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Integrating GNN and Neural ODEs for Estimating Non-Reciprocal Two-Body Interactions in Mixed-Species Collective Motion
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Integrates a dynamic-edge GNN with neural ODE/SDE solvers to infer pairwise (including non-reciprocal) forces in toy harmonic and mixed-species slime-mold-like collective motion models from trajectories.
Problem
Estimating the equations of motion and two-body interaction functions from observed trajectories of many interacting agents, including mixed-species systems with non-reciprocal interactions, where prior system-identification methods (SINDy, Bayesian optimization) scale poorly or are untested.
Approach
Treat the system as a graph whose edges are defined dynamically by a distance threshold at each time step, compute self-propulsion and pairwise forces with neural networks inside a GNN, and embed this GNN in a neural ODE/SDE solver so the force networks are trained by matching multi-step-ahead simulated states to data, avoiding precomputed edge time series and dense graphs.
Key results
- On the mixed-species model, the estimated F(1)_r has normalized MSE on the order of 10^-3 to 10^-4 across parameter sets (e.g., 1.5e-3 for set (i)).
- Interaction-force errors for the mixed-species model are higher, with normalized MSE for F(2)_r ranging from 0.19 to 0.99 and F(2)_phi from 0.05 to 0.24.
- With the harmonic model, estimation with 3 training trajectories failed for gamma=1e-2 (F(1)_v MSE 1.7 and 9.4) but succeeded for gamma=1e-1 (MSE 1.2e-2 and 5.4e-2).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Increase the training set from Mtra=3 to Mtra=270 | Proposed method · Harmonic interaction, γ=1×10⁻² | MSE(F¹ᵥ): 1.7 -> 3.6\times10^{-3}; 9.4 -> 3.6\times10^{-3} | Helped |
| Increase friction from γ=1×10⁻² to γ=1×10⁻¹ | Proposed method · Harmonic interaction, Mtra=3 | MSE(F¹ᵥ): 1.7 -> 1.2\times10^{-2}; 9.4 -> 5.4\times10^{-2} | Helped |
| Increase friction from γ=1×10⁻² to γ=1×10⁻¹ | Proposed method · Harmonic interaction, Mtra=3 | MSE(F²ᵥ): 2.2\times10^{-2} -> 2.2\times10^{-2}; 2.2\times10^{-2} -> 2.3\times10^{-2} | No clear effect |
| Estimate forces from mixed-species trajectories with species-dependent interactions | Proposed method · Mixed-species model, parameter sets (i) and (ii) | Force recovery | Helped |
| Train with species-independent parameters as a negative control | Proposed method · Mixed-species model, parameter sets (iii)–(v) | Species-dependent force recovery | No clear effect |
Surprising findings
- Estimated models trained on mixed-species data reproduced species-dependent behavior even though the species dependence was learned from only 3 trajectories, and negative-control models with identical species gave minimal dependence on species type.
- Training was very slow: 300 epochs took 15 days on the harmonic model and 20,000 epochs took 25 days for the mixed-species model.
Limitations
- Estimation requires very long training times (days per model), making it costly.
- Only deterministic pairwise interactions are estimated; three-body interactions and noise effects are not handled.
- Weak effects such as very small friction are not reliably estimated.
Experimental setup
Datasets: simulated harmonic interaction model trajectories simulated mixed-species slime-mold-like collective motion trajectories
Benchmarks: simulated harmonic interaction model simulated mixed-species collective motion model
Metrics: normalized MSE normalized MAE position and orientation prediction loss
Baselines: SINDy
Builds on: neural ODE neural SDE SINDy Vicsek model LAMB optimizer
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