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 

机构地区：[1]Beijing International Center for Mathematical Research,Peking University,Beijing 100871,China [2]School of Computer Science and Technology,University of Science and Technology of China,Hefei 230027,Anhui,China [3]Central Software Institute,Huawei Technologies Co.Ltd,Hangzhou 310007,Zhejiang,China [4]Center for Machine Learning Research,Peking University,Beijing 100871,China

出　　处：《Communications on Applied Mathematics and Computation》2024年第2期1096-1130,共35页应用数学与计算数学学报（英文）

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

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,etc.Typical reduced order modeling techniques accelerate the solution of the parametric PDEs by projecting them onto a linear trial manifold constructed in the ofline stage.These 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 ansatz.Utilizing 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 width.Based on the meta-learning concept,the trial manifold can be learned in a mesh-free and unsupervised way during the pre-training stage.Fast 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 time.Extensive numerical experiments show that the MAD method exhibits a faster convergence speed without losing the accuracy than other deep learning-based methods.

关 键 词：Parametric partial differential equations(PDEs) META-LEARNING Reduced order modeling Neural networks(NNs) Auto-decoder 

分 类 号：O241.8[理学—计算数学] TP39[理学—数学]