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. However, as 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. We address this issue by proposing a novel method that leverages batch matrix multiplications to efficiently construct Vandermonde-structured matrices and compute forward and inverse transforms, on arbitrarily distributed points. An efficient implementation of such structured matrix methods 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 one to extend neural operators to very general point distributions with significant gains in training speed over baselines, while retaining or improving accuracy.

翻译：许多广泛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）进行谱域计算。然而，由于FFT仅限于等距（矩形）网格，当输入和输出函数需要在一般非等距点分布上处理时，此类神经算子的效率受到限制。针对这一问题，我们提出了一种新方法，该方法利用批量矩阵乘法高效构建范德蒙德结构矩阵，并在任意分布的点上实现正向和逆向变换。将这种结构化矩阵方法的高效实现与现有神经算子模型相结合，能够处理任意非等距点分布上的数据。通过广泛的实证评估，我们证明所提出的方法可将神经算子扩展到非常一般的点分布，在保证或提升精度的同时，在训练速度上较基线方法有显著提升。