Meta-Auto-Decoder:a Meta-Learning-Based Reduced Order Model for Solving Parametric Partial Differential Equations

作     者：Zhanhong Ye Xiang Huang Hongsheng Liu Bin Dong Zhanhong Ye;Xiang Huang;Hongsheng Liu;Bin Dong

作者机构：Beijing International Center for Mathematical ResearchPeking UniversityBeijing 100871China School of Computer Science and TechnologyUniversity of Science and Technology of ChinaHefei 230027AnhuiChina Central Software InstituteHuawei Technologies Co.LtdHangzhou 310007ZhejiangChina Center for Machine Learning ResearchPeking UniversityBeijing 100871China 

出 版 物：《Communications on Applied Mathematics and Computation》 (应用数学与计算数学学报（英文）)

年 卷 期：2024年第6卷第2期

页      面：1096-1130页

核心收录：

学科分类：07[理学] 0714[理学-统计学（可授理学、经济学学位）] 0701[理学-数学] 0812[工学-计算机科学与技术（可授工学、理学学位）] 070101[理学-基础数学] 

基　　金：supported by the National Key R&D Program of China under Grant No.2021ZD0110400 

主　　题：Parametric partial differential equations(PDEs) Meta-learning Reduced order modeling Neural networks(NNs) Auto-decoder 

摘      要：Many important problems in science and engineering require solving the so-called parametric partial differential equations(PDEs),i.e.,PDEs with different physical parameters,boundary conditions,shapes of computational domains,*** reduced order modeling techniques accelerate the solution of the parametric PDEs by projecting them onto a linear trial manifold constructed in the ofline *** methods often need a predefined mesh as well as a series of precomputed solution snapshots,and may struggle to balance between the efficiency and accuracy due to the limitation of the linear *** the nonlinear representation of neural networks(NNs),we propose the Meta-Auto-Decoder(MAD)to construct a nonlinear trial manifold,whose best possible performance is measured theoretically by the decoder *** on the meta-learning concept,the trial manifold can be learned in a mesh-free and unsupervised way during the pre-training *** adaptation to new(possibly heterogeneous)PDE parameters is enabled by searching on this trial manifold,and optionally fine-tuning the trial manifold at the same *** numerical experiments show that the MAD method exhibits a faster convergence speed without losing the accuracy than other deep learning-based methods.