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Geometric Trajectory Diffusion Models
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Proposes GeoTDM, the first diffusion model for generating 3D geometric trajectories, using an SE(3)-equivariant denoiser (EGTN) with temporal attention and a learnable equivariant prior for conditional generation.
Problem
Generative modeling of 3D geometric trajectories (dynamic physical systems) while preserving SE(3) symmetry and capturing temporal correlations, which prior generative models for geometric data (static molecules/proteins) do not address.
Approach
Diffusion over geometric trajectories with SE(3)-equivariant transition kernels built from equivariant graph convolution (EGCL) for spatial interaction and relative-time-encoded temporal attention, plus a learnable equivariant prior built from transformed conditioning frames to improve temporal conditioning.
Key results
- On N-body conditional forecasting, GeoTDM reaches ADE 0.110 / FDE 0.258 on Charged Particles versus 0.141 / 0.310 for the prior best EqMotion.
- On MD17 conditional generation, GeoTDM improves average ADE/FDE by 23.1%/15.3% over EqMotion across eight molecules.
- On ETH-UCY pedestrian forecasting, GeoTDM achieves the lowest average minADE/minFDE (0.23/0.40) compared with SVAE (0.24/0.42).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| use GeoTDM instead of EqMotion | conditional trajectory forecasting · Charged Particles | ADE: 0.141 -> 0.110 | Helped |
| use GeoTDM instead of EqMotion | conditional trajectory forecasting · Spring | ADE: 0.0134 -> 0.0030 | Helped |
| use GeoTDM instead of EqMotion | conditional trajectory forecasting · Gravity | ADE: 0.302 -> 0.256 | Helped |
| use GeoTDM instead of SVAE | ETH-UCY forecasting · ETH-UCY average | minADE: 0.24 -> 0.23 | Helped |
| use GeoTDM instead of SVAE | unconditional trajectory generation · Charged Particles | Marginal: 0.0668 -> 0.0055 | Helped |
| replace the learnable equivariant prior with N(0,I) (ablation) | GeoTDM · Charged Particles | ADE: 0.110 -> 0.220 | Hurt |
| replace the learnable equivariant prior with a center-of-mass prior (ablation) | GeoTDM · Charged Particles | ADE: 0.110 -> 0.135 | Hurt |
| replace the learnable equivariant prior with a point-wise conditioning-frame prior (ablation) | GeoTDM · Charged Particles | ADE: 0.110 -> 0.123 | Hurt |
| remove equivariance from the denoising network (ablation) | GeoTDM · Charged Particles | ADE: 0.110 -> 0.251 | Hurt |
| replace temporal attention with equivariant convolutions (ablation) | GeoTDM · Charged Particles | ADE: 0.110 -> 0.133 | Hurt |
| replace relative temporal embeddings with absolute temporal embeddings (ablation) | GeoTDM · Charged Particles | FDE: 0.258 -> 0.330 | Hurt |
| use GeoTDM instead of linear interpolation | trajectory interpolation · Charged Particles | ADE: 0.171 -> 0.055 | Helped |
Surprising findings
- Non-equivariant DDPM-style Gaussian prior performs markedly worse than equivariant priors on N-body (ADE 0.220 vs 0.110), and the fixed CoM prior is inferior to the learnable prior.
Limitations
- Multi-step sampling of diffusion models may require substantial compute.
- Evaluated only on small systems (N-body with 5-10 particles, MD17 small molecules, pedestrian data).
Experimental setup
Datasets: Charged Particles Spring Dynamics Gravity System MD17 ETH-UCY
Benchmarks: Charged Particles Spring Dynamics Gravity System MD17 ETH-UCY
Metrics: ADE FDE minADE minFDE Marginal score Classification score Prediction score
Baselines: Radial Field Tensor Field Network SE(3)-Transformer EGNN EqMotion SVAE SGAN EGVAE SoPhie PECNet Traj++ BiTraP
Builds on: DDPM EGNN SVAE EqMotion
Closest papers by meaning
- Bridging Geometric States via Geometric Diffusion Bridge (NeurIPS 2024 · similarity 0.91)
- GeoAda: Efficiently Finetune Geometric Diffusion Models with Equivariant Adapters (NeurIPS 2025 · similarity 0.90)
- GeoSPRINT: Geometric Redundancy-Aware Step Pruning for Inference in Diffusion Trajectories (NeurIPS 2026 · similarity 0.88)
- Equivariant Neural Diffusion for Molecule Generation (NeurIPS 2024 · similarity 0.88)
- Unified Generative Modeling of 3D Molecules with Bayesian Flow Networks (ICLR 2024 · similarity 0.87)
- Scalable Non-Equivariant 3D Molecule Generation via Rotational Alignment (ICML 2025 · similarity 0.87)
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