# Geometric Trajectory Diffusion Models

arXiv: https://arxiv.org/abs/2410.13027 · HTML: https://mlsanity.com/paper/2410.13027/

## Key facts

- Cited by 15 papers in the map (data through 14 Oct 2026).
- Placed in Flow matching, time series and science, one of 14 directions in Diffusion models.
- Builds on 57 of its 71 references that are in the map.
- First posted on arXiv 16 Oct 2024.

## Placed in

- Diffusion models / [Flow matching, time series and science](https://mlsanity.com/diffusion/flow-matching-time-series-and-science/)

## Builds on (most cited first, 30 of 57)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Denoising Diffusion Implicit Models](https://mlsanity.com/paper/2010.02502/) (2020)
- [Diffusion Models Beat GANs on Image Synthesis](https://mlsanity.com/paper/2105.05233/) (2021)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Neural Message Passing for Quantum Chemistry](https://mlsanity.com/paper/1704.01212/) (2017)
- [Video Diffusion Models](https://mlsanity.com/paper/2204.03458/) (2022)
- [DPM-Solver: A Fast ODE Solver for Diffusion Probabilistic Model Sampling in Around 10 Steps](https://mlsanity.com/paper/2206.00927/) (2022)
- [SDEdit: Guided Image Synthesis and Editing with Stochastic Differential Equations](https://mlsanity.com/paper/2108.01073/) (2021)
- [Consistency Models](https://mlsanity.com/paper/2303.01469/) (2023)
- [Spatio-Temporal Graph Convolutional Networks: A Deep Learning Framework for Traffic Forecasting](https://mlsanity.com/paper/1709.04875/) (2017)
- [Self-Attention with Relative Position Representations](https://mlsanity.com/paper/1803.02155/) (2018)
- [Human Motion Diffusion Model](https://mlsanity.com/paper/2209.14916/) (2022)
- [Improved Techniques for Training Score-Based Generative Models](https://mlsanity.com/paper/2006.09011/) (2020)
- [E(n) Equivariant Graph Neural Networks](https://mlsanity.com/paper/2102.09844/) (2021)
- [Learning to Simulate Complex Physics with Graph Networks](https://mlsanity.com/paper/2002.09405/) (2020)
- [Tensor field networks: Rotation- and translation-equivariant neural networks for 3D point clouds](https://mlsanity.com/paper/1802.08219/) (2018)
- [Social GAN: Socially Acceptable Trajectories with Generative Adversarial Networks](https://mlsanity.com/paper/1803.10892/) (2018)
- [Equivariant Diffusion for Molecule Generation in 3D](https://mlsanity.com/paper/2203.17003/) (2022)
- [SE(3)-Transformers: 3D Roto-Translation Equivariant Attention Networks](https://mlsanity.com/paper/2006.10503/) (2020)
- [CSDI: Conditional Score-based Diffusion Models for Probabilistic Time Series Imputation](https://mlsanity.com/paper/2107.03502/) (2021)
- [Directional Message Passing for Molecular Graphs](https://mlsanity.com/paper/2003.03123/) (2020)
- [GeoDiff: a Geometric Diffusion Model for Molecular Conformation Generation](https://mlsanity.com/paper/2203.02923/) (2022)
- [Interaction Networks for Learning about Objects, Relations and Physics](https://mlsanity.com/paper/1612.00222/) (2016)
- [Equivariant message passing for the prediction of tensorial properties and molecular spectra](https://mlsanity.com/paper/2102.03150/) (2021)
- [PhysDiff: Physics-Guided Human Motion Diffusion Model](https://mlsanity.com/paper/2212.02500/) (2022)
- [Neural Relational Inference for Interacting Systems](https://mlsanity.com/paper/1802.04687/) (2018)
- [Autoregressive Denoising Diffusion Models for Multivariate Probabilistic Time Series Forecasting](https://mlsanity.com/paper/2101.12072/) (2021)

## Built on by (most cited first, 15 of 15)

- [Discrete Diffusion Trajectory Alignment via Stepwise Decomposition](https://mlsanity.com/paper/2507.04832/) (2025)
- [STFlow: Data-Coupled Flow Matching for Geometric Trajectory Simulation](https://mlsanity.com/paper/2505.18647/) (2025)
- [Zero-Shot Cyclic Peptide Design via Composable Geometric Constraints](https://mlsanity.com/paper/2507.04225/) (2025)
- [RealTraj: Towards Real-World Pedestrian Trajectory Forecasting](https://mlsanity.com/paper/2411.17376/) (2024)
- [Beyond Ensembles: Simulating All-Atom Protein Dynamics in a Learned Latent Space](https://mlsanity.com/paper/2509.02196/) (2025)
- [Universally Invariant Learning in Equivariant GNNs](https://mlsanity.com/paper/2510.13169/) (2025)
- [PAINET: A Principled Efficient Transformer for 3D Dynamics Modeling](https://mlsanity.com/paper/2510.04233/) (2025)
- [GeoAda: Efficiently Finetune Geometric Diffusion Models with Equivariant Adapters](https://mlsanity.com/paper/2507.02085/) (2025)
- [CHORDS: Diffusion Sampling Accelerator with Multi-core Hierarchical ODE Solvers](https://mlsanity.com/paper/2507.15260/) (2025)
- [SEA: State-Exchange Attention for High-Fidelity Physics Based Transformers](https://mlsanity.com/paper/2410.15495/) (2024)
- [Follow the MEP: Scalable Neural Representations for Minimum-Energy Path Discovery in Molecular Systems](https://mlsanity.com/paper/2504.16381/) (2025)
- [All-Atom GPCR-Ligand Dynamics Simulation via a Residual Latent Flow Mode](https://mlsanity.com/paper/2602.03902/) (2026)
- [Align Your Structures: Generating Trajectories with Structure Pretraining for Molecular Dynamics](https://mlsanity.com/paper/2604.03911/) (2026)
- [Graph Mamba Operator: A Latent Simulator for Interacting Particle Systems](https://mlsanity.com/paper/2606.09432/) (2026)
- [Probabilistic Simulation of Gas-Liquid Interfaces with Coupled Flow Matching on Spherical Harmonics](https://mlsanity.com/paper/2610.16327/) (2026)

## Abstract (arXiv)

Generative models have shown great promise in generating 3D geometric systems, which is a fundamental problem in many natural science domains such as molecule and protein design. However, existing approaches only operate on static structures, neglecting the fact that physical systems are always dynamic in nature. In this work, we propose geometric trajectory diffusion models (GeoTDM), the first diffusion model for modeling the temporal distribution of 3D geometric trajectories. Modeling such distribution is challenging as it requires capturing both the complex spatial interactions with physical symmetries and temporal correspondence encapsulated in the dynamics. We theoretically justify that diffusion models with equivariant temporal kernels can lead to density with desired symmetry, and develop a novel transition kernel leveraging SE(3)-equivariant spatial convolution and temporal attention. Furthermore, to induce an expressive trajectory distribution for conditional generation, we introduce a generalized learnable geometric prior into the forward diffusion process to enhance temporal conditioning. We conduct extensive experiments on both unconditional and conditional generation in various scenarios, including physical simulation, molecular dynamics, and pedestrian motion. Empirical results on a wide suite of metrics demonstrate that GeoTDM can generate realistic geometric trajectories with significantly higher quality.