-observation NLL is reported above each column. Profile colors are not comparable across columns.
Figure B3 : Selection of the behavioral variable set. Columns correspond to models estimated with behavioral sets , , and . The top row shows the profile eigenvalue spectrum, the middle row shows the profile dominant-mode locations using the two variables in that ablation, and the bottom row shows context-activation coefficients. The per-observation NLL is reported above each column and is not comparable across columns because each model represents a different number of behavioral variables. Profile colors are not comparable across columns.
Figure B4 : Selection of the context variable set. Columns correspond to models estimated with density alone, density and speed entropy, and the full set of density, speed entropy, and acceleration entropy. The top row shows the profile eigenvalue spectrum, the middle row shows the profile dominant-mode locations in speed–spacing space, and the bottom row shows context-activation coefficients. The best per-observation NLL is reported above each column. Profile colors are not comparable across columns.
Figure B5 : Selection of the feature dimension . Columns correspond to models estimated with . The top row shows the profile eigenvalue spectrum , the middle row shows the profile dominant-mode locations in speed–spacing space, and the bottom row shows context-activation coefficients. The per-observation NLL is reported above each column. Projected to the training scale used for the main results in this paper ( egos, epochs), the wall-clock cost is approximately , , and for respectively. Profile colors are not comparable across columns.