Physics-Informed Diffusion Models for Vehicle Speed Trajectory Generation

Synthetic vehicle speed trajectory generation is essential for evaluating vehicle control algorithms and connected vehicle technologies. Traditional Markov chain approaches suffer from discretization artifacts and limited expressiveness. This paper proposes a physics-informed diffusion framework for conditional micro-trip synthesis, combining a dual-channel speed-acceleration representation with soft physics constraints that resolve optimization conflicts inherent to hard-constraint formulations. We compare a 1D U-Net architecture against a transformer-based Conditional Score-based Diffusion Imputation (CSDI) model using 6,367 GPS-derived micro-trips. CSDI achieves superior distribution matching (Wasserstein distance 0.30 for speed, 0.026 for acceleration), strong indistinguishability from real data (discriminative score 0.49), and validated utility for downstream energy assessment tasks. The methodology enables scalable generation of realistic driving profiles for intelligent transportation systems (ITS) applications without costly field data collection.

💡 Research Summary

This paper introduces a physics‑informed diffusion modeling framework for the conditional synthesis of vehicle speed trajectories at the micro‑trip level. Traditional approaches, such as Markov chain models, suffer from discretization artifacts, limited expressiveness, and difficulty incorporating physical constraints. Recent deep generative methods—including GANs and normalizing flows—have shown promise but are plagued by training instability, mode collapse, and challenges in enforcing hard kinematic limits.

To address these gaps, the authors propose a dual‑channel representation that simultaneously models speed and acceleration, allowing the diffusion process to learn the intrinsic relationship between the two signals. Physical constraints essential for realistic driving—zero initial and final speeds, acceleration bounds, and conditioning on aggregate trip statistics (average speed, duration)—are incorporated as soft penalty terms in the loss function rather than as hard constraints, thereby avoiding the optimization conflicts observed in earlier attempts.

Two diffusion‑based architectures are evaluated: a standard 1‑D U‑Net adapted from image generation, and a transformer‑based Conditional Score‑based Diffusion Imputation (CSDI) model originally designed for missing‑value imputation in multivariate time series. In the CSDI formulation, the entire trajectory is treated as “missing” and generated conditionally on the provided trip‑level statistics, leveraging self‑attention to capture long‑range temporal dependencies while the diffusion framework supplies a principled uncertainty model.

The experimental dataset consists of 6,367 GPS‑derived micro‑trips collected from the 2007 Chicago Metropolitan Agency for Planning (CMAP) Household Travel Survey. After preprocessing, each trip is sampled at 1 Hz and includes both speed and derived acceleration profiles. The trips exhibit wide variability: durations from 34 s to 12,841 s, average speeds from 5 m/s to 32 m/s, and distances spanning two orders of magnitude. K‑means clustering (K = 4) reveals distinct driving regimes (urban, suburban, highway, mixed), which are used to stratify evaluation.

Training experiments show that hard‑constraint variants quickly become unstable, producing unrealistic acceleration spikes and violating boundary conditions. In contrast, the soft‑constraint CSDI model achieves superior distributional fidelity: Wasserstein distances of 0.30 for speed and 0.026 for acceleration, and a discriminative score of 0.49, indicating near‑indistinguishability from real data. Downstream utility is demonstrated through a Train‑on‑Synthetic‑Test‑on‑Real (TSTR) experiment in which synthetic trajectories are fed into an energy consumption model; the resulting energy estimates are statistically indistinguishable from those obtained with real trajectories.

Baseline comparisons include traditional Markov chains, GANs, normalizing flows, DoppelGANger, and the Synthetic Data Vault (SDV) autoregressive model. While each baseline offers certain advantages—interpretability for Markov chains, multimodal capacity for GANs—their overall performance lags behind the physics‑informed diffusion approach, especially in maintaining kinematic plausibility over long horizons.

Key contributions of the work are: (1) the first application of diffusion models specifically to vehicle speed micro‑trip generation for transportation energy assessment, (2) the successful integration of kinematic physics constraints via soft penalties into a transformer‑based diffusion architecture, (3) a comprehensive documentation of design iterations and failure modes (hard‑constraint diffusion, DoppelGANger, SDV), providing practical guidance for future researchers, (4) a rigorous evaluation framework combining distributional metrics (Wasserstein, MMD), kinematic validity (smoothness, boundary violations), and downstream utility (discriminative score, TSTR), and (5) an open‑source release of code, pretrained weights, processed data, and figure‑generation scripts to ensure reproducibility.

The authors conclude that physics‑informed diffusion models deliver stable training, high‑quality multimodal samples, and faithful adherence to vehicle dynamics, making them a powerful tool for generating large‑scale synthetic driving data. Such data can directly support a range of ITS applications, including electric‑vehicle range analysis, emission and fuel‑consumption microsimulation, connected‑vehicle scenario testing, and policy evaluation, without the prohibitive cost of extensive field data collection.