基于两重网格的深度学习方法求解定常偏微分方程  

Deep Learning Method Based on Two-Grid for Solving Stationary Partial Differential Equations

作　　者：彭湃 冯新龙[1] PENG Pai;FENG Xinlong(School of Mathematics and System Sciences,Xinjiang University,Urumqi Xinjiang 830017,China)

机构地区：[1]新疆大学数学与系统科学学院,新疆乌鲁木齐830017

出　　处：《新疆大学学报（自然科学版）（中英文）》2022年第4期412-420,共9页Journal of Xinjiang University(Natural Science Edition in Chinese and English)

基　　金：新疆维吾尔自治区重点实验室开放课题(2020D04002).

摘　　要：随着机器学习在多个领域的研究取得进展,物理信息神经网络为偏微分方程的求解提供了新思路,但该方法难以获得高精度的数值解.结合物理信息神经网络与两重网格求解偏微分方程的思想,提出了基于两重网格的深度学习方法求解定常偏微分方程.针对神经网络求解多目标问题,采取了动态权重策略平衡损失函数中各项之间的数值差异,有效缓解了梯度病态现象.最后,给出了若干数值实验,验证了结合动态权重策略的深度学习方法在提高计算精度上的有效性.With the progress of machine learning in many fields,physics-informed neural networks provide new ideas for solving partial differential equations,but this method is difficult to obtain high-precision numerical solu-tions.Absorbing the philosophy of physics-informed neural network and the two-grid solution of partial differential equations,this paper puts forward the deep learning method based on two-grid for solving stationary partial dif-ferential equations.For the neural network to solve the multi-objective problem,the dynamic weight strategy is adopted to balance the numerical difference between the items in the loss function,and alleviate the gradient ill-conditioned phenomenon.Finally,this paper gives several numerical experiments to verify the effectiveness of the deep learning method combined with dynamic weight strategy in improving the calculation accuracy.

关 键 词：定常偏微分方程 深度学习 动态权重 高精度 两重网格 

分 类 号：O242.21[理学—计算数学]