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Thickness-aware E(3)-Equivariant 3D Mesh Neural Networks
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Proposes T-EMNN, a thickness-aware E(3)-equivariant mesh GNN that links each surface node to its opposing-surface "thickness pair" via a learned, thresholded thickness edge, plus PCA-based data-driven invariant coordinates, evaluated on an injection-molding deformation dataset.
Problem
Surface-mesh GNN surrogates for 3D static analysis ignore the thickness of objects, so interactions between opposing surfaces are not modeled, and spatial coordinates are hard to use without breaking E(3)-equivariance.
Approach
Adds thickness edges connecting each node to its normal-direction opposing-surface pair, gated by a learnable thickness threshold via a sigmoid activation, and encodes coordinates with a PCA-based data-driven canonical frame that is E(3)-invariant (with an inverse transform for equivariance).
Key results
- T-EMNN reaches RMSE 0.2132, MAE 0.0892, R^2 0.9228 in-distribution, and RMSE 0.2131, R^2 0.9513 out-of-distribution, the best among compared models.
- Removing thickness edges raises in-distribution RMSE to 0.2156 and drops R^2 to 0.9148 in the ablation.
- On the Deforming Plate dataset, T-EMNN improves R^2 from 0.7007 to 0.7579 when thickness edges are included.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Add thickness edges and the thickness processor | T-EMNN · Injection molding, in-distribution (original) | RMSE: 0.2210 (0.0057) -> 0.2132 (0.0046) | Helped |
| Add thickness edges and the thickness processor | T-EMNN · Injection molding, out-of-distribution (rotated) | RMSE: 0.2210 (0.0057) -> 0.2131 (0.0046) | Helped |
| Add thickness edges (ablation) | T-EMNN · Deforming Plate, next timestep | RMSE (×10³): 17.420 -> 14.903 | Helped |
| Add thickness edges (ablation) | T-EMNN · Deforming Plate, next timestep | R²: 0.7007 -> 0.7579 | Helped |
| Add thickness distance to thickness-edge features (ablation) | T-EMNN · Injection molding, in-distribution (original) | RMSE: 0.2156 (0.0064) -> 0.2132 (0.0046) | Helped |
| Add normal-vector dot product to thickness-edge features (ablation) | T-EMNN · Injection molding, in-distribution (original) | RMSE: 0.2191 (0.0143) -> 0.2132 (0.0046) | Helped |
| Use invariant data-driven coordinates instead of original coordinates | MGN · Injection molding, out-of-distribution (rotated) | RMSE: 0.4514 (0.0190) -> 0.2241 (0.0042) | Helped |
| Use invariant data-driven coordinates instead of original coordinates | EGNN · Injection molding, out-of-distribution (rotated) | RMSE: 0.4576 (0.0184) -> 0.2271 (0.0019) | Helped |
| Use invariant data-driven coordinates instead of original coordinates | EMNN · Injection molding, out-of-distribution (rotated) | RMSE: 0.4668 (0.0180) -> 0.2210 (0.0057) | Helped |
| Set the thickness threshold to zero, removing thickness edges (ablation) | T-EMNN · Injection molding | performance | Hurt |
| Use a thickness threshold above 10 (ablation) | T-EMNN · Injection molding | performance | Hurt |
Surprising findings
- Original-coordinate EGNN and EMNN variants reach extremely negative R^2 (about -14000 to -34000) on the rotated test set in the unnormalized baselines, showing that non-invariant coordinate use fails badly under rotation.
Limitations
- Evaluated on a single proprietary-style industrial injection-molding dataset with a single material assumption.
- The PCA sign-disambiguation criterion is not well-defined for perfectly symmetric shapes, per the appendix.
- Dynamic-setting evaluation uses a single benchmark and the coordinate system is switched to the original frame.
Experimental setup
Datasets: injection molding dataset (504 samples, 28 geometries) Deforming Plate dataset
Benchmarks: injection molding dataset (in-distribution and rotated out-of-distribution) Deforming Plate dataset
Metrics: RMSE MAE R^2
Baselines: MGN EGNN EMNN MLP
Builds on: MeshGraphNets EGNN EMNN
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