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SEGNO: Generalizing Equivariant Graph Neural Networks with Physical Inductive Biases
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Proposes SEGNO, a second-order equivariant graph neural ODE that wraps an Equiv-GNN backbone (EGNN or GMN) with Euler-type integrators to learn continuous, second-order trajectories for N-body and motion dynamics, with theory on uniqueness and bounded error.
Problem
Equivariant GNNs for multi-object physical dynamics learn discrete mappings between adjacent states and use only first-order velocity information, which limits generalization to longer horizons and intermediate states.
Approach
Parameterize acceleration with an Equiv-GNN and integrate it through a numerical ODE solver (Neural ODE with second-order motion equations) so the model learns instantaneous acceleration and a continuous latent trajectory while preserving E(3) equivariance of the backbone; it is plug-and-play with any Equiv-GNN backbone.
Key results
- On Charged N-body at 1000/1500/2000 ts, SEGNO reaches MSE 0.433/1.858/4.285 (x1e-2) vs best baseline SEGNN 0.448/2.573/5.972.
- On MD22 seven molecules, SEGNO (GMN backbone) gives 15.6% average relative improvement over GMN.
- On CMU motion capture at 50 ts, SEGNO MSE 29.264 (x1e-2) vs runner-up GMN 47.883.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Replace the EGNN baseline with SEGNO's continuous second-order EGNN model | EGNN on simulated N-body systems · Charged and Gravity, 1000, 1500, and 2000 ts | MSE (×10^-2): Charged: 0.711 -> 0.433; Gravity: 0.766 -> 0.338 | Helped |
| Apply SEGNO with a GMN backbone | GMN · MD22, seven molecules | MSE (×10^-3): average relative improvement ... 15.6% | Helped |
| Apply SEGNO with a GMN backbone | GMN · CMU motion capture, 30, 40, and 50 ts | MSE (×10^-2): 16.005 -> 14.462; 38.193 -> 22.229; 47.883 -> 29.264 | Helped |
| Add continuous parameter sharing across solver iterations (ablation) | First-order discrete and second-order discrete SEGNO variants · Simulated N-body systems, all reported settings | MSE (×10^-2) | Helped |
| Add second-order acceleration modeling instead of first-order velocity modeling (ablation) | First-order SEGNO variants · Simulated N-body systems, all reported settings | MSE (×10^-2) | Helped |
| Increase solver iteration count, reducing the step interval Δt (ablation) | SEGNO · Charged and Gravity simulated N-body systems | MSE | Helped |
| Continue increasing solver iterations beyond approximately 10 (ablation) | SEGNO · Charged and Gravity simulated N-body systems | MSE: approximately 10 | No clear effect |
Surprising findings
- Discrete models trained with the same backbone show large error and variance at intermediate states, while continuous models do not, suggesting discrete models fail to learn true latent dynamics.
Limitations
- Theoretical bounds rely on strong assumptions such as analyticity, boundedness, and Lipschitz continuity, and on near-perfect approximation.
- Some baselines (TFN, SE(3) Transformer) ran out of memory on MD22 and were omitted.
Experimental setup
Datasets: Charged N-body simulation Gravity N-body simulation MD22 CMU motion capture
Benchmarks: Charged N-body Gravity N-body MD22 CMU motion capture
Metrics: mean squared error
Baselines: Linear GNN GDE TFN SE(3) Transformer Radial Field EGNN GMN SEGNN
Builds on: EGNN GMN Neural ODE GDE SEGNN
Closest papers by meaning
- Equivariant Graph Neural Operator for Modeling 3D Dynamics (ICML 2024 · similarity 0.92)
- Permutation Equivariant Neural Controlled Differential Equations for Dynamic Graph Representation Learning (NeurIPS 2025 · similarity 0.88)
- CSG-ODE: ControlSynth Graph ODE For Modeling Complex Evolution of Dynamic Graphs (ICML 2025 · similarity 0.88)
- Signed Graph Neural Ordinary Differential Equation for Modeling Continuous-Time Dynamics (AAAI 2024 · similarity 0.87)
- Rethink GraphODE Generalization within Coupled Dynamical System (ICML 2025 · similarity 0.87)
- ControlSynth Neural ODEs: Modeling Dynamical Systems with Guaranteed Convergence (NeurIPS 2024 · similarity 0.87)
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