Synthetic Trajectory Generation

• Synthetic trajectory generation is the algorithmic creation of artificial movement paths using classical models and modern deep learning to replicate real-world spatiotemporal patterns.
• It is applied in urban mobility, biomolecular simulations, reinforcement learning, and privacy-preserving data publication by leveraging techniques like Markov models, GANs, and diffusion processes.
• Evaluation methodologies use spatial, temporal, and multimodal metrics to ensure that synthetic trajectories closely match real data in both accuracy and utility for downstream applications.

Synthetic trajectory generation refers to the algorithmic construction of artificial movement paths representing agents—vehicles, humans, biological molecules, or information carriers—in continuous or discrete space-time. Its applications span urban mobility modeling, biomolecular simulations, reinforcement learning (RL), privacy-preserving data publication, and simulation for downstream tasks such as optimization or forecasting. The domain encompasses classical models (e.g., Markov chains), modern generative frameworks (e.g., deep neural networks, GANs, diffusion models), and rigorous frameworks for joint spatial-temporal control, data augmentation, privacy, and benchmarking. This article provides a technical overview emphasizing the state-of-the-art models and methodologies, as well as their evaluation protocols in high-complexity environments.

1. Controlled Spatiotemporal Infilling: Transformer-Based Generative Models

A recent advance is controlled infilling, where the goal is to generate entire sequences, partially specified via "known" or "blanked-out" visits, such that generated trajectories respect both fixed regions and temporal consistency. TrajGPT (Hsu et al., 7 Nov 2024) formalizes each trajectory as a sequence of visit tuples $x_i = (r_i, t_i^a, t_i^d)$, with region, arrival, and departure. The core controlled generation task is, given a partially observed trajectory $X'$ with BLANK spans, to predict missing visits in correct temporal order, such that: $$P(x | X') = P(r | X')\,P(t^a | X', r)\,P(t^d | X', r, t^a)$$ TrajGPT encodes locations via Space2Vec, times via Time2Vec, and regions via a learnable embedding; the concatenated sequence is processed by a multi-layer Transformer encoder. Its multi-task head jointly predicts:

• Discrete region (softmax classification)
• Travel time ($\Delta t^T = t^a_i - t^d_{i-1}$) via Gaussian Mixture Model (GMM), conditioned on predicted region
• Visit duration ($\Delta t^D = t^d_i - t^a_i$) via GMM, conditioned on region and arrival time

Training employs infilling-style data perturbations and teacher forcing. Inference proceeds by autoregressive prediction over missing spans. Bayesian integration guarantees instant joint modeling of region and timing. Experimental results on GeoLife and MobilitySim datasets demonstrate dramatic improvement in temporal accuracy (e.g., arrival time error reductions by >26-fold while preserving >98% spatial accuracy). Ablations confirm that joint spatiotemporal modeling and GMM time heads are essential for fidelity.

2. Probabilistic Bézier Curves for Synthetic Ground-Truth Distribution Generation

Probabilistic composite Bézier curves (Hug et al., 5 Apr 2024) enable generation of closed-form probabilistic trajectory datasets, which are crucial for evaluating multi-step predictors by distributional metrics. Here, each trajectory is a realization of a deterministic Bézier curve (with Bernstein polynomials), elevated to a stochastic process by modeling each control point as a Gaussian, so all points along the curve are multivariate Gaussian: $$X_t \sim \mathcal{N}\Big( \sum b_{l,L}(t) \mu_l,\sum b_{l,L}(t)^2 \Sigma_l \Big)$$ More expressive path families are built by stitching together multiple such segments and introducing mixture models for multi-modal ground truth. Arbitrary discretizations yield finite-length trajectories. Posterior conditioning is exact, exploiting the block-Gaussian conditional formula. This approach enables expressive metrics such as sliced Wasserstein distance, rewarding both mean and covariance matching. Empirical results show that negative log-likelihood does not always reflect full-distribution recovery, whereas Wasserstein is more sensitive to error in statistical variance. Limitations arise from increased covariance complexity and high computational demands for large-scale problems.

3. Trajectory Generation for Biomolecular Systems via Markov State Models

Synthetic molecular dynamics (synMD) (Russo et al., 2022) leverages Markov state modeling to generate atomistic biomolecular trajectories efficiently. Configuration space is discretized into thousands of states, each mapped to a representative atomistic structure. The MSM is constructed by stratified k-means clustering in tICA-reduced space to enhance kinetic fidelity even at short lag times, and the detailed-balance transition matrix $P_{ij}$ is inferred from MD data. Trajectories are sampled via categorical propagation through $P$. Backmapping to atomistic coordinates is trivial and highly parallelizable. This framework provides >2000× computational speedup relative to explicit MD, with equilibrium populations, MFPTs, and tICA projections matching reference MD within statistical error.

4. Diffusion, GAN, and Sequence Models for Mobility and Social Network Data

Categorical Diffusion Models

Diffusion probabilistic models (DPMs) have recently been adapted for sequences of categorical location variables (Dirmeier et al., 19 Feb 2024). An embedding matrix maps categorical region IDs to continuous space; the diffusion process proceeds by forward Gaussian noising and learned reverse denoising. At each step, masking permits conditional infilling (partial sequence observation), and final discrete decoding projects to location indices via softmax over embedding dot-products. Empirical Shannon entropy, visit-count histograms, and pairwise spatial statistics are faithfully replicated in both conditional and unconditional sampling settings.

GANs and CNNs

CNN-based GANs (DCGAN architecture) have been introduced for spatially coherent synthetic generation by implementing a reversible trajectory-to-CNN transformation (RTCT) to embed trajectories in a spatial grid suitable for convolution (Merhi et al., 24 Jul 2024). The generator maps random noise to grid-formatted tensors encoding spatial and temporal features; the inverse mapping reconstructs variable-length trajectory sequences. While DCGAN captures high-density spatial clusters better than RNN models, sequential and temporal coherence is sacrificed (with high time-reversal ratios indicating poor realism).

Coarse-to-Fine Latent Diffusion

GeoGen (Xu et al., 9 Oct 2025) employs a two-stage pipeline for LBSN check-in data: reconstructing temporally regular, spatially continuous movement sequences (coarse), then applying a sparsity-aware spatio-temporal diffusion model (S$^2$TDiff) to learn their structure. The second stage expands coarse sequences to fine-grained POI/timestamp trajectories via a context-fusion Transformer, dynamically blending spatial and temporal attention. Multi-task hybrid-head decoding jointly predicts POIs and timestamps under neural TPP objectives. On FS-TKY, GeoGen achieves ≥69% and 55% improvement over baselines in distance and radius metrics.

5. RL-Based and Diffusion Approaches for Path Coverage and Optimization

Link flow estimation in road networks (Zhong et al., 2022) can be formulated as an MDP, where link-to-link actions generate synthetic vehicle trajectories consistent with observed volumes and sparse tracking data. Maximum Entropy Inverse RL (IRL-F) infers a linear reward maximizing agreement in visitation between sampled and synthetic populations, incorporating the cLAD estimator for population counts. Constrained RL (CRL-F, via APPROPO) ensures feature visitation matches observed measurement vectors as convex constraints. The resultant synthetic data yields robust, accurate link-flow estimations even when detector coverage or trajectory penetration is low.

In offline model-based optimization, GTG employs guided trajectory generation via conditional diffusion models, with locality bias and score-based classifier-free guidance (Yun et al., 29 Jun 2024). Context conditioning enables targeted high-score exploration. Guided sampling and proxy-based design selection yield superior performance against baselines on both discrete and continuous benchmarks under sparse and noisy settings.

6. Privacy-Protecting Synthetic Trajectory Generation

Recent systematization-of-knowledge (SoK) work (Buchholz et al., 12 Mar 2024) identifies five critical design goals for privacy-friendly trajectory generators: formal DP guarantees, explicit unit of privacy (UoP) selection, rigorous empirical attack evaluation, utility on both pointwise and sequence statistics, and practical computational tractability. Most current deep/sequential generative models fail to meet all these, typically lacking trajectory-level DP or semantic guarantees. Promising strategies include "no-real-input" ensembles trained on DP-SGD, with utility validated against state-of-the-art attacks and metrics.

Mean-field Langevin dynamics enables private, continuous-time synthetic generation (Gu et al., 13 Jun 2025) where each user contributes only one timepoint, not an entire trajectory. The system solves for marginal distributions using noisy SGD particle updates on path-space functional objectives, with DP guarantees following from standard privacy amplification and composition analysis. Statistical rates and utility guarantees are established for convergence to true marginals (e.g., average $W_2$ distortion ≈ 0.02–0.05), enabling high-quality privacy-friendly synthetic paths for sensitive domains such as healthcare.

7. Benchmarking, Multi-Agent, and Context-Aware Generative Architectures

For urban vehicle trajectories, TrajGAIL (Choi et al., 2020) formulates generation as adversarial imitation learning in a POMDP, with an RNN policy and discriminator. At each timestep, the generator selects discrete maneuvers; the discriminator provides a reward for RL updates. In multiscenario evaluation against Markov and RNN models, TrajGAIL achieves the lowest sequence BLEU/METEOR and route-frequency Jensen-Shannon distances.

HiD² (Yang et al., 3 Oct 2025) synthesizes high-density traffic scenarios by gridifying continuous maps, explicit multi-agent conflict handling, rule-based maneuvers, and Frenet trajectory smoothing with dynamic feasibility constraints. This ensures realistic behavioral diversity and agent density, reducing collision and off-road rates relative to original data. Downstream predictors trained on HiD² outperform low-density baselines under overcrowded scenario conditions.

Zero-shot, map-conditioned diffusion models (Map2Traj (Tao et al., 29 Jul 2024)) generate mobility trajectories strictly from street network images, requiring no observed real trajectories in the target region. Fully conditional U-Net architectures guide the denoising process to spatial priors derived from the underlying map. Quantitative metrics (EDR, DTW, cosine and Wasserstein distances) confirm close replication of mobility patterns; when applied to wireless network DRL, synthetic data achieves ≈92% of real-trained agent utility.

Temporally adaptive generative models, such as MIRAGE (Deng et al., 20 Sep 2024), couple temporal point processes (neural TPP) with exploration/preferential-return location decoders. Ablations reveal that intensity-free TPP plus neural EPR markedly increase both distributional fidelity and downstream task performance (as measured by metrics on location recommendation, next-location prediction, semantic labeling, and epidemic simulation).

8. Evaluation Methodologies and Quantitative Results

Synthetic trajectory evaluation strictly relies on:

• Spatial metrics (Acc@K, JSD, Hausdorff, Wasserstein)
• Temporal metrics (P_{±t}, interval/interval-distribution, entropy)
• Multimodal and distributional metrics (variance, autocorrelation, kurtosis, skewness)
• Downstream utility: "Train on Synthetic, Test on Real" (TSTR), next-location/POI prediction accuracy, controller/passenger Turing tests

Benchmarks demonstrate that modern transformer, diffusion, and GAN-based models now approach near-parity with real data in both statistical and utility metrics.

Model Framework	Spatial Fidelity	Temporal Fidelity	Utility Gain
TrajGPT	>98%	×26 improvement	GeoLife, MobilitySim
ATRADA	43%–25% ↑ vs. baselines	36% ↑ vs. baselines	ATCo Turing test, TSTR
GeoGen	69%–55% ↓ JSD	10–20% of oracle	POI prediction
MIRAGE	59–71% ↓ JSD	11–33% task ↑	Multi-task
TimeVQVAE	0.0029 FID	80% flyable	Statistical suite

All claims directly trace to referenced papers or their metrics.

9. Future Directions and Open Problems

Open technical challenges include:

• Integrating strict trajectory-level differential privacy without utility collapse (Buchholz et al., 12 Mar 2024)
• Extending current models to capture complex agent interactions, heterogeneous semantic activities, or physics-informed constraints (Yang et al., 3 Oct 2025, Murad et al., 12 Apr 2025)
• Formal benchmarking with closed-form posterior evaluation for model selection (Hug et al., 5 Apr 2024)
• Scaling to high-dimensional, high-frequency data with computational tractability (Hug et al., 5 Apr 2024, Xu et al., 9 Oct 2025)
• Robustness against out-of-domain and sparse-scenario evaluation (Yun et al., 29 Jun 2024, Tao et al., 29 Jul 2024)

A plausible implication is that hybrid architectures (ensemble CNN+RNN, transformer+GMM, diffusion+attention) subject to strict privacy analysis and equipped with expressive loss functions and distributional evaluation protocols will define next-generation synthetic trajectory generators.