Dilated convolution neural operator for multiscale partial differential equations
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. We show that DCNO strikes an optimal balance between accuracy and computational cost and offers a promising solution for multiscale operator learning.
Cite
@article{arxiv.2408.00775,
title = {Dilated convolution neural operator for multiscale partial differential equations},
author = {Bo Xu and Xinliang Liu and Lei Zhang},
journal= {arXiv preprint arXiv:2408.00775},
year = {2024}
}