Geometric Trajectory Diffusion Models (2410.13027v1)

Abstract: 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.

• The paper presents GeoTDM, a pioneering approach that models 3D geometric trajectories with a diffusion framework preserving inherent physical symmetries.
• It employs a novel combination of SE(3)-equivariant spatial convolution and temporal attention, achieving up to 56.7% lower prediction errors in certain scenarios.
• GeoTDM's demonstrated success in molecular dynamics and pedestrian motion prediction highlights its practical value for generating realistic, dynamic trajectories.

Geometric Trajectory Diffusion Models: An Overview

This paper addresses a significant challenge in the field of generative models: the dynamic nature of 3D geometric structures over time. While previous methods have focused on generating static structures, this work introduces the Geometric Trajectory Diffusion Models (GeoTDM), which aim to model the temporal distribution of 3D geometric trajectories. This advancement is particularly relevant for applications in natural sciences, such as molecular and protein dynamics.

Key Contributions

1. Introduction of GeoTDM: GeoTDM is designed to capture the complex spatial interactions and temporal correlations inherent in 3D geometric trajectories. It leverages diffusion models with SE(3)-equivariant kernels to ensure that the generated trajectories preserve physical symmetries.
2. Equivariant Temporal Diffusion: The model utilizes a novel transition kernel that combines SE(3)-equivariant spatial convolution with temporal attention mechanisms. This ensures that the generated trajectories conform to the desired physical symmetries, both spatially and temporally.
3. Learnable Equivariant Prior: For conditional generation, the paper introduces a generalized learnable geometric prior that enhances temporal conditioning. This approach allows for more expressive trajectory distributions, catering to scenarios where initial temporal data are provided.
4. Diverse Applications: GeoTDM is evaluated in various scenarios, including physical simulations, molecular dynamics, and pedestrian motion prediction. The model demonstrates superior performance in generating realistic trajectories, outperforming existing methods with up to 56.7% lower prediction scores in unconditional settings and 16.8% lower forecasting error in conditional scenarios.

Theoretical Insights

The paper provides a rigorous theoretical foundation for the use of diffusion models in trajectory generation. By ensuring that both the prior and the transition kernels are equivariant, the model guarantees that the generated distributions respect the inherent symmetries of the physical processes being modeled. This approach not only maintains the symmetry but also enables the model to capture temporal dynamics more accurately.

Practical Implications and Future Work

GeoTDM's ability to generate high-quality geometric trajectories opens new avenues for various applications. In molecular dynamics, for example, it aids in simulating molecular behavior over time, providing insights into binding activities that are crucial for drug discovery. The model's adaptability also makes it suitable for robotics and animation, where understanding movement dynamics is vital.

Looking ahead, the research suggests several directions for further exploration. These include optimizing the computational efficiency of GeoTDM, possibly integrating methods like DDIM or latent space modeling. Additionally, extending the model to more complex systems, such as protein folding or robotic path planning, could enhance its applicability.

In summary, GeoTDM represents a formal and structured leap towards modeling dynamic 3D geometric systems. By combining theoretical rigor with empirical validation across multiple domains, the paper sets a foundation for future advancements in the generative modeling of dynamic structures.