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Physics-Embedded Neural Networks: Graph Neural PDE Solvers with Mixed Boundary Conditions
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Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| use NIsoGCN (Neumann-aware gradient) instead of original IsoGCN | IsoGCN · Gradient dataset | MSE gradient (x1e-3): 192.72 -> 6.70 | Helped |
| use NIsoGCN instead of IsoGCN, loss on Neumann boundary only | IsoGCN · Gradient dataset | MSE Neumann boundary (x1e-3): 1390.95 -> 3.52 | Helped |
| use E(n)-equivariant PENN instead of MP-PDE on rotated/translated test data | MP-PDE · Incompressible flow, transformed | MSE u (x1e-4): 1953.62 -> 4.36 | Helped |
| PENN vs MP-PDE on untransformed test data | MP-PDE · Incompressible flow | MSE u (x1e-4): 1.30 -> 4.36 | Hurt |
| add boundary encoder (encoded boundary) (ablation) | PENN · Incompressible flow | MSE: Divergent -> 4.36 | Helped |
| add boundary conditions inside the neural nonlinear solver (ablation) | PENN · Incompressible flow | MSE u (x1e-4): 65.10 -> 4.36 | Helped |
| add neural nonlinear solver (ablation) | PENN · Incompressible flow | MSE u (x1e-4): 31.03 -> 4.36 | Helped |
| add boundary condition input (ablation) | PENN · Incompressible flow | MSE u (x1e-4): 20.08 -> 4.36 | Helped |
| add Dirichlet layer (ablation) | PENN · Incompressible flow | MSE u (x1e-4): 8.22 -> 4.36 | Helped |
| add pseudoinverse decoder (ablation) | PENN · Incompressible flow | MSE u (x1e-4): 8.91 -> 4.36 | Helped |
| pseudoinverse decoder instead of Dirichlet layer after decoding (ablation) | PENN · Incompressible flow | MSE u (x1e-4): 6.65 -> 4.36 | Helped |
Closest papers by meaning
- Graph Neural PDE Solvers with Conservation and Similarity-Equivariance (ICML 2024 · similarity 0.93)
- PhyMPGN: Physics-encoded Message Passing Graph Network for spatiotemporal PDE systems (ICLR 2025 · similarity 0.89)
- BENO: Boundary-embedded Neural Operators for Elliptic PDEs (ICLR 2024 · similarity 0.87)
- Hyperbolic-PDE GNN: Spectral Graph Neural Networks in the Perspective of A System of Hyperbolic Partial Differential Equations (ICML 2025 · similarity 0.87)
- PINNacle: A Comprehensive Benchmark of Physics-Informed Neural Networks for Solving PDEs (NeurIPS 2024 · similarity 0.87)
- RoPINN: Region Optimized Physics-Informed Neural Networks (NeurIPS 2024 · similarity 0.86)
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