Meta-Auto-Decoder for Solving Parametric Partial Differential Equations
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 parametri...
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Zusammenfassung: | 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. The project page with code is available:
https://gitee.com/mindspore/mindscience/tree/master/MindElec/. |
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DOI: | 10.48550/arxiv.2111.08823 |