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Neural Green’s Functions
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Proposes Neural Green's Function, a neural solution operator that predicts eigendecomposition-based Green's function features and per-vertex masses from domain point clouds, then evaluates solutions by numerical integration, generalizing across unseen source and boundary functions.
Problem
Learning solution operators for linear PDEs (e.g., Poisson, Biharmonic) over irregular 3D geometries that generalize to unseen source and boundary functions without retraining or meshing.
Approach
Predict features from domain geometry alone that approximate the eigendecomposition of the Green's function (G = Phi Lambda^-1 Phi^T) plus per-vertex mass and boundary coupling terms, so the network is independent of source/boundary functions; solutions are then computed by numerical integration with these predicted terms.
Key results
- Average 13.9% error reduction over Transolver on five MCB shape categories (e.g., Screws & Bolts 0.189 vs 0.221; Gear 0.243 vs 0.281).
- 2D Poisson test relative L2 error 0.012 vs Transolver 0.372; Biharmonic test 0.009 vs 0.337.
- Up to 350x faster than FEM solver with mesh generation, with runtimes comparable to neural operator baselines.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| use geometry-derived Green's function features instead of direct function-conditioned prediction | Neural Green's Function vs. Transolver · 2D Poisson and Biharmonic test sets | relative L2 error: 0.372 -> 0.012; 0.337 -> 0.009 | Helped |
| use geometry-derived Green's function features instead of direct function-conditioned prediction | Neural Green's Function vs. Transolver · MCB thermal analysis, five shape categories | relative L2 error: 0.221 -> 0.189; 0.320 -> 0.275; 0.407 -> 0.338; 0.180 -> 0.160; 0.281 -> 0.243 | Helped |
| use Neural Green's Function instead of FEM | MCB thermal analysis · MCB thermal analysis, five shape categories | runtime: up to a $350\times$ speedup | Helped |
| add mass regularization during training (ablation) | Neural Green's Function · MCB Screws & Bolts test set | relative L2 error: 0.285 -> 0.189 | Helped |
| add mass regularization during training (ablation) | Neural Green's Function · MCB Gear test set | relative L2 error: 0.411 -> 0.243 | Helped |
| use feature dimension 64 instead of 128 (ablation) | Neural Green's Function · MCB Screws & Bolts test set | relative L2 error: 0.189 -> 0.180 | No clear effect |
| use feature dimension 256 instead of 128 (ablation) | Neural Green's Function · MCB Screws & Bolts test set | relative L2 error: 0.189 -> 0.206 | No clear effect |
Surprising findings
- Without mass regularization, errors rise substantially (e.g., Screws & Bolts 0.285 vs 0.189), since per-vertex masses are on the order of 1e-4.
Limitations
- Only Dirichlet boundary conditions; Neumann and Robin not handled.
- Limited to linear PDEs whose operators admit eigendecompositions (e.g., Poisson, Biharmonic).
- Numerical integration step adds forward-pass cost; no error bars reported.
Experimental setup
Datasets: MCB FEM-generated Poisson solutions (fTetWild meshes)
Benchmarks: MCB-based steady-state thermal analysis benchmark (5 categories) 2D Poisson and Biharmonic examples
Metrics: relative L2 error runtime (s)
Baselines: Transolver Latent Neural Operator (LNO) Universal Physics Transformer (UPT)
Builds on: Transolver Green's function theory finite element method
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