Looking Ahead and Back: Anticipatory Generative Modeling for Road-Network Trajectories

Zhenhao Huang; Li Sun; Philip S. Yu

cs.AIOct 9, 2026

Looking Ahead and Back: Anticipatory Generative Modeling for Road-Network Trajectories

Authors: Zhenhao Huang, Li Sun, Philip S. Yu

Organizations: North China Electric Power University · Beijing University of Posts and Telecommunications · University of Illinois Chicago

Abstract

Trajectory generation underpins urban applications such as traffic management and urban planning. Within road networks, vehicle movement along road segments is largely deterministic, whereas transitions at intersections are critical--multiple outgoing paths may appear equally plausible, yet they can lead to substantially different downstream routes. Existing generative methods such as diffusion models simulate location sequences and fail short of next transition selection, even producing trajectories that drift off the road network. To fill this gap, we advocate a fundamental shift: moving beyond location simulation toward behavior modeling on the road network. In this paper, we revisit trajectory generation through the lens of generative route planning, and introduce a new graph-constrained generative paradigm, Anticipatory Generative Modeling, which resolves next transition selection by incorporating the Anticipation of how the future route may unfold toward the destination. Concretely, we propose a novel generative model named Foresight that formulates a probabilistic model for an "anticipate--select--update--re-anticipate" loop that enables longer-term foresight while continuing sequential generation along the historical trajectory. Extensive experiments demonstrate that Foresight wins state-of-the-art baselines.

Figures & tables

Panel of Figure 1 — Figure 1: Motivated example for anticipatory generative modeling.

Figure 2: Overview of Foresight . Further architectural details are provided in Appendix C .

	Porto				San Francisco				Beijing
Method	D-JSD	R-JSD	HD	EDR	D-JSD	R-JSD	HD	EDR	D-JSD	R-JSD	HD	EDR
Dijkstra	2.47	1.31	0.481	0.358	2.18	1.29	0.399	0.305	2.64	1.56	0.449	0.341
Yen- $k$	2.36	1.29	0.503	0.386	2.34	1.39	0.497	0.377	2.78	1.67	0.552	0.414
Markov	2.61	1.46	0.613	0.462	2.39	1.42	0.602	0.458	2.88	1.72	0.681	0.503
SemRoute	1.68	1.07	0.514	0.384	1.69	0.91	0.481	0.363	2.16	1.19	0.548	0.414
HOSER	1.57	0.96	0.431	0.317	1.82	0.98	0.412	0.306	2.04	1.09	0.463	0.351

Table 1: Overall performance across cities. D-JSD/R-JSD denote Distance/Radius JSD and are reported in units of

10^{-2}

. HD is Hausdorff distance and EDR is edit distance on real sequences. Cells report OD-bootstrap means under the common evaluation protocol (see Appendix E ). Lower is better.

	Porto				San Francisco
Variant	D-JSD	R-JSD	HD	EDR	D-JSD	R-JSD	HD	EDR
Foresight -F	1.53	0.79	0.401	0.312	1.39	0.71	0.386	0.298
Foresight -M	1.01	0.71	0.375	0.285	2.68	0.65	0.542	0.415
w/o $\mathcal{L}_{F,t}$	2.25	1.35	0.500	0.390	2.10	1.25	0.480	0.380
w/o $\ell_{t}^{\mathrm{comp}}$	2.55	1.55	0.550	0.420	2.40	1.45	0.530	0.410
w/o $E_{\mathrm{fut}}$	2.05	1.20	0.460	0.360	1.95	1.15	0.450	0.350

Table 2: Ablation results for Porto and San Francisco. JSD values are scaled by

10^{-2}

Panel of Figure 3 — Figure 3: Porto spatial comparison. Top: road network and test-set trajectory distribution. Bottom: trajectory distributions generated by Foresight and Dijkstra.

Figure 4: Matched Porto route comparison for two fixed test OD conditions.

Appendix figures & tables26 assets

Supplementary material from the paper’s appendix.

Appendix

Symbol	Meaning
Trajectory, state, and legal events
$\mathcal{G}=(\mathcal{V},\mathcal{E})$	Directed road graph.
$X_{t}\in\mathcal{V}$	Location at step $t$ .
$X_{0:T}$	Location trajectory; $T$ is an index (and may be the endpoint of a realized trajectory).
$g\in\mathcal{V}$	Destination location.
$A_{t}=\psi(X_{t},X_{t-1})$	Direction/fiber attribute induced by the incoming movement; $A_{0}$ is the initialization token.

Appendix

Table 3: Core notation used in the paper.

Figure 5: Future Anticipation Decoder. The structured Transformer maps the causal history $c_{t}$ and destination $g$ to global-slot latents, future hidden representations, and global road-space predictions. The downstream hidden tail is passed to the Transition Network; candidate-conditioned re-decoding reuses the same decoder after a hypothetical legal movement.

row $\backslash$ key	$C$	$F$	$g$	$G$	$T$
$C$	prefix	no	no	no	no
$F_{h}$	yes	$F_{\leq h}$	yes	yes	yes
$g$	no	no	self	no	no
$G$	yes	no	yes	all $G$	no
$T$	no	no	no	no	self

Appendix

Table 4: Visibility mask for the Future Anticipation Decoder. Padding masks additionally remove invalid rows and keys.

Figure 6: Topology-constrained Transition Network. A candidate-independent readout summarizes $c_{t}$ , $g$ , and the downstream Future hidden tail. Complete legal node–fiber candidates are scored afterward with a projected dot product and normalized only over $\mathcal{U}_{t}$ ; the STOP event uses its independent scalar head. Candidate-conditioned future evaluation is a separate branch performed after the base distribution.

	context	query 1	query 2	query 3	$\cdots$	query $H$
Current	$c_{t}$	$t+1$	$t+2$	$t+3$	$\cdots$	$t+H$
Branch	$c_{t}\\|s^{\prime}$	$t+2$	$t+3$	$t+4$	$\cdots$	$t+H+1$

Appendix

Figure 7: Overlap alignment for candidate-conditioned re-anticipation. Predictions are compared only when they refer to the same absolute future position.

Figure 8: Closed-loop generation. At each decision, Foresight anticipates a fixed future window from the realized history and goal, scores the graph-legal events, optionally evaluates candidate-conditioned future consistency, and selects one movement or STOP. A movement updates the history and triggers a fresh Future Anticipation Decoder call; STOP terminates only at the goal, while budget exhaustion is reported as a timeout.

Current condition	Event support and outcome
Non-goal with legal outgoing edges	Movement candidates only; STOP is masked.
Goal with legal outgoing edges	Movement candidates and STOP compete under $q_{t}$ (or $\pi_{t}$ ).
Goal with no outgoing edges	STOP is the unique legal event.
Non-goal with no outgoing edges	Empty support; return invalid_transition .
STOP selected at the goal	End without appending a state; success=true .
Budget exhausted before STOP	End with timeout ; success=false .

Appendix

Table 5: Termination and boundary semantics during generation.

Scale	$d_{X}=d_{F}=d_{T}$	$d_{V}$	base hidden	$M$	$H$	$L_{E}$	$L_{D}$	$L_{T}$	FFN ratio
s1	72	64	144	16	64	2	2	2	$4\times$
s2	96	64	192	24	64	3	2	2	$4\times$
s3	152	64	304	32	64	4	3	3	$4\times$
s4	240	64	480	48	64	6	4	4	$4\times$
s5	400	64	800	64	64	8	6	6	$4\times$

Appendix

Table 6: Resolved architecture dimensions for the five model scales.

H

is fixed at 64 in every configuration.

City ( $\|\mathcal{V}\|$ )	s1	s2	s3	s4	s5
Beijing (40,060)	3,300,102	4,628,670	8,448,638	17,366,670	44,881,710
Porto (11,024)	1,180,474	1,812,178	4,006,130	10,368,994	33,238,274
San Francisco (27,187)	2,360,373	3,379,989	6,479,069	14,264,277	39,719,637

Appendix

Table 7: Total trainable parameters (Future plus Transition) by city and model scale. Counts include the Future vocabulary head and exclude optimizer state, gradients, activations, and runtime caches.

Dataset	Roads	Trajectories	Avg. distance	Avg. time
Porto	11,024	687,656	3.66 km	7.99 min
Beijing	40,060	899,115	5.16 km	12.87 min
San Francisco	27,187	293,023	3.41 km	9.29 min

Appendix

Table 8: Dataset statistics for the three evaluation cities, following the public HOSER dataset report [ Cao et al., 2025 ] .

Scale	Future fixed	Future head	Transition	Total formula
s1	199,401	$73V$	176,321	$375,722+73V$
s2	438,657	$97V$	304,193	$742,850+97V$
s3	1,303,273	$153V$	1,016,185	$2,319,458+153V$
s4	4,526,385	$241V$	3,185,825	$7,712,210+241V$
s5	16,201,585	$401V$	12,616,065	$28,817,650+401V$

Appendix

Table 9: Parameter decomposition for the five resolved model scales. The Future vocabulary head is

(d_{X}+1)V

for vocabulary size

V

City ( $V$ )	s1	s2	s3	s4	s5
Beijing (40,060)	3.300	4.629	8.449	17.367	44.882
Porto (11,024)	1.180	1.812	4.006	10.369	33.238
San Francisco (27,187)	2.360	3.380	6.479	14.264	39.720

Appendix

Table 10: Total trainable parameters by city and model scale, reported in millions (M).

V

is the graph vocabulary size; counts include the Future vocabulary head.

Setting	Value
Optimizer	AdamW
Learning rate	5e-4
Weight decay	1e-4
Learning-rate schedule	constant
Warmup steps	3000
Total optimizer updates	60000

Appendix

Table 12: Training configuration used for the reported experiments.

Method / Variant	Distance JSD	Radius JSD	DTW	EDR	Hausdorff
Dijkstra	2.47 $\mathord{\pm}$ 0.082	1.31 $\mathord{\pm}$ 0.052	12.74 $\mathord{\pm}$ 0.27	0.358 $\mathord{\pm}$ 0.009	0.481 $\mathord{\pm}$ 0.011
Yen- $k$	2.36 $\mathord{\pm}$ 0.116	1.29 $\mathord{\pm}$ 0.077	13.18 $\mathord{\pm}$ 0.34	0.386 $\mathord{\pm}$ 0.013	0.503 $\mathord{\pm}$ 0.016
Markov	2.61 $\mathord{\pm}$ 0.238	1.46 $\mathord{\pm}$ 0.171	15.42 $\mathord{\pm}$ 0.71	0.462 $\mathord{\pm}$ 0.021	0.613 $\mathord{\pm}$ 0.025
SemRoute	1.68 $\mathord{\pm}$ 0.204	1.07 $\mathord{\pm}$ 0.143	13.42 $\mathord{\pm}$ 0.68	0.384 $\mathord{\pm}$ 0.019	0.514 $\mathord{\pm}$ 0.022
HOSER	1.57 $\mathord{\pm}$ 0.183	0.96 $\mathord{\pm}$ 0.137	12.28 $\mathord{\pm}$ 0.66	0.317 $\mathord{\pm}$ 0.017	0.431 $\mathord{\pm}$ 0.020
TS-TrajGen	3.24 $\mathord{\pm}$ 0.287	1.83 $\mathord{\pm}$ 0.219	16.18 $\mathord{\pm}$ 0.89	0.481 $\mathord{\pm}$ 0.024	0.641 $\mathord{\pm}$ 0.029

Appendix

Table 13: Full Results on Porto. Entries are mean

\pm

standard deviation with exact-OD bootstrap, Distance JSD and Radius JSD use

10^{-2}

units.

Method / Variant	Distance JSD	Radius JSD	DTW	EDR	Hausdorff
Dijkstra	2.18 $\mathord{\pm}$ 0.076	1.29 $\mathord{\pm}$ 0.049	11.18 $\mathord{\pm}$ 0.24	0.305 $\mathord{\pm}$ 0.008	0.399 $\mathord{\pm}$ 0.010
Yen- $k$	2.34 $\mathord{\pm}$ 0.123	1.39 $\mathord{\pm}$ 0.082	13.16 $\mathord{\pm}$ 0.37	0.377 $\mathord{\pm}$ 0.013	0.497 $\mathord{\pm}$ 0.016
Markov	2.39 $\mathord{\pm}$ 0.244	1.42 $\mathord{\pm}$ 0.176	15.31 $\mathord{\pm}$ 0.73	0.458 $\mathord{\pm}$ 0.020	0.602 $\mathord{\pm}$ 0.024
SemRoute	1.69 $\mathord{\pm}$ 0.204	0.91 $\mathord{\pm}$ 0.148	12.74 $\mathord{\pm}$ 0.70	0.363 $\mathord{\pm}$ 0.018	0.481 $\mathord{\pm}$ 0.021
HOSER	1.82 $\mathord{\pm}$ 0.187	0.98 $\mathord{\pm}$ 0.139	11.47 $\mathord{\pm}$ 0.67	0.306 $\mathord{\pm}$ 0.017	0.412 $\mathord{\pm}$ 0.019
TS-TrajGen	3.06 $\mathord{\pm}$ 0.294	1.86 $\mathord{\pm}$ 0.216	15.43 $\mathord{\pm}$ 0.87	0.462 $\mathord{\pm}$ 0.023	0.608 $\mathord{\pm}$ 0.028

Appendix

Table 14: Full Results on San Francisco. Entries are mean

\pm

standard deviation with exact-OD bootstrap; Distance JSD and Radius JSD use

10^{-2}

units.

Method / Variant	Distance JSD	Radius JSD	DTW	EDR	Hausdorff
Dijkstra	2.64 $\mathord{\pm}$ 0.084	1.56 $\mathord{\pm}$ 0.053	16.28 $\mathord{\pm}$ 0.29	0.341 $\mathord{\pm}$ 0.009	0.449 $\mathord{\pm}$ 0.011
Yen- $k$	2.78 $\mathord{\pm}$ 0.127	1.67 $\mathord{\pm}$ 0.086	18.92 $\mathord{\pm}$ 0.38	0.414 $\mathord{\pm}$ 0.013	0.552 $\mathord{\pm}$ 0.017
Markov	2.88 $\mathord{\pm}$ 0.263	1.72 $\mathord{\pm}$ 0.187	22.34 $\mathord{\pm}$ 0.82	0.503 $\mathord{\pm}$ 0.022	0.681 $\mathord{\pm}$ 0.026
SemRoute	2.16 $\mathord{\pm}$ 0.218	1.19 $\mathord{\pm}$ 0.157	18.21 $\mathord{\pm}$ 0.76	0.414 $\mathord{\pm}$ 0.019	0.548 $\mathord{\pm}$ 0.022
HOSER	2.04 $\mathord{\pm}$ 0.196	1.09 $\mathord{\pm}$ 0.146	16.73 $\mathord{\pm}$ 0.72	0.351 $\mathord{\pm}$ 0.017	0.463 $\mathord{\pm}$ 0.020
TS-TrajGen	3.42 $\mathord{\pm}$ 0.314	2.16 $\mathord{\pm}$ 0.226	22.19 $\mathord{\pm}$ 0.92	0.503 $\mathord{\pm}$ 0.024	0.684 $\mathord{\pm}$ 0.030

Appendix

Table 15: Full Results on Beijing. Entries are mean

\pm

standard deviation with exact-OD bootstrap; Distance JSD and Radius JSD use

10^{-2}

units.

Method / Variant	Distance JSD	Radius JSD	DTW	EDR	Hausdorff
Foresight -F	1.53 $\mathord{\pm}$ 0.18	0.79 $\mathord{\pm}$ 0.11	11.84 $\mathord{\pm}$ 0.64	0.312 $\mathord{\pm}$ 0.014	0.401 $\mathord{\pm}$ 0.018
Foresight -M	1.01 $\mathord{\pm}$ 0.08	0.71 $\mathord{\pm}$ 0.05	11.58 $\mathord{\pm}$ 0.25	0.285 $\mathord{\pm}$ 0.008	0.375 $\mathord{\pm}$ 0.010
Foresight w/o $\mathcal{L}_{F,t}$	2.25 $\mathord{\pm}$ 0.24	1.35 $\mathord{\pm}$ 0.18	13.90 $\mathord{\pm}$ 0.72	0.390 $\mathord{\pm}$ 0.018	0.500 $\mathord{\pm}$ 0.022
Foresight w/o $\ell_{t}^{\mathrm{comp}}$	2.55 $\mathord{\pm}$ 0.27	1.55 $\mathord{\pm}$ 0.20	14.80 $\mathord{\pm}$ 0.80	0.420 $\mathord{\pm}$ 0.020	0.550 $\mathord{\pm}$ 0.024
Foresight w/o $E_{\mathrm{fut}}$	2.05 $\mathord{\pm}$ 0.22	1.20 $\mathord{\pm}$ 0.16	13.20 $\mathord{\pm}$ 0.68	0.360 $\mathord{\pm}$ 0.017	0.460 $\mathord{\pm}$ 0.020

Appendix

Table 16: Full ablation results on Porto. Entries are mean

\pm

standard deviation values; Distance JSD and Radius JSD use

10^{-2}

units.

Method / Variant	Distance JSD	Radius JSD	DTW	EDR	Hausdorff
Foresight -F	1.39 $\mathord{\pm}$ 0.16	0.71 $\mathord{\pm}$ 0.10	11.01 $\mathord{\pm}$ 0.59	0.298 $\mathord{\pm}$ 0.013	0.386 $\mathord{\pm}$ 0.017
Foresight -M	2.68 $\mathord{\pm}$ 0.180	0.65 $\mathord{\pm}$ 0.110	22.92 $\mathord{\pm}$ 0.64	0.415 $\mathord{\pm}$ 0.014	0.542 $\mathord{\pm}$ 0.018
Foresight w/o $\mathcal{L}_{F,t}$	2.10 $\mathord{\pm}$ 0.22	1.25 $\mathord{\pm}$ 0.16	13.20 $\mathord{\pm}$ 0.68	0.380 $\mathord{\pm}$ 0.017	0.480 $\mathord{\pm}$ 0.020
Foresight w/o $\ell_{t}^{\mathrm{comp}}$	2.40 $\mathord{\pm}$ 0.25	1.45 $\mathord{\pm}$ 0.18	14.20 $\mathord{\pm}$ 0.76	0.410 $\mathord{\pm}$ 0.019	0.530 $\mathord{\pm}$ 0.022
Foresight w/o $E_{\mathrm{fut}}$	1.95 $\mathord{\pm}$ 0.20	1.15 $\mathord{\pm}$ 0.15	12.60 $\mathord{\pm}$ 0.64	0.350 $\mathord{\pm}$ 0.016	0.450 $\mathord{\pm}$ 0.019

Appendix

Table 17: Full ablation results on San Francisco. Entries are mean

\pm

standard deviation values; Distance JSD and Radius JSD use

10^{-2}

units.

Method / Variant	Distance JSD	Radius JSD	DTW	EDR	Hausdorff
Foresight -F	1.68 $\mathord{\pm}$ 0.19	0.88 $\mathord{\pm}$ 0.12	16.34 $\mathord{\pm}$ 0.88	0.326 $\mathord{\pm}$ 0.015	0.418 $\mathord{\pm}$ 0.019
Foresight -M	1.87 $\mathord{\pm}$ 0.21	1.01 $\mathord{\pm}$ 0.13	17.90 $\mathord{\pm}$ 0.88	0.344 $\mathord{\pm}$ 0.016	0.438 $\mathord{\pm}$ 0.020
Foresight w/o $\mathcal{L}_{F,t}$	2.55 $\mathord{\pm}$ 0.27	1.50 $\mathord{\pm}$ 0.20	19.50 $\mathord{\pm}$ 0.95	0.400 $\mathord{\pm}$ 0.020	0.520 $\mathord{\pm}$ 0.024
Foresight w/o $\ell_{t}^{\mathrm{comp}}$	2.90 $\mathord{\pm}$ 0.30	1.75 $\mathord{\pm}$ 0.23	20.80 $\mathord{\pm}$ 1.02	0.440 $\mathord{\pm}$ 0.022	0.580 $\mathord{\pm}$ 0.027
Foresight w/o $E_{\mathrm{fut}}$	2.40 $\mathord{\pm}$ 0.25	1.35 $\mathord{\pm}$ 0.18	18.70 $\mathord{\pm}$ 0.90	0.380 $\mathord{\pm}$ 0.019	0.490 $\mathord{\pm}$ 0.023

Appendix

Table 18: Full ablation results on Beijing. Entries are mean

\pm

standard deviation values; Distance JSD and Radius JSD use

10^{-2}

units.

Panel of Figure 9 — Figure 9: Porto road network and trajectory density. Generated trajectories recover the dominant and secondary spatial modes of the real test set while suppressing isolated noise-like artifacts. Both heatmaps use the same spatial extent, road geometry, traversal-count definition, and color normalization.

Panel of Figure 10 — Figure 10: San Francisco road network and trajectory density. The generated samples recover the dominant and secondary spatial modes under the common plotting protocol.

Panel of Figure 11 — Figure 11: Beijing road network and trajectory density. Generated trajectories track the observed corridors and remain sparse in weakly supported regions.

Panel of Figure 12 — Figure 12: Matched Porto path examples. Each row uses one origin–destination condition, with the real trajectory on the left and the generated trajectory on the right. Generated routes preserve connectivity and destination awareness while allowing benign local variation.

Panel of Figure 13 — Figure 13: Matched San Francisco path examples, with real trajectories on the left and generated trajectories on the right.

Panel of Figure 14 — Figure 14: Matched San Francisco path examples (continued).

Panel of Figure 15 — Figure 15: Matched Beijing path examples, with real trajectories on the left and generated trajectories on the right.

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