Dilated convolution neural operator for multiscale partial differential equations

Bo Xu, Xinliang Liu, Lei Zhang

Abstract: This paper introduces a data-driven operator learning method for multiscale partial differential equations, with a particular emphasis on preserving high-frequency information. Drawing inspiration from the representation of multiscale parameterized solutions as a combination of low-rank global bases (such as low-frequency Fourier modes) and localized bases over coarse patches (analogous to dilated convolution), we propose the Dilated Convolutional Neural Operator (DCNO). The DCNO architecture effectively captures both high-frequency and low-frequency features while maintaining a low computational cost through a combination of convolution and Fourier layers. We conduct experiments to evaluate the performance of DCNO on various datasets, including the multiscale elliptic equation, its inverse problem, Navier–Stokes equation, and Helmholtz equation. Our results demonstrate that DCNO achieves an optimal balance between accuracy and computational efficiency, making it a promising approach for multiscale operator learning.