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Energy-conserving equivariant GNN for elasticity of lattice architected metamaterials
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Introduces a dataset of ~635k strut-based lattice geometries with FE-computed stiffness tensors and an SE(3)-equivariant MACE-based GNN whose output is made positive semi-definite via a matrix-square layer in Mandel notation, enforcing energy conservation.
Problem
Machine learning surrogates for lattice metamaterial stiffness tensors lack physical grounding: they are not guaranteed to be equivariant to rotations and can predict negative deformation energy (non-positive-semi-definite stiffness).
Approach
Combine an SE(3)-equivariant higher-order message-passing network (MACE) with a positive semi-definite output layer applied to the 6x6 stiffness matrix in Mandel notation (matrix square), which the authors prove preserves equivariance, unlike Voigt notation.
Key results
- MACE+ve (equivariant with PSD layer) reaches L_comp 3.61 vs 8.37 for base CGC and 3.88 for base MACE, with 0% lattices having negative eigenvalues versus 30% for MACE.
- The matrix-square PSD layer gives the best L_comp (3.61) among exponential, power, and truncated-exponential variants.
- All GNN models are about three orders of magnitude faster than FE (5-15 s vs 1.1e4 s for 5000 lattices).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| add an equivariant MACE backbone instead of non-equivariant CGC | stiffness prediction on 1imp · 1imp | L_comp: 9.31 -> 3.88 | Helped |
| add rotation augmentation to CGC (ablation) | CGC on 1imp · 1imp | L_comp: 8.37 -> 4.47 | Helped |
| add rotation augmentation to CGC (ablation) | CGC on 1imp · 1imp | L_equiv: 0.33 -> 0.15 | Helped |
| add rotation augmentation to CGC (ablation) | CGC on 1imp · 1imp | lambda_%^-: 4 -> 26 | Hurt |
| add rotation augmentation to MACE (ablation) | MACE on 1imp · 1imp | L_comp: 3.88 -> 3.47 | Helped |
| add rotation augmentation to MACE (ablation) | MACE on 1imp · 1imp | lambda_%^-: 30 -> 34 | Hurt |
| add a PSD layer to MACE using matrix square (ablation) | MACE on 1imp · 1imp | lambda_%^-: 30 -> 0 | Helped |
| add a PSD layer to MACE using matrix square (ablation) | MACE on 1imp · 1imp | L_comp: 3.88 -> 3.61 | Hurt |
| add a directional-projection penalty for negative stiffness (ablation) | MACE on 1imp · 1imp | lambda_%^-: 30 -> 29 | No clear effect |
| use matrix square rather than matrix exponential in the PSD layer (ablation) | MACE+ve on 1imp · 1imp | L_comp: 4.58 -> 3.61 | Helped |
| use matrix square rather than fourth matrix power in the PSD layer (ablation) | MACE+ve on 1imp · 1imp | L_comp: 4.41 -> 3.61 | Helped |
| increase message body-order from 2-body to three- and four-body terms (ablation) | MACE · perturbed test dataset | test error | Helped |
Surprising findings
- Adding a positive semi-definite layer to CGC and NNConv models increased stiffness errors, and rotation augmentation made CGC and NNConv more prone to negative eigenvalues.
- A learning-bias penalty on negative directional stiffness was not effective at removing negative eigenvalues (29% vs 30%).
Limitations
- Positive semi-definite layer slightly degraded or did not improve stiffness errors for CGC/NNConv, and MACE+ve's gain over MACE is modest in component loss.
- Test set is a single perturbation level (0.1) for OOD evaluation; generalization to other lattice families is not shown beyond the dataset.
- Dataset limited to strut-based lattices with cylindrical struts and fixed relative densities.
Experimental setup
Datasets: lattice dataset from Bastek et al. built on crystallographic databases FE-computed stiffness tensors for perturbed lattices
Benchmarks: held-out perturbed lattice test set (level 0.1, OOD)
Metrics: L_comp L_dir L_dir,rel L_equiv percentage of lattices with negative eigenvalues inference time
Baselines: CGC CGC+tr NNConv NNConv+tr MACE mCGC finite element method
Builds on: MACE E(3)-equivariant message passing Crystal Graph Convolutions NNConv
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