A General and Efficient SE(3)-Equivariant Graph Framework: Encoding Symmetries with Complete Differential Invariants and Frames

Equivariant graph neural networks (Equiv-GNNs) have demonstrated effectiveness in modeling dynamics of multi-object systems via explicit symmetry encoding. Scalarization-based Equiv-GNNs are widely adopted in practice for their superior computational efficiency, as opposed to high-steerable models. However, most existing scalarization-based approaches rely on the empirical design of invariant scalars, which fail to fully and systematically capture system dynamics, particularly when object differential components like velocities are involved, leading to limited prediction performance. To address these issues, we propose a general, efficient, and lightweight SE(3)-equivariant graph framework with Complete Differential Invariants and Frames (CDIF). Specifically, we first systematically construct a maximal functionally independent set of basic invariant scalars derived from system states, providing a theoretically complete representational basis for arbitrary SE(3)-invariant functions. We then design local frames between objects that integrate both positional and differential components to recover directional information from the constructed invariants, which enables robust recovery of directional information while enhancing the model’s capability of dynamic modeling. Extensive experiments across diverse domains including molecular dynamics, formation control, human motion capture and particle simulation, validate that our CDIF framework outperforms state-of-the-art baselines on most benchmarks, and exhibits strong scalability for large-scale multi-object scenarios.