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HAMLET: Graph Transformer Neural Operator for Partial Differential Equations
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Proposes PhysGTN (HAMLET), a graph transformer neural operator with modular input encoders, circular-radius graph construction, and a CrossFormer query decoder for solving PDEs on arbitrary geometries, evaluated on PDEBench and Airfoil.
Problem
Learning solution operators for parametric PDEs that generalise across geometries, discretisations, input formats, and limited data, where PINNs and existing neural operators have limitations.
Approach
Uses a graph transformer built on radius-based (circular) graph connectivity with RoPE positional encoding as the input encoder, a Galerkin-style cross-attention CrossFormer to fuse input features with arbitrary query locations, and an MLP recurrent propagator for time-dependent decoding; claims the graph transformer block is a special case of a neural operator integral kernel.
Key results
- On PDEBench Darcy Flow, PhysGTN achieves best nRMSE at beta=0.1, 1.0, 10.0 and 100.0 (e.g., 1.40E-02 at beta=1.0 vs 2.05E-02 for OFormer), but not at beta=0.01 where OFormer is best (3.04E-01 vs 3.11E-01).
- On Diffusion Reaction, PhysGTN reports nRMSE 9.02E-02 versus 1.20E+00-range baselines such as FNO at 1.20E-01 and OFormer at 3.28E+00.
- On Airfoil relative L2, PhysGTN obtains 3.03E-02 versus 3.49E-02 for OFormer and 1.19E-01 for EAGLE.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| use HAMLET instead of the strongest listed baseline | Darcy Flow β=0.01 · Darcy Flow β=0.01 | nRMSE: 7.71E-02 -> 3.11E-01 | Hurt |
| use HAMLET instead of the strongest listed baseline | HAMLET · Darcy Flow β=0.1 | nRMSE: 8.10E-02 -> 7.65E-02 | Helped |
| use HAMLET instead of the strongest listed baseline | HAMLET · Darcy Flow β=1.0, 10.0, and 100.0 | nRMSE: 2.05E-02 -> 1.40E-02; 6.34E-03 -> 4.77E-03; 4.19E-03 -> 3.46E-03 | Helped |
| use HAMLET instead of the strongest listed baseline | HAMLET · Shallow Water | nRMSE: 2.35E-03 -> 2.04E-03 | Helped |
| use HAMLET instead of the strongest listed baseline | HAMLET · Diffusion Reaction | nRMSE: 1.20E-01 -> 9.02E-02 | Helped |
| use HAMLET instead of the strongest listed baseline | HAMLET · Airfoil | relative L2 error: 3.486E-02 -> 3.030E-02 | Helped |
| replace circular graph construction with KNN connectivity (ablation) | HAMLET · Darcy Flow β=1.0, 64 × 64 grid | nRMSE: 0.01402 -> 0.02018; 0.01996; 0.02054; 0.02094 | Hurt |
| increase the circular graph radius (ablation) | HAMLET · Darcy Flow | nRMSE | Helped |
| use RoPE rather than coordinate concatenation or no position encoding (ablation) | HAMLET · Position encoding comparison | nRMSE | Helped |
| reduce training data from 9K to 1K (ablation) | HAMLET · Darcy Flow β=1.0 | nRMSE: 1.402E-02 -> 2.779E-02 | Hurt |
| reduce training data from 900 to 100 (ablation) | HAMLET · Shallow Water | nRMSE: 2.044E-03 -> 4.746E-03 | Hurt |
| use HAMLET instead of OFormer with reduced training data | HAMLET · Darcy Flow β=1.0 and Shallow Water across reported training sizes | nRMSE | Helped |
Surprising findings
- In the appendix RMSE table, OFormer outperforms PhysGTN on most Darcy Flow settings (e.g., beta=0.01, 0.1, 1.0, 10.0, 100.0 for OFormer 2.21E-03 vs 2.45E-03 at beta=0.01), contradicting the main-text claim of superiority.
- MAgNet training failed to converge on Shallow Water and Diffusion Reaction, so those results are omitted.
Limitations
- Graph construction time is a noted cost of the graph-based approach.
- Does not outperform all baselines (e.g., OFormer is best on Darcy beta=0.01 in nRMSE and most RMSE settings).
- Evaluation limited to 2D problems and a few benchmarks; 3D extension left to future work.
Experimental setup
Datasets: PDEBench Darcy Flow PDEBench Shallow Water PDEBench Diffusion Reaction Airfoil (MeshGraphNets)
Benchmarks: PDEBench Darcy Flow PDEBench Shallow Water PDEBench Diffusion Reaction Airfoil
Metrics: nRMSE RMSE relative L2 error
Baselines: U-Net FNO PINN DeepONet Geo-FNO MAgNet GNOT EAGLE OFormer
Builds on: Fourier Neural Operator OFormer Geo-FNO PDEBench Rotary Position Embedding
Closest papers by meaning
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- Graph Neural PDE Solvers with Conservation and Similarity-Equivariance (ICML 2024 · similarity 0.86)
- BENO: Boundary-embedded Neural Operators for Elliptic PDEs (ICLR 2024 · similarity 0.86)
- Geometry Aware Operator Transformer as an efficient and accurate neural surrogate for PDEs on arbitrary domains (NeurIPS 2025 · similarity 0.86)
- PDE-Transformer: Efficient and Versatile Transformers for Physics Simulations (ICML 2025 · similarity 0.85)
- RIGNO: A Graph-based Framework For Robust And Accurate Operator Learning For PDEs On Arbitrary Domains (NeurIPS 2025 · similarity 0.85)
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