面向多尺度偏微分方程的扩张卷积神经算子
Dilated convolution neural operator for multiscale partial differential equations
AI总结：
本文提出扩张卷积神经算子DCNO，通过卷积与傅里叶层结合捕获多尺度问题中的高频和低频特征，在椭圆方程、Navier-Stokes方程等任务上实现了精度与计算成本的最优平衡。
摘要：
本文提出了一种面向多尺度偏微分方程的数据驱动算子学习方法，特别强调保留高频信息。受多尺度参数化解可表示为低秩全局基（如低频傅里叶模式）与粗网格块上的局部基（类似于扩张卷积）组合这一思想的启发，我们提出了扩张卷积神经算子（DCNO）。DCNO架构通过卷积层和傅里叶层的组合，在保持较低计算成本的同时有效捕获高频和低频特征。我们进行了实验以评估DCNO在多个数据集上的性能，包括多尺度椭圆方程、其反问题、Navier-Stokes方程和Helmholtz方程。结果表明，DCNO在精度与计算成本之间达到了最优平衡，为多尺度算子学习提供了一种有前景的解决方案。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
