Meta-Auto-Decoder for Solving Parametric Partial Differential Equations
This addresses the need for efficient and accurate solvers for parametric PDEs in science and engineering, offering a novel approach that combines the benefits of existing methods while overcoming their limitations.
The paper tackles the problem of solving parametric partial differential equations (PDEs) by proposing Meta-Auto-Decoder (MAD), a mesh-free and unsupervised deep learning method that enables fast adaptation to new equation instances without retraining, achieving faster convergence speed without losing accuracy compared to other methods.
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/.