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Beyond Regular Grids: Fourier-Based Neural Operators on Arbitrary Domains
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Proposes Direct Spectral Evaluation (DSE), which computes truncated Fourier/spherical-harmonic transforms on arbitrary point clouds as a precomputed matrix-vector product, and plugs it into FNO, UFNO, FFNO and SFNO.
Problem
FFT-based neural operators such as FNO are limited to equispaced grids, so they cannot efficiently process inputs and outputs sampled on non-equispaced points or point clouds.
Approach
Replace the FFT inside spectral convolution layers with a direct evaluation of a truncated (few-mode) transform: build a matrix of basis functions evaluated at the actual sample points and apply it via batched matmul, with the inverse given by the conjugate transpose. Cost is O(mN) because only a small fixed number of modes is kept.
Key results
- On the contracting-expanding Burgers grid, FNO-DSE reaches 0.184% L1 error at 0.11 s/epoch versus 0.195% and 1.00 s for cubic interpolation plus FFT.
- On the 2D shear layer lattice, FNO-DSE gets 5.53% error at 57 s/epoch versus 6.76% at 251 s for FFT on the equispaced grid.
- On the spherical shallow water task, SFNO-DSE reaches 3.88% error at 15 s/epoch versus 7.29% for Gaussian interpolation (86 s) and 12.7% for linear interpolation (71 s).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Replace FFT with direct spectral evaluation | FNO · Burgers' Equation, equispaced | L1 test error: 0.0575% -> 0.0551% | Helped |
| Replace FFT plus cubic interpolation with direct spectral evaluation | FNO · Burgers' Equation, contracting-expanding | L1 test error: 0.195% -> 0.184% | Helped |
| Replace FFT plus cubic interpolation with direct spectral evaluation | FNO · Burgers' Equation, contracting-expanding | training time per epoch: 1.00s -> 0.11s | Helped |
| Replace FFT with direct spectral evaluation | FNO · Shear Layer, 2D non-equispaced lattice | L1 test error: 6.76% -> 5.53% | Helped |
| Replace FFT with direct spectral evaluation | UFNO · Shear Layer, 2D non-equispaced lattice | L1 test error: 6.61% -> 5.61% | Helped |
| Replace FFT with direct spectral evaluation | FFNO · Shear Layer, 2D non-equispaced lattice | L1 test error: 12.4% -> 12.4% | No clear effect |
| Replace FFT with direct spectral evaluation | FNO · Specific Humidity, 2D non-equispaced lattice | L1 test error: 5.09% -> 4.65% | Helped |
| Replace FFT with direct spectral evaluation | UFNO · Specific Humidity, 2D non-equispaced lattice | L1 test error: 4.34% -> 3.91% | Helped |
| Replace FFT with direct spectral evaluation | FFNO · Specific Humidity, 2D non-equispaced lattice | L1 test error: 5.02% -> 4.41% | Helped |
| Replace geometric layer with direct spectral evaluation | FNO · Flow Past Airfoil, 2D point cloud | L1 test error: 1.20% -> 0.220% | Helped |
| Replace geometric layer with direct spectral evaluation | UFNO · Flow Past Airfoil, 2D point cloud | L1 test error: 0.679% -> 0.380% | Helped |
| Replace geometric layer with direct spectral evaluation | FFNO · Elasticity, 2D point cloud | L1 test error: 2.20% -> 1.73% | Helped |
Surprising findings
- Direct evaluation of the truncated transform remains faster than FFT-based approaches on equispaced grids for 8 to 64 modes, despite its O(mN) cost versus the FFT's O(N log N).
- Geometric-layer diffeomorphisms were difficult to train jointly with the Fourier layers, often requiring loss tuning and freezing.
Limitations
- The transform matrix depends on sample positions, so each distinct point set requires building a new matrix at run time, which slows training.
- Only Fourier and spherical harmonic transforms are demonstrated; wavelets and Laplace transforms are left to future work.
- The number of modes must be chosen by model selection and is not shown to be independent of problem complexity.
Experimental setup
Datasets: Burgers' equation benchmark data Shear layer Navier-Stokes simulations (1024^2 spectral viscosity) MERRA-2 surface specific humidity Airfoil transonic Euler data (Geo-FNO setup) Elasticity hyperelastic data (Geo-FNO setup) Shallow water equations on sphere
Benchmarks: Burgers' equation Shear layer Surface-level specific humidity Flow past airfoil Elasticity Spherical shallow water equations
Metrics: relative L1 test error training time per epoch
Baselines: FFT with cubic interpolation KB-NUFFT Toeplitz-NUFFT Geometric Layer (Geo-FNO) Gaussian RBF interpolation linear interpolation
Builds on: Fourier Neural Operator Geo-FNO Spherical Fourier Neural Operator Factorized FNO U-shaped Neural Operator
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