Discretization Error of Fourier Neural Operators
<|reference_start|>Discretization Error of Fourier Neural Operators: Operator learning is a variant of machine learning that is designed to approximate maps between function spaces from data. The Fourier Neural Operator (FNO) is a common model architecture used for operator learning. The FNO combines pointwise linear and nonlinear operations in physical space with pointwise linear operations in Fourier space, leading to a parameterized map acting between function spaces. Although FNOs formally involve convolutions of functions on a continuum, in practice the computations are performed on a discretized grid, allowing efficient implementation via the FFT. In this paper, the aliasing error that results from such a discretization is quantified and algebraic rates of convergence in terms of the grid resolution are obtained as a function of the regularity of the input. Numerical experiments that validate the theory and describe model stability are performed.<|reference_end|>
@article{lanthaler2024discretization,
    title={Discretization Error of Fourier Neural Operators},
    author={Samuel Lanthaler, Andrew M. Stuart, and Margaret Trautner},
    journal={arXiv preprint arXiv:2405.02221},
    year={2024},
    archivePrefix={arXiv},
    eprint={2405.02221},
    primaryClass={math.NA cs.LG cs.NA}
}