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BENO: Boundary-embedded Neural Operators for Elliptic PDEs
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Proposes BENO, a dual-branch GNN neural operator with a Transformer boundary encoder that embeds complex boundary geometry and inhomogeneous boundary values to solve elliptic (Poisson) PDEs.
Problem
Neural operators for elliptic PDEs fail to handle complex boundary geometries and inhomogeneous Dirichlet boundary values, which create severe high-frequency solution fluctuations and global influence on the interior solution.
Approach
Inspired by Green's function, split the solution into an interior-source branch and a boundary-value branch, each a GNN message-passing network, with a Transformer encoding the global boundary into a latent vector that conditions every message passing layer.
Key results
- On 4-Corners training, BENO reaches relative L2 of 0.3523 on the 4-Corners test set while GKN, FNO, GNN-PDE and MP-PDE all sit near 1.0 L2.
- BENO trained on 4-Corners with homogeneous boundary values reduces L2 to about 0.09-0.24 across shapes, versus about 0.26-0.55 for MP-PDE, roughly 60.96% average improvement claimed.
- Ablations show removing the dual-branch design (w/o D) or replacing the Transformer with GCN (w. G) degrades performance; for example, w. G yields L2 near 0.9 on 4-Corners.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| use BENO instead of the strongest baseline, MP-PDE | Elliptic PDE solver trained on 4-Corners · 4-Corners test set | relative L2: 1.0007 ± 0.0677 -> 0.3523 ± 0.1245 | Helped |
| use BENO instead of the strongest baseline, MP-PDE | Elliptic PDE solver trained on 4-Corners · 3-Corners, 2-Corners, 1-Corner, and No-Corner test sets | relative L2: 52.18%, 52.43%, 47.38%, and 52.94% | Helped |
| replace boundary-embedded message passing with vanilla message passing (ablation) | BENO · 4-Corners test set | relative L2: 0.3523 ± 0.1245 -> 1.0130 ± 0.0858 | Hurt |
| remove dual-branch structure (ablation) | BENO · 4-Corners test set | relative L2: 0.3523 ± 0.1245 -> 0.4058 ± 0.1374 | Hurt |
| add Transformer output to edge message passing (ablation) | BENO · 4-Corners test set | relative L2: 0.3523 ± 0.1245 -> 0.4113 ± 0.1236 | Hurt |
| replace Transformer with vanilla graph convolution (ablation) | BENO · 4-Corners test set | relative L2: 0.3523 ± 0.1245 -> 0.9037 ± 0.1104 | Hurt |
Surprising findings
- All re-implemented baselines (GKN, FNO, GNN-PDE, MP-PDE) fail to learn the solution operator with inhomogeneous boundary values, with relative L2 near 1.0.
- Injecting the global boundary embedding into edge messages (w. E) did not help relative to node-level injection.
Limitations
- Only Dirichlet boundary conditions are studied; Neumann and Robin are not addressed in the main method.
- Generalization is tested on synthetic 2D Poisson problems on square domains with corner variations.
- Baselines were re-implemented by the authors with boundary nodes added, which may affect fairness.
Experimental setup
Datasets: 4-Corners 3-Corners 2-Corners 1-Corner No-Corner mixed corner dataset (generated with finite volume / Gauss-Seidel solver)
Benchmarks: 4-Corners 3-Corners 2-Corners 1-Corner No-Corner 64x64 resolution test zero boundary value test
Metrics: relative L2 error MAE
Baselines: GKN FNO GNN-PDE MP-PDE
Builds on: Graph Network Simulator MP-PDE GNN-PDE Fourier Neural Operator
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