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Fourier Neural Operator with Learned Deformations for PDEs on General Geometries
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Proposes Geo-FNO, which learns a deformation from irregular physical domains to a uniform computational grid so that FFT-based Fourier neural operators can solve PDEs on general geometries.
Problem
FNO relies on FFT and is limited to rectangular domains with uniform grids, so irregular geometries and non-uniform meshes require wasteful embedding or interpolation that adds error.
Approach
Learn a coordinate map phi^{-1}(x,a)=x+f(x,a) (MLP with sinusoidal features) end-to-end with the neural operator, so Fourier transforms are computed in a uniform latent space with deformed basis and mesh weights, retaining FFT speed.
Key results
- On elasticity with point-cloud inputs, learned Geo-FNO reaches 0.0229 test relative L2 error versus 0.0508 for FNO with interpolation and 0.0531 for UNet with interpolation.
- On airfoil, Geo-FNO test error is 0.0138 versus 0.0421 for FNO interpolation; on pipe 0.0067 vs 0.0151.
- Geo-FNO is reported to be about 10^5 times faster than numerical solvers on the airfoil problem, with at least 10^4x speed-up at equal accuracy in the cost-accuracy study.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use learned deformation instead of fixed R-mesh deformation (ablation) | Geo-FNO on hyper-elasticity · Hyper-elasticity test | relative L2 error: 0.0536 -> 0.0229 | Helped |
| Use learned deformation instead of fixed O-mesh deformation (ablation) | Geo-FNO on hyper-elasticity · Hyper-elasticity test | relative L2 error: 0.0363 -> 0.0229 | Helped |
| Apply learned geometry deformation rather than direct interpolation | FNO on hyper-elasticity · Hyper-elasticity test | relative L2 error: 0.0508 -> 0.0229 | Helped |
| Apply learned geometry deformation rather than direct interpolation | UNet on hyper-elasticity · Hyper-elasticity test | relative L2 error: 0.0531 -> 0.0229 | Helped |
| Apply Geo-FNO instead of FNO interpolation | Airfoil solver · Airfoil test | relative L2 error: 0.0421 -> 0.0138 | Helped |
| Apply Geo-FNO instead of UNet interpolation | Airfoil solver · Airfoil test | relative L2 error: 0.0519 -> 0.0138 | Helped |
| Apply Geo-FNO instead of FNO interpolation | Pipe solver · Pipe test | relative L2 error: 0.0151 -> 0.0067 | Helped |
| Apply Geo-FNO instead of UNet interpolation | Pipe solver · Pipe test | relative L2 error: 0.0182 -> 0.0067 | Helped |
| Use domain decomposition on the sphere rather than a 2D spherical-map FNO | FNO2D · Advection on unit sphere test | relative L2 error: 0.0381 -> 0.0332 | Helped |
| Use domain decomposition on the sphere rather than a 3D embedding | FNO3D · Advection on unit sphere test | relative L2 error: >100% -> 0.0332 | Helped |
| Evaluate the trained low-resolution Geo-FNO at higher resolution | Geo-FNO on airfoil · Airfoil test at 56 × 51, 111 × 51, and 221 × 51 | relative L2 error: 0.0147, 0.0329, and 0.0428 | Hurt |
Surprising findings
- Learned deformation on elasticity outperforms heuristic R-mesh and O-mesh deformations in test error (0.0229 vs 0.0536 and 0.0363).
- UNet with interpolation achieves low training error (0.0039 on airfoil) but much higher test error (0.0519), indicating overfitting.
Limitations
- Domain decomposition for general genus-n topologies is left as future work; only sphere via hemispheres was demonstrated.
- Theoretical approximation guarantees are not provided.
- Experiments are on relatively small datasets (about 1000 training samples) and a single GPU, with only heuristic comparisons to some baselines.
Experimental setup
Datasets: elasticity (FEM-generated) plasticity (ABAQUS-generated) advection on sphere (finite volume) airfoil Euler (finite volume) pipe Navier-Stokes (FEM)
Benchmarks: hyper-elastic elasticity plasticity forging advection on unit sphere airfoil Euler pipe Navier-Stokes
Metrics: relative L2 error training time model size inference speed-up vs numerical solver
Baselines: FNO with interpolation UNet with interpolation GNO DeepONet Geo-FNO (R mesh) Geo-FNO (O mesh) FNO3D numerical solvers (Euler finite volume)
Builds on: Fourier neural operator adaptive moving mesh methods graph kernel network DeepONet U-Net
Closest papers by meaning
- Operator Learning with Domain Decomposition for Geometry Generalization in PDE Solving (ICLR 2026 · similarity 0.87)
- Beyond Regular Grids: Fourier-Based Neural Operators on Arbitrary Domains (ICML 2024 · similarity 0.87)
- Reference Neural Operators: Learning the Smooth Dependence of Solutions of PDEs on Geometric Deformations (ICML 2024 · similarity 0.85)
- Neural Operators with Localized Integral and Differential Kernels (ICML 2024 · similarity 0.84)
- Adaptive Coordinate Transforms for Neural Operators (NeurIPS 2026 · similarity 0.83)
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