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Meta-Auto-Decoder for Solving Parametric Partial Differential Equations
Xiang Huang · Zhanhong Ye · Hongsheng Liu · Shi Ji · Zidong Wang · Kang Yang · Yang Li · Min Wang · Haotian CHU · Fan Yu · Bei Hua · Lei Chen · Bin Dong

Wed Dec 07 05:00 PM -- 07:00 PM (PST) @
in Lightning Talks 4B-1 »

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 computation domains, etc. Recently, building learning-based numerical solvers for parametric PDEs has become an emerging new field. One category of methods such as the Deep Galerkin Method (DGM) and Physics-Informed Neural Networks (PINNs) aim to approximate the solution of the PDEs. They are typically unsupervised and mesh-free, but require going through the time-consuming network training process from scratch for each set of parameters of the PDE. Another category of methods such as Fourier Neural Operator (FNO) and Deep Operator Network (DeepONet) try to approximate the solution mapping directly. Being fast with only one forward inference for each PDE parameter without retraining, they often require a large corpus of paired input-output observations drawn from numerical simulations, and most of them need a predefined mesh as well. In this paper, we propose Meta-Auto-Decoder (MAD), a mesh-free and unsupervised deep learning method that enables the pre-trained model to be quickly adapted to equation instances by implicitly encoding (possibly heterogenous) PDE parameters as latent vectors. The proposed method MAD can be interpreted by manifold learning in infinite-dimensional spaces, granting it a geometric insight. Extensive numerical experiments show that the MAD method exhibits faster convergence speed without losing accuracy than other deep learning-based methods.

Keywords: Meta-Learning · Physics-Informed Learning · Parametric PDE · Auto-decoder

Author Information

Xiang Huang (University of Science and Technology of China)
Zhanhong Ye (Peking University)
Hongsheng Liu (Huawei Technologies Ltd.)
Shi Ji (University of Chinese Academy of Sciences)
Zidong Wang (Zhejiang University)
Kang Yang (Huazhong University of Science and Technology)
Yang Li (University of Chinese Academy of Sciences)
Min Wang (Huawei Technologies Ltd.)
Haotian CHU (Huawei)
Fan Yu (University of Science and Technology of China)
Bei Hua (University of Science and Technology of China)
Lei Chen (Hong Kong University of Science and Technology)
Bin Dong (Peking University)

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