  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. We show that DCNO strikes an optimal balance between accuracy and
computational cost and offers a promising solution for multiscale operator
learning.
