DC-DeepONet:一种求解多尺度偏微分方程的算子学习方法  

DC-DeepONet:An operator learning method for multiscale differential equations

作　　者：汪璐 邹舒帆 邓小刚 WANG Lu;ZOU Shu-Fan;DENG Xiao-Gang(College of Computer Science,Sichuan University,Chengdu 610065,China;National Key Laboratory of Fundemental Algorithms and Models for Engineering Simulation,Chengdu 610065,China;College of Aerospace Science and Engineering,National University of Defense Technology,Changsha 410073,China;Institute of Defense Science and Technology Innovation,Academy of Military Sciences,Beijing 100071,China)

机构地区：[1]四川大学计算机学院,成都610065 [2]工程数值模拟基础算法与模型全国重点实验室,成都610065 [3]国防科技大学空天科学学院,长沙410073 [4]军事科学院国防科技创新研究院,北京100071

出　　处：《四川大学学报（自然科学版）》2024年第6期59-70,共12页Journal of Sichuan University(Natural Science Edition)

基　　金：国家重大专项(GJXM92579);四川省科技计划资助(2023YFG0329)。

摘　　要：自然界中存在各种各样的多尺度现象,如湍流、微结构材料、气泡与颗粒相互耦合等问题,针对此类问题建模,使用的偏微分方程模型也会表现出多尺度特征.算子学习作为一种利用深度神经网络求解偏微分方程的新兴范式,在泛化性和计算效率方面展现出巨大优势.然而现有的算子学习模型在针对多尺度偏微分方程进行求解时存在频率选择问题,即无法有效学习高频信号.为此本文基于卷积神经网络和DeepONet,提出了两种新的网络模型CDeepONet和DC-DeepONet,两种模型分别采用普通卷积和空洞卷积的方式,专门用于多尺度偏微分方程的求解,可有效捕捉多尺度问题中的高频特征.本文以多尺度泊松方程和多尺度达西流方程为例进行实验,实验结果表明,相对于原始的DeepONet,我们提出的DCDeepONet的平均相对误差减少了88%.Various multiscale phenomena,such as turbulence,microstructured materials,and the coupled in⁃teraction between bubbles and particles,are present in nature.The modeling of these phenomena often uses partial differential equations(PDEs)that exhibit multiscale characteristics.Operator learning,a novel para⁃digm utilizing deep neural networks for solving PDEs,has shown significant advantages in terms of generaliz⁃ability and computational efficiency.However,existing operator learning models face challenges in solving multiscale PDEs,particularly with the frequency principle,due to their inability to effectively learn highfrequency signals.To address this issue,our paper introduces two new network models:C-DeepONet and DC-DeepONet,which are based on convolutional neural networks and DeepONet.The two models utilize standard and dilated convolution respectively,specifically designed for addressing multiscale PDEs,enabling effective capture of high-frequency features in multiscale scenarios.Experiments are performed using multi⁃scale Poisson equations and Darcy flow equations as illustrative examples.The results demonstrate that the proposed DC-DeepONet achieves an 88%reduction in average relative error compared to the baseline DeepONet model.

关 键 词：算子学习 多尺度 偏微分方程 空洞卷积 

分 类 号：TP391[自动化与计算机技术—计算机应用技术]