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Equivariant Graph Neural Operator for Modeling 3D Dynamics
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Proposes EGNO, an SE(3)-equivariant Fourier neural operator that maps a current 3D state to its full future trajectory in one forward pass, evaluated on N-body, motion capture, and molecular dynamics.
Problem
Learned models of 3D relational dynamics usually predict only the next step, so they fail to capture temporal correlations along trajectories, and existing equivariant methods do not model trajectories as functions of time.
Approach
Treat dynamics as a function over time and learn an equivariant neural operator: temporal convolutions implemented as Fourier-space kernels with block-diagonal weights on scalar features and directional vectors keep SO(3)/SE(3) equivariance, stacked with EGNN layers to decode the whole trajectory in parallel.
Key results
- On N-body simulation, EGNO reaches F-MSE 0.0054 versus 0.0071 for EGNN (about 18% relative improvement).
- On MD17 Aspirin, EGNO achieves MSE 9.18 (x10^-2) versus 14.41 for EGNN, a 36% improvement; it is best on all eight molecules.
- On motion capture, EGNO gives roughly 52% average F-MSE improvement over baselines (e.g., Walk 8.1 vs 28.7 for EGNN).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Replace EGNN's single-state prediction with EGNO trajectory modeling | EGNN · N-body S2S | F-MSE: 0.0071 -> 0.0054 | Helped |
| Replace EGNN-S sequential trajectory readout with EGNO | EGNN · N-body S2T | A-MSE: 0.0045 -> 0.0022 | Helped |
| Replace EGNN-S with EGNO trajectory modeling | EGNN · CMU Motion Capture Walk and Run | F-MSE (× 10^-2): 26.4 -> 8.1; 54.2 -> 33.9 | Helped |
| Add equivariant temporal convolution to EGHN | EGHN · AdK equilibrium trajectory | F-MSE: 2.034 -> 1.801 | Helped |
| Add velocity to temporal convolution (ablation) | EGNO · N-body | F-MSE: 0.0057 -> 0.0055 | Helped |
| Add velocity to temporal convolution (ablation) | EGNO · CMU Motion Capture Run | F-MSE (× 10^-2): 35.6 -> 33.9 | Helped |
| Add directional position features to temporal convolution (ablation) | EGNO · N-body | F-MSE: 0.0061 -> 0.0055 | Helped |
| Add directional position features to temporal convolution (ablation) | EGNO · CMU Motion Capture Run | F-MSE (× 10^-2): 39.1 -> 33.9 | Helped |
| Add temporal convolution (ablation) | EGNO · N-body | F-MSE: 0.0072 -> 0.0055 | Helped |
| Add temporal convolution (ablation) | EGNO · CMU Motion Capture Run | F-MSE (× 10^-2): 48.2 -> 33.9 | Helped |
| Increase temporal modes from 2 to 3 (ablation) | EGNO · N-body and CMU Motion Capture Run | F-MSE | Mixed |
| Increase the number of training timesteps from P=5 to P=10 (ablation) | EGNO · N-body | F-MSE: 0.0055 -> 0.0054 | No clear effect |
Surprising findings
- EGNO-T2T, which conditions on multiple past states, achieves better A-MSE (0.0020) than the standard EGNO on N-body, which the authors note uses more input information.
- Removing velocity from the Fourier temporal convolution hurts, and removing all directional information degrades performance further, indicating directional features must be processed in the temporal layer.
Limitations
- Evaluated only on simulated or captured trajectories in a few domains; scaling to fluids or deformable materials is left to future work.
- Temporal convolution is equivariant to rotations, with translation equivariance handled by centering on center of mass, a heuristic rather than a built-in symmetry.
Experimental setup
Datasets: N-body charged particle simulation CMU Motion Capture MD17 AdK equilibrium trajectory (MDAnalysis)
Benchmarks: N-body simulation CMU Motion Capture MD17 AdK protein trajectory
Metrics: F-MSE A-MSE
Baselines: Linear SE(3)-Transformer TFN MPNN Radial Field ClofNet EGNN EGNN-R EGNN-S EGHN
Builds on: Fourier neural operator EGNN temporal bundling
Closest papers by meaning
- SEGNO: Generalizing Equivariant Graph Neural Networks with Physical Inductive Biases (ICLR 2024 · similarity 0.92)
- Equivariant Spatio-Temporal Attentive Graph Networks to Simulate Physical Dynamics (NeurIPS 2023 · similarity 0.91)
- Non-stationary Equivariant Graph Neural Networks for Physical Dynamics Simulation (NeurIPS 2025 · similarity 0.89)
- Large Language-Geometry Model: When LLM meets Equivariance (ICML 2025 · similarity 0.88)
- Equivariant Neural Tangent Kernels (ICML 2025 · similarity 0.87)
- Equivariant Spherical Transformer for Efficient Molecular Modeling (NeurIPS 2026 · similarity 0.87)
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