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arXiv · 2408.00775

Dilated convolution neural operator for multiscale partial differential equations

Bo Xu·Xinliang Liu·Lei Zhang
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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.

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Bo Xu, Xinliang Liu, Lei Zhang. 2024-07-16. Dilated convolution neural operator for multiscale partial differential equations. https://arxiv.org/abs/2408.00775

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