Fourier Neural Operators (FNOs) have emerged as very popular machine learning architectures for learning operators, particularly those arising in PDEs. However, as FNOs rely on the fast Fourier transform for computational efficiency, the architecture can be limited to input data on equispaced Cartesian grids. Here, we generalize FNOs to handle input data on non-equispaced point distributions. Our proposed model, termed as Vandermonde Neural Operator (VNO), utilizes Vandermonde-structured matrices to efficiently compute forward and inverse Fourier transforms, even on arbitrarily distributed points. We present numerical experiments to demonstrate that VNOs can be significantly faster than FNOs, while retaining comparable accuracy, and improve upon accuracy of comparable non-equispaced methods such as the Geo-FNO.

翻译：傅里叶神经算子（FNOs）已成为处理算子学习（尤其是偏微分方程中的算子学习）的广泛流行的机器学习架构。然而，由于FNO依赖快速傅里叶变换以实现计算效率，其架构可能局限于处理等距笛卡尔网格上的输入数据。本文中，我们将FNO推广至能够处理非等距点分布上的输入数据。我们提出的模型称为范德蒙德神经算子（VNO），它利用范德蒙德结构矩阵，即使在任意分布的点集上也能高效计算正向和逆向傅里叶变换。数值实验表明，VNO在保持可比精度的同时，计算速度显著快于FNO，并且相比Geo-FNO等同类非等距方法，其精度也有所提升。