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.

翻译：算子学习是机器学习的一个分支，旨在通过数据近似函数空间之间的映射。傅里叶神经算子（FNO）是用于算子学习的常见模型架构。FNO将物理空间中的逐点线性与非线性操作与傅里叶空间中的逐点线性操作相结合，从而生成作用于函数空间之间的参数化映射。尽管FNO形式上涉及连续统上函数的卷积，但在实践中，计算是在离散网格上执行的，从而可通过快速傅里叶变换高效实现。本文量化了这种离散化导致的混叠误差，并基于输入的正则性，推导了关于网格分辨率的代数收敛速率。同时进行了数值实验，验证了理论结果并描述了模型稳定性。