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Neural Operators with Localized Integral and Differential Kernels
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Proposes adding local differential kernels (rescaled, mean-subtracted CNN kernels) and local integral kernels (DISCO convolutions) as parallel branches in FNO/SFNO layers, yielding resolution-agnostic local neural operators that reduce relative L2 error on PDE benchmarks.
Problem
Fourier neural operators use global spectral convolutions that over-smooth and capture local details poorly, while CNN-based operators are tied to a fixed grid and lose information when downsampled to change resolution.
Approach
Two resolution-independent local layers added as branches to FNO/SFNO: (i) a differential layer obtained by subtracting the mean kernel value and scaling convolution kernels by 1/h so they converge to a differential operator; (ii) an integral layer using DISCO convolutions (learnable piecewise-linear hat-basis kernels with local support) that works on general meshes and spheres.
Key results
- Adding local differential and integral kernels to FNO reduces Darcy flow relative L2 error from 5.867e-2 to 9.03e-3 (combined) or 7.36e-3 (differential only), an improvement of about 87%.
- On Kolmogorov Navier-Stokes (Re=5000), FNO + local integral + differential kernels reaches relative L2 error 9.022e-2 vs 1.381e-1 for FNO (about 34% reduction).
- On spherical shallow water equations, SFNO + local integral kernel reaches relative L2 error 2.624e-4 vs 9.220e-4 for SFNO (about 72% reduction).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| add differential kernel branch (3x3 mean-subtracted, 1/h scaled) in parallel to FNO layers | FNO · Darcy flow | relative L2-error: 5.867e-2 -> 7.357e-3 (about 87% lower) | Helped |
| add local integral kernel branch to FNO (ablation) | FNO · Darcy flow | relative L2-error: 6.034e-2 vs 5.867e-2 baseline | No clear effect |
| add differential kernel branch to FNO (ablation) | FNO · Navier-Stokes Re=5000 | relative L2-error (1 step): 1.381e-1 -> 1.073e-1 | Helped |
| add local integral kernel branch to FNO (ablation) | FNO · Navier-Stokes Re=5000 | relative L2-error (1 step): 1.381e-1 -> 1.110e-1 | Helped |
| add both local integral and differential kernel branches to FNO | FNO · Navier-Stokes Re=5000 | relative L2-error (1 and 5 steps): 1 step: 1.381e-1 -> 9.022e-2; 5 steps: 2.360e-1 -> 1.956e-1 | Helped |
| add local integral kernel (DISCO) branch to SFNO | SFNO · Spherical shallow water | relative L2-error (1 and 5 steps): 1 step: 9.220e-4 -> 2.624e-4; 5 steps: 3.185e-3 -> 5.392e-4 | Helped |
| add differential kernel branch to FNO (ablation) | FNO · 2D diffusion-reaction (PDEBench) | relative L2-error: 8.3e-2 -> 3.4e-2 | Helped |
| add local integral kernel branch to FNO (ablation) | FNO · 2D diffusion-reaction (PDEBench) | relative L2-error: 8.3e-2 -> 6.3e-2 | Helped |
| add both local integral and differential kernels to FNO | FNO · 2D diffusion-reaction (PDEBench) | relative L2-error: 8.3e-2 -> 3.1e-2 | Helped |
| add local integral encoder/decoder and local integral kernels in FNO processor for unstructured mesh | FNO · Flow past cylinder (250 samples) | MSE: best baseline GNN 5.00e-3 -> 2.88e-3 | Helped |
Limitations
- Hyperparameters were tuned per architecture; comparisons rely on approximately matched parameter counts rather than exhaustive tuning.
- Differential kernels are restricted to regular grids, so extension to unstructured meshes relies on integral kernels.
Experimental setup
Datasets: Darcy flow (generated) Kolmogorov Navier-Stokes (Re=5000) PDEBench 2D diffusion-reaction shallow water equations on sphere (torch-harmonics) flow past a cylinder dataset
Benchmarks: Darcy flow Navier-Stokes (Kolmogorov, Re=5000) PDEBench 2D diffusion-reaction shallow water equations on sphere flow past a cylinder
Metrics: relative L2 error fRMSE RMSE MSE
Baselines: FNO U-Net SFNO Spherical U-Net GINO DeepONet GNN ViT
Builds on: Fourier neural operator DISCO convolutions Spherical Fourier neural operator graph neural operator
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