The computational efficiency of many neural operators, widely used for learning solutions of PDEs, relies on the fast Fourier transform (FFT) for performing spectral computations. As the FFT is limited to equispaced (rectangular) grids, this limits the efficiency of such neural operators when applied to problems where the input and output functions need to be processed on general non-equispaced point distributions. Leveraging the observation that a limited set of Fourier (Spectral) modes suffice to provide the required expressivity of a neural operator, we propose a simple method, based on the efficient direct evaluation of the underlying spectral transformation, to extend neural operators to arbitrary domains. An efficient implementation* of such direct spectral evaluations is coupled with existing neural operator models to allow the processing of data on arbitrary non-equispaced distributions of points. With extensive empirical evaluation, we demonstrate that the proposed method allows us to extend neural operators to arbitrary point distributions with significant gains in training speed over baselines while retaining or improving the accuracy of Fourier neural operators (FNOs) and related neural operators.

翻译：众多常用于学习偏微分方程解的神经算子的计算效率，依赖于快速傅里叶变换（FFT）来执行谱域计算。由于FFT仅适用于等间距（矩形）网格，当此类神经算子应用于需要在一般非等间距点分布上处理输入与输出函数的问题时，其效率受到限制。基于只需有限数量的傅里叶（谱）模态即可提供神经算子所需表达力的观察，我们提出了一种简单方法，通过高效直接计算底层谱变换，将神经算子扩展到任意域。我们将这种直接谱评估的高效实现与现有神经算子模型相结合，以允许在任意非等间距点分布上处理数据。通过广泛的实证评估，我们证明所提方法能够将神经算子扩展到任意点分布，在训练速度上相比基线方法取得显著提升，同时保持或提高傅里叶神经算子（FNO）及相关神经算子的准确性。