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

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

扫码查看
原文链接
  • NETL
  • NSTL
  • 万方数据
Many important problems in science and engineering require solving the so-called para-metric partial differential equations(PDEs),i.e.,PDEs with different physical parameters,boundary conditions,shapes of computational domains,etc.Typical reduced order mod-eling techniques accelerate the solution of the parametric PDEs by projecting them onto a linear trial manifold constructed in the offline stage.These methods often need a pre-defined 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.Uti-lizing 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-train-ing 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 con-vergence speed without losing the accuracy than other deep learning-based methods.

Parametric partial differential equations(PDEs)Meta-learningReduced order modelingNeural networks(NNs)Auto-decoder

Zhanhong Ye、Xiang Huang、Hongsheng Liu、Bin Dong

展开 >

Beijing International Center for Mathematical Research,Peking University,Beijing 100871,China

School of Computer Science and Technology,University of Science and Technology of China,Hefei 230027,Anhui,China

Central Software Institute,Huawei Technologies Co.Ltd,Hangzhou 310007,Zhejiang,China

Center for Machine Learning Research,Peking University,Beijing 100871,China

展开 >

2024

10.1007/s42967-023-00293-7

应用数学与计算数学学报
上海大学

应用数学与计算数学学报

影响因子：0.165
ISSN：1006-6330
年,卷(期)：2024.6(2)