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Component Fourier Neural Operator for Singularly Perturbed Differential Equations
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Proposes ComFNO, an FNO extension that adds asymptotic-analysis-derived exponential layer blocks to learn boundary/inner layers of singularly perturbed differential equations, improving accuracy and few-shot performance over vanilla FNO.
Problem
Neural operators such as FNO struggle to learn solution operators for singularly perturbed differential equations, whose solutions have rapid transitions in thin boundary or inner layers, leading to underfitting and oscillatory residuals.
Approach
Decompose the solution into a smooth FNO part plus layer blocks of the form NN_i * exp(FNO_i) with coordinate-stretched inputs f(xi), xi=(x0-x)/eps, guided by asymptotic expansions of the layer structure; the framework is also said to extend to DeepONet.
Key results
- On the 1D no-turning-point problem, ComFNO reduces mean residual from 5.0e-04 (FNO) to 1.0e-04 and infinity norm from 3.8e-03 to 6.0e-04.
- On the 1D initial-boundary parabolic problem, ComFNO mean residual is 4.2e-06 versus 9.0e-05 for FNO.
- In the few-shot setting (100 training samples), ComFNO mean residual is 3.0e-4 versus 1.2e-3 for FNO.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Replace vanilla FNO with ComFNO layer blocks | FNO · 1D no turning point | mean residual: 5.0e-04 -> 1.0e-04 | Helped |
| Replace vanilla FNO with ComFNO layer blocks | FNO · 1D turning point | mean residual: 1.6e-03 -> 6.0e-04 | Helped |
| Replace vanilla FNO with ComFNO layer blocks | FNO · 1D initial-boundary | mean residual: 9.0e-05 -> 4.2e-06 | Helped |
| Replace vanilla FNO with ComFNO layer blocks | FNO · 2D | mean residual: 5.0e-4 -> 2.0e-04 | Helped |
| Replace vanilla FNO with ComFNO layer blocks | FNO · Multiple epsilon | mean residual: 7.0e-04 -> 6.0e-04 | Helped |
| Replace vanilla FNO with ComFNO layer blocks | FNO · Few-shot | mean residual: 1.2e-03 -> 3.0e-4 | Helped |
| Replace vanilla FNO with ComFNO layer blocks | FNO · 1D no turning point | L-infinity residual: 3.8e-03 -> 6.0e-04 | Helped |
| Replace vanilla FNO with ComFNO layer blocks | FNO · 1D turning point | L-infinity residual: 8.5e-03 -> 2.1e-03 | Helped |
| Replace vanilla FNO with ComFNO layer blocks | FNO · 1D initial-boundary | L-infinity residual: 1.7e-03 -> 4.0e-05 | Helped |
| Replace vanilla FNO with ComFNO layer blocks | FNO · 2D | L-infinity residual: 1.67e-02 -> 3.0e-03 | Helped |
| Replace vanilla FNO with ComFNO layer blocks | FNO · Multiple epsilon | L-infinity residual: 1.7e-02 -> 2.1e-03 | Helped |
| Replace vanilla FNO with ComFNO layer blocks | FNO · Few-shot | L-infinity residual: 4.5e-03 -> 1.3e-03 | Helped |
Surprising findings
- Omitting the corner layer block in the 2D problem still yielded accurate results, the authors attributing this to the two layer blocks covering the corner.
Limitations
- Focuses only on exponential-type boundary layers; other layer types require adapting the layer block operations.
- Corner layer in 2D is not explicitly modeled, relying on two layer blocks.
- DeepONet extension is claimed but not experimentally demonstrated in the text.
Experimental setup
Datasets: synthetic SPDE solutions from Gaussian random field source terms, computed with upwind scheme on Shishkin mesh and Crank
Benchmarks: 1D convection-diffusion (no turning point) 1D turning point problem 1D initial-boundary parabolic 2D elliptic convection-diffusion multiple epsilon few-shot
Metrics: mean absolute residual infinity norm of residual variance of residual
Baselines: FNO
Builds on: Fourier Neural Operator DeepONet matched asymptotic expansion
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