PhyGeoNet: Physics-informed geometry-adaptive convolutional neural networks for solving parameterized steady-state PDEs on irregular domain
PhyGeoNet: Physics-informed geometry-adaptive convolutional neural networks for solving parameterized steady-state PDEs on irregular domain
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DOI：
10.1016/j.jcp.2020.110079
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发表时间：
2021-01-07
影响因子：
4.1
通讯作者：
Wang, Jian-Xun
中科院分区：
文献类型：
--
作者：
Gao, Han;Sun, Luning;Wang, Jian-Xun
最近，深度学习的出现激发了人们对开发物理信息神经网络（PINN）的兴趣，以有效地解决偏微分方程（PDE），特别是在参数设置中。在所有不同类别的深度神经网络中，卷积神经网络（CNN）在科学机器学习社区中引起了越来越多的关注，因为CNN中的参数共享功能可以有效地学习大规模时空领域的问题。然而，最大的挑战之一是CNN只能处理具有图像格式的规则几何形状（即，具有均匀网格的矩形域）。在本文中，我们提出了一种新的物理约束CNN学习架构，旨在学习不规则域上参数偏微分方程的解，而无需任何标记数据。为了利用强大的经典CNN骨干，引入了椭圆坐标映射来实现不规则物理域和规则参考域之间的坐标变换。所提出的方法已被评估通过求解一些稳态偏微分方程的不规则域，包括热方程，Navier-Stokes方程，泊松方程参数化的边界条件，不同的几何形状，和空间变化的源场。此外，所提出的方法也进行了比较，对国家的最先进的PINN与全连接神经网络（FC-NN）制定。数值结果表明，该方法的有效性，并表现出显着的优越性，基于FC-NN的PINN在效率和精度。(c)2020爱思唯尔公司All rights reserved.
Recently, the advent of deep learning has spurred interest in the development of physics informed neural networks (PINN) for efficiently solving partial differential equations (PDEs), particularly in a parametric setting. Among all different classes of deep neural networks, the convolutional neural network (CNN) has attracted increasing attention in the scientific machine learning community, since the parameter-sharing feature in CNN enables efficient learning for problems with large-scale spatiotemporal fields. However, one of the biggest challenges is that CNN only can handle regular geometries with image like format (i.e., rectangular domains with uniform grids). In this paper, we propose a novel physics-constrained CNN learning architecture, aiming to learn solutions of parametric PDEs on irregular domains without any labeled data. In order to leverage powerful classic CNN backbones, elliptic coordinate mapping is introduced to enable coordinate transforms between the irregular physical domain and regular reference domain. The proposed method has been assessed by solving a number of steady-state PDEs on irregular domains, including heat equations, Navier-Stokes equations, and Poisson equations with parameterized boundary conditions, varying geometries, and spatially-varying source fields. Moreover, the proposed method has also been compared against the state-of-the-art PINN with fully-connected neural network (FC-NN) formulation. The numerical results demonstrate the effectiveness of the proposed approach and exhibit notable superiority over the FC-NN based PINN in terms of efficiency and accuracy. (c) 2020 Elsevier Inc. All rights reserved.