A Decision-Focused Neural Optimization Framework for Personalized Route Reproduction from Vehicle Trajectories

1. Introduction

• The framework treats each observed route as a minimum-cost path conditional on traveler-specific perceived link costs, rather than deriving route-choice probabilities.The introduction distinguishes direct shortest-path matching from likelihood-based route-choice approaches.Source: §1 · pp. 3–4
• Unlike approaches that smooth the argmax, the framework keeps the shortest-path layer discrete in the forward pass and uses an iMLE-based surrogate gradient for backpropagation.The approximate gradient addresses the non-differentiability of the constrained optimization layer while preserving the optimization structure.Source: §1 · p. 4
• The model learns individual-specific perceived link costs from context and uses a full-network shortest-path layer to reproduce observed trajectories without enumerating candidate routes.Inputs include individual characteristics, trip attributes, and network-wide traffic states.Source: §1 · p. 4
• The framework is reported to outperform five baseline models, especially where route diversity is high and heterogeneous choices must be reproduced.The authors frame it as route reproduction, not a likelihood-based route-choice probability estimator.Source: §1 · p. 4
• Because multiple perceived-cost sets can yield the same shortest path, training adds a regularizer that anchors estimated costs to average observed travel times.The stated aim is to align perceived costs with the scale of observed travel costs and penalize extreme deviations.Source: §1 · p. 4
• Retrieved latent cost structures are presented as a way to interpret heterogeneous route choices under identical traffic conditions.The paper contrasts this granular interpretation with discrete choice models relying on aggregate stochasticity.Source: §1 · p. 5

2.1 Discrete choice models of route choice

• Path-based models require advance candidate-set generation and can be sensitive to that set, while substantial route overlap can produce unrealistic substitution under simple multinomial logit.Source: §2.1 · p. 6
• Existing discrete route-choice models often use linear, link-additive utilities, limiting their ability to capture nonlinear effects, richer trip context, and traveler heterogeneity.Source: §2.1 · p. 6

2.2 Learning-based route choice

• Neural path-size models add nonlinear flexibility but retain candidate-set dependence, including dependence on route generation and candidate coverage.Source: §2.2 · pp. 6–7
• Most learning-based route-choice methods target OD-conditional distributions, link-choice policies, or latent rewards rather than a driver’s perceived link-cost distribution.Source: §2.2 · p. 7

2.3 Similar Studies with Latent Travel Costs

• PURC predicts expected link-use or flow vectors for OD pairs, not deterministic individual paths, and does not retrieve individual-specific perceived link costs explaining realized trajectories.Source: §2.3 · p. 7
• The framework learns trip-specific perceived link costs from traveler, trip, and network context, then trains them so the minimum-cost path matches the observed trajectory.The shortest-path solution is the supervised object, rather than an enumerated set of candidate routes.Source: §2.3 · p. 8
• The iMLE method supplies a surrogate gradient by repeatedly evaluating the discrete shortest-path layer, avoiding explicit differentiation of a large-dimensional equilibrium or assignment problem.Source: §2.3 · p. 8

2.4 Decision-Focused Learning and Differentiable Constrained Optimization

• Decision-focused learning is motivated by the fact that higher prediction accuracy need not improve decisions, since estimation errors differ in their effects on the optimal solution.Source: §2.4 · p. 9
• Decision-focused learning combines prediction and optimization in an end-to-end framework, updating model parameters based on downstream task performance rather than prediction error alone.Source: §2.4 · p. 9
• Discrete, non-differentiable optimization prevents ordinary backpropagation through the solver, prompting surrogate-gradient, continuous-relaxation, and perturbation-based methods (Amos and Kolter, 2017).These approaches make it possible to train predictive models through optimization layers.Source: §2.4 · p. 9
• Perturbation-based iMLE preserves the original discrete optimization problem in the forward pass while supplying usable gradient estimates for end-to-end learning.This supports systems evaluated through structured decisions rather than prediction error alone.Source: §2.4 · p. 9

3.1 Behavioral Assumptions

• The model separates deriving subjective link-level perceptions from selecting a path, replacing probabilistic choice modeling with a deterministic solver on latent costs.The resulting route-generation rule directly represents the assumed route-choosing process rather than computing choice probabilities.Source: §3.1 · p. 10
• Perceived link costs vary across travelers and links and are modeled using individual characteristics, trip-specific attributes, and network-level traffic states.The latent cost distribution is allowed to be lognormal.Source: §3.1 · p. 10
• Conditional on perceived link costs, travelers are assumed to choose the deterministic minimum-cost path, with path cost equal to the sum of its link costs.The observed route is treated as the solution to a shortest-path optimization problem.Source: §3.1 · p. 10

3.2 Problem set-ups

• The road network is a directed graph whose nodes and links index trip origins, destinations, and route choices.Each trip is associated with an origin, destination, and departure-time index.Source: §3.2 · p. 10
• Each trip receives its own perceived link-cost vector, and route generation uses shortest-path finding rather than computing choice probabilities.The model-generated route is compared with the observed route in the learning objective.Source: §3.2 · pp. 10–11
• The trip covariate vector parameterizes the perceived travel-cost distribution, with Table 1 summarizing the framework’s core notation.The setup identifies the observed route and generated route separately.Source: §3.2 · p. 11, Table 1 · p. 11

3.3.1 Establishing model architecture

• The perception network maps context covariates to Gaussian link-cost parameters, exponentiates samples to ensure positive perceived costs, and feeds them to the shortest-path layer.The diagonal covariance assumption and exponential transformation reflect a lognormal model for perceived link travel times.Source: §3.3.1 · p. 12

3.3.2 Setting up the original loss function

• The original route-matching loss is the complement of link-based intersection-over-union, but the discrete shortest-path operator makes it piecewise constant and non-differentiable.Its gradient is zero almost everywhere and undefined at path-switching boundaries, preventing direct ordinary backpropagation.Source: §3.3.2 · pp. 12–13

3.3.3 iMLE surrogate loss and gradient

• iMLE replaces direct differentiation with a surrogate whose approximate gradient uses structured link perturbations and repeated shortest-path evaluations instead of enumerating feasible routes.Perturb-and-MAP samples approximate the needed route moments; the surrogate preserves the original objective’s learning signal.Source: §3.3.3 · pp. 14–15
• Route-matching supervision cannot identify a unique set of perceived link costs because scaling all costs, such as doubling them, leaves the shortest path unchanged.The resulting scale ambiguity permits infinitely many perceived-cost sets to reproduce one observed route, a scaling issue the passage says prior iMLE studies had not discussed.Source: §3.3.3 · p. 16
• The model adds regularization to stabilize estimation and reduce scale ambiguity while recovering behaviorally meaningful perceived link costs.Source: §3.3.3 · p. 16

3.3.4 Regularization for cost-scale anchoring

• A KL divergence between Gaussian cost distributions anchors perception-model outputs to the mean and standard deviation of log observed link travel times.The observed travel-time statistics define the baseline distribution that the learned perceived costs are constrained not to stray from.Source: §3.3.4 · p. 16
• The reinforced loss adds the KL term to the original objective with weight ξ, and its derivative with respect to model parameters is analytically computable.Source: §3.3.4 · p. 16
• Training combines perceived-cost sampling, perturb-and-MAP inference, iMLE-based surrogate-gradient estimation, and KL regularization.Source: §3.3.4 · p. 17

3.4 Architecture of the perception model

• Three parallel subnetworks encode personal information, trip context, and network state; their representations are consolidated into the mean and variance of the perceived-cost distribution.The input covariates are partitioned among the three encoders.Source: §3.4 · p. 18

3.4.1 Personal encoder

• With no explicit sociodemographic data, the personal encoder uses past same-driver kinematic summaries, including jerk, acceleration, and speed, to represent driving habits.The features are computed only from that driver’s trajectories collected before each trip’s departure.Source: §3.4.1 · p. 19

3.4.2 Trip context encoder

• The trip-context encoder conditions trip attributes on the personal representation and uses endpoint context, including land use within 500 meters; its simple spatial scheme reportedly outperforms rigorous 2-D positional encoding.Inputs include temporal variation and origin-destination information.Source: §3.4.2 · p. 19

3.4.3 Network state encoder

• A graph convolutional network encodes network conditions, combining static link characteristics with dynamic traffic measures such as observed travel-time means and variances.The encoder is intended to capture spatiotemporal traffic propagation across the urban network.Source: §3.4.3 · p. 20

3.4.4 The final encoder

• The final encoder maps consolidated features to perceived-cost distribution parameters, samples link costs through reparameterization, and regularizes them toward observed travel-cost distributions.Reparameterization keeps sampling differentiable for end-to-end training.Source: §3.4.4 · pp. 20–21

4.1 Testbed and Data Preparation

• The testbed covers three Seoul districts on a directed network of 7,723 nodes and 14,769 links, with GPS trajectories map-matched to links.Trips were restricted to those with both endpoints inside the study area.Source: §4.1 · p. 22
• Drivers were assumed to know departure-time ground-truth link travel times, while analysis suggests stable subjective perceptions especially within origin–destination sub-networks.Observed travel times serve as model inputs; the study also prepared individual speed-profile, endpoint land-use, and static and dynamic network covariates.Source: §4.1 · p. 23
• The refined test dataset contains 3,668 trips across 546 OD pairs, selected with at least five observed trips per pair and a unique-route-to-trip ratio exceeding 0.9.The complete dataset was randomly partitioned into training (80%) and testing (20%) sets.Source: §4.1 · p. 24

4.2 Baseline Models for Comparative Analysis

• Five reference models span path-based PSL and DNN-PSL and link-based ReL, AIRL, and PURC formulations.Path-based models use k=10 candidate routes; ReL avoids path enumeration, AIRL learns context-dependent rewards and policies, and PURC uses static inverse optimization.Source: §4.2 · p. 24, ReL · p. 25

ReL

• Because PSL and DNN-PSL predict distributions over candidate paths, their route-reproduction scores aggregate top-10 path metrics by estimated choice probabilities; candidate sets may omit actual routes.This expectation-based post-processing supports comparison with the proposed individual-route framework rather than relying on stochastic sampling.Source: ReL · p. 25
• Thirty sampled routes are matched to each observed route by maximum overlap, giving stochastic link-based baselines favorable, upper-bound scores.The comparison uses a best-matching sampled route rather than deterministic route-level prediction accuracy.Source: ReL · p. 26
• PURC’s link flows are normalized into transition probabilities; thirty random-walk routes are sampled, and the most frequent route is its prediction.This converts PURC’s nondeterministic flow output into a route-level prediction for comparison.Source: ReL · p. 26

4.3 Evaluation Metrics

• Route reproduction is evaluated with IoU as the primary metric, supplemented by TLLA and edit distance, using test-set average scores unless noted.IoU and TLLA reward higher values, while edit distance rewards lower values.Source: §4.3 · p. 26

Intersection-over-Union on links (IoU)

• IoU measures set-level overlap between predicted and observed links on a 0–1 scale, with 1 indicating perfect overlap.It is the primary route-reproduction metric.Source: Intersection-over-Union on links (IoU) · p. 26

Trip Length by Links Accuracy (TLLA)

• TLLA measures the proportion of observed route length covered by the predicted route, with higher values indicating better reproduction.It accounts for link lengths in the overlapping route segments.Source: Trip Length by Links Accuracy (TLLA) · p. 27

Edit Distance (ED) on link sequences

• Normalized edit distance measures order-sensitive dissimilarity between link sequences using insertions, deletions, and substitutions; lower values indicate better reproduction.Normalization makes scores comparable across trips of different lengths.Source: Edit Distance (ED) on link sequences · p. 27

5.1 Route Matching Performance and Comparative Analysis

• IoU 0.666, TLLA 0.926, and ED 0.302; the proposed framework outperforms all reference models on the test set.Path-based baselines use a pre-constrained candidate route set, whereas the proposed model predicts directly over the entire network.Source: Table 3 · p. 28, §5.1 · p. 28
• IoU 0.835, TLLA 0.939, and ED 0.153; the proposed framework outperforms all competing methods on the diversity subset.Its improvement over baselines is larger in this more complex choice setting, supporting the value of individual-specific perceived travel times for distinguishing route choices.Source: Table 4 · p. 29, §5.1 · p. 29
• The proposed model outperforms baselines across all tested dimensions and levels, and its performance gap widens as path length increases.For TLLA, the proposed model and PURC remain stable as paths lengthen, while link- and path-based baselines deteriorate; the proposed model slightly improves and is slightly better than PURC on long, complex paths.Source: §5.1 · p. 30, Figure 6 · p. 31

5.2 Ablation Analysis

• Removing KL regularization causes the largest degradation, while replacing the GCN with feed-forward layers also reduces route reproduction performance.The GCN-based encoder is described as helping preserve network topology.Source: Table 5 · p. 31, §5.2 · p. 32
• Removing personal, trip-context, or network-state inputs has a larger performance cost on the diversity subset and as route complexity increases.The three types of contextual information play complementary roles in resolving more ambiguous route choices.Source: §5.2 · p. 32, Table 6 · p. 32, Figure 7 · p. 33

5.3 Interpretation of Perceived Travel Times at Network Level

• The framework externalizes latent perceived travel times to interpret individual route-choice motives, and findings suggest learning naturally regularizes a driver’s perception.With calibration, the model estimates personalized perceived link travel times within the testbed.Source: §5.3 · p. 33
• Perceived-cost deviations concentrate near trip origins, destinations, and connecting corridors, while remote links remain near baseline; the perception area expands with OD distance.This localized pattern emerged without explicit spatial constraints or priors.Source: §5.3 · pp. 34–35, Figure 8 · p. 34
• Perception-area radius generally increases with OD distance, and its variance grows for longer trips, indicating more heterogeneous spatial configurations and route-choice strategies.The spatial pattern emerged autonomously from route-data training, without explicit spatial priors.Source: §5.3 · p. 35, Figure 9 · p. 35
• Perceived travel-time variability is greater on freeways than arterials, yet perceived variations across links are generally less pronounced than variations in observed traffic states.Observed travel-time variance does not distinctly differ between freeways and arterials.Source: §5.3 · pp. 35–36, Figure 10 · p. 36

5.4 Interpretation of Individual-Specific Route Choice Behaviors

• Under observed network conditions, the objective shortest path took 11.5 minutes, while traveler A chose a 13.6-minute route and traveler B a 12.6-minute route, both longer than that path.Traveler A perceived the chosen route at 17.7 minutes and the objective shortest path at 25.9 minutes; traveler B underestimated the chosen route’s cost at 12.0 minutes.Source: §5.4 · pp. 37–38
• Route deviations reflect localized perceptions: both travelers overestimated an on-ramp’s delay, while different perceived costs at a weaving area led A and B to diverge.A avoided the weaving area; B considered it endurable but detoured earlier to avoid the on-ramp.Source: §5.4 · p. 38
• The model distinguishes within-driver perceptual variance from between-driver mean perceived travel times, revealing different perceptions for D and E despite identical observed routes and disjoint alternatives.Source: §5.4 · p. 39
• Despite sampling across the network, the model generates no irrational alternatives far from the OD corridor, indicating that decision-focused learning constrains stochastic routes to remain behaviorally plausible.Source: §5.4 · p. 40
• Under identical disruptions, H rerouted at 1.5 minutes, F at 6 minutes, and G only at 9 minutes, showing traveler-specific resistance to network changes.Source: §5.4 · pp. 40–41
• H detours near the origin to avoid disruption, whereas F and G stay on the original corridor longer and divert near affected links.Source: §5.4 · p. 41
• The framework predicts who reroutes, at what threshold, and where, supporting personalized traffic management when incidents or other events cause sudden travel-time surges.Source: §5.4 · p. 41

• The individual-level framework bypasses path enumeration and overlapping problems, while decision-focused learning aligns shortest-path solutions with observed behavior and calibrates latent behavioral parameters end to end.Source: §6 · p. 42
• The framework consistently outperforms five baseline models, with particular strength in reproducing heterogeneous choices across OD pairs with high route diversity.Source: §6 · p. 42
• As path length and travel time increase, the framework’s predictive capability is reinforced, maintaining high accuracy across long-distance paths.Source: §6 · p. 42
• Retrieved latent perceived travel times offer a way to interpret route-choice motives; simulations find divergent choices under identical traffic conditions and a spatially bounded cognitive range focused on the immediate trip corridor.The analysis connects individualized perception fields with heterogeneous decisions and reports that travelers’ cognitive range is primarily spatially bound to the trip corridor.Source: §6 · p. 42
• The study lacks socio-demographic covariates, constraining granular retrieval of personalized travel times; the framework currently targets individualized navigation guidance rather than downstream traffic assignment.Source: §6 · p. 42

Appendix A. Hyperparameters and experimental settings

• Table A1 lists the hyperparameter settings used for training.Source: Table A1 · p. 48
• κ was fixed to an empirical constant reflecting the average number of links in observed routes, for computational simplicity.Source: Appendix A · p. 48
• Table A2 lists constant values and covariate dimensions.Source: Table A2 · p. 48

Appendix C. Model specifications of reference models

• Table C1 presents the model specifications of the reference models.Source: Table C1 · p. 50