Reproduce “Discretization Error of Fourier Neural Operators” on OpenResearch
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About this paper
Samuel Lanthaler, Andrew M. Stuart, Margaret Trautner
This work rigorously quantifies the discretization error of Fourier Neural Operators (FNOs), demonstrating that the error decays algebraically as O(N^-s) with grid resolution N and input Sobolev regularity s. The study also introduces an adaptive subsampling algorithm, leveraging this understanding to reduce FNO training time by dynamically adjusting grid resolution.
Key insights
- The discretization error of FNOs is directly proportional to the inverse of the grid resolution raised to the power of the input function's Sobolev regularity (i.e., faster decay for smoother inputs and finer grids).
- Globally Lipschitz and C-infinity smooth activation functions (e.g., GeLU) are crucial for preserving Sobolev regularity across FNO layers, enabling the theoretical error decay rates to hold in practice.
- Low-regularity additional inputs, such as non-periodic positional encodings, can bottleneck the overall discretization error convergence rate, limiting it to the regularity of the least smooth input component.
Summary by alphaXiv.
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