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Curvature-aware Graph Attention for PDEs on Manifolds
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Proposes Curv-GT, a curvature-aware graph attention for learning time-dependent PDEs on discrete 2D manifolds, using parallel transport, tensor fields, closed-form transport on constant-curvature surfaces, and subtree-based parameter sharing.
Problem
Neural operators for time-dependent PDEs on discrete manifolds (graphs with node coordinates) mostly ignore intrinsic geometry such as curvature, relying on spatial coordinates or Euclidean assumptions.
Approach
Generalizes vanilla graph attention to manifolds by moving neighbor features via parallel transport and replacing linear maps with (1,1) and (0,2) tensor fields; edges are embedded in sphere, plane, or pseudosphere according to local Gaussian curvature so transport has closed form, and subtree partitioning shares parameters across nodes via transport.
Key results
- Curv-GT achieves the lowest relative L2 error on all three wrinkle benchmarks, e.g., 0.025% on isotropic diffusion vs. 0.040% for GNOT and 0.681% for Transolver.
- On p-Laplacian diffusion, curvature-aware attention lowers GT's L2 error from 9.70% to 1.58% and GNOT's from 4.39% to 0.92%.
- Curvature-aware GAT reaches 3.41% L2 error vs. 8.47% for non-curvature-aware GAT on p-Laplacian diffusion.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| replace vanilla attention with curvature-aware attention (ablation) | GAT · Wrinkle manifold, p-Laplacian diffusion | L2 relative error (%): 8.47±2.82 -> 3.41±0.59 | Helped |
| directly concatenate point curvatures to node features (ablation) | GAT · Wrinkle manifold, p-Laplacian diffusion | L2 relative error (%): 8.47±2.82 -> 6.92±1.86 | Helped |
| aggregate information with a shared linear mapping (ablation) | GAT · Wrinkle manifold, p-Laplacian diffusion | L2 relative error (%): 8.47±2.82 -> 4.57±0.70 | Helped |
| replace vanilla attention with curvature-aware attention (ablation) | GraphTransformer · Wrinkle manifold, p-Laplacian diffusion | L2 relative error (%): 9.70±4.24 -> 1.58±0.84 | Helped |
| replace vanilla attention with curvature-aware attention (ablation) | GNOT · Wrinkle manifold, p-Laplacian diffusion | L2 relative error (%): 4.39±1.92 -> 0.92±0.31 | Helped |
| increase the number of curvature-aware attention heads (ablation) | Curvature-aware attention · Wrinkle manifold | square loss on test set | Helped |
| use too large a maximum subtree depth (ablation) | Curvature-aware attention · Wrinkle manifold | square loss on test set | Hurt |
Surprising findings
- Directly concatenating point curvatures to node features (6.92%) or using a shared linear mapping (4.57%) performs much worse than curvature-aware attention (3.41%) for GAT.
Limitations
- Complexity grows with subtree depth d, and performance drops when depth is too large.
- Method restricted to 2-dimensional manifolds in the study; generalization to higher dimensions not shown.
- Subtree partitioning can be imbalanced due to random source selection.
Experimental setup
Datasets: wrinkle manifold PDE data (isotropic diffusion, p-Laplacian diffusion, wave equation) torus elliptic paraboloid
Benchmarks: wrinkle manifold isotropic diffusion wrinkle manifold p-Laplacian diffusion wrinkle manifold wave equation
Metrics: relative L2 error (%) relative H1 error (%) training time per epoch
Baselines: ResNet GCN DeepONet GraphTransformer FourierType GalerkinType Graph U-Net MKGN GNOT GINO Geo-FNO Transolver
Builds on: Graph Transformer Graphormer Geo-FNO Transolver Graph Attention Networks
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