Flow for Future: Geometric SE(3)-Equivariant Flow Matching for 3D Trajectory Prediction

Abstract

Predicting 3D geometric trajectory requires capturing complex spatiotemporal dependencies while preserving physical symmetries. While flow matching offers a powerful generative paradigm, extending it to SE(3)-equivariant dynamics is challenging due to the inherent gap between deterministic history and stochastic evolving flows. To address this, we introduce GSE-Flow, an SE(3)-equivariant flow matching framework. We first propose a Coherent Sequence Encoding and Time-Modulated Embedding strategy that unifies historical and evolving streams, incorporating velocity and flow time via equivariant affine transformations to guide continuous evolution. We further design a Geometry-Feature Tensorization mechanism that projects node states into a tensor product space, enabling Context-Flow Fusion to guide trajectory evolution with historical context. GSE-Flow guarantees theoretical SE(3)-equivariance and achieves SOTA accuracy on MD17, MD22, and CMU MoCap benchmarks for geometric trajectory prediction, while demonstrating generality by enhancing deterministic baselines.