PhyGeoNet: Physics-informed geometry-adaptive convolutional neural networks for solving parameterized steady-state PDEs on irregular domain

By Han Gao, Luning Sun, Jian-Xun Wang. 2020

646 citations recorded by the source corpus.

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 \emph{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 PDEs on irregular domains, including heat equations and steady Navier-Stokes equations with parameterized boundary conditions and varying geometries. 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.

Subfields: Computational Mechanics, Mechanical Engineering, Statistical and Nonlinear Physics

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Citations in the corpus

Physics informed deep learning for computational elastodynamics without labeled data (2020, background): “…found vast applications, within a short time, in a wide range of physical problems including modeling ﬂuid ﬂows and Navier-Stokes equations [10, 17–21], solving stochastic PDEs [22], ﬂows transport in porous media [23–25], cardiovascular systems [26–28], design of metamaterials [29– 31],…”
Physics-informed neural networks (PINNs) for fluid mechanics: a review (2021, methodology): “, using multi-fidelity framework [35], adaptive activation functions [36] and dynamic weights of the loss function [37], hard constraints [38] and CNN-based network architectures [39], which can improve the performance of PINNs on different problems.”
Scientific Machine Learning Through Physics–Informed Neural Networks: Where we are and What’s Next (2022, methodology): “A physics-informed geometry-adaptive convolutional neural network (PhyGeoNet) was introduced in Gao et al (2021).”
Neural operators for accelerating scientific simulations and design (2023, neutral): “Physics-Informed Neural Operator (PINO): PINNs and variants [63] are highly flexible, and have been effective in solving steady-state PDEs [64–67].”
A comprehensive review of digital twin — part 1: modeling and twinning enabling technologies (2022, methodology): “This approach is the most common way to build hybrid physics-ML models and has found success in many digital twin applications such as solving PDEs and surrogate modeling (Raissi et al. 2019a; Zhu et al. 2019b; Haghighat et al. 2021; Gao et al. 2021; Zhou et al. 2022a), inverse modeling (Raissi et a”
Finite basis physics-informed neural networks (FBPINNs): a scalable domain decomposition approach for solving differential equations (2021, neutral): “Zhu et al. [2019], Geneva and Zabaras [2020] and Gao et al. [2021] instead used ﬁnite difference ﬁlters to construct the differential equation residual in a PINN-like loss function when training CNNs and RNNs which approximate the solution of a differential equation on a mesh.”
Physics-informed graph neural Galerkin networks: A unified framework for solving PDE-governed forward and inverse problems (2021, methodology): “To reduce training costs and enable efficient learning, discrete PINNs that leverage convolution operations and numerical discretizations have begun to spur interests due to their better efficiency and scalability [19,20].”
CAN-PINN: A Fast Physics-Informed Neural Network Based on Coupled-Automatic-Numerical Differentiation Method (2021, methodology): “Different architectures, from fully connected neural networks [2,3,9–12] to convolutional neural networks [13– 16], to recurrent neural networks [17–20], and generative adversarial neural networks [21,22], have been explored in the context of PINN.”
Physics-Informed Neural Operator for Learning Partial Differential Equations (2021, positive): “A simple but efficient approach is to use conventional numerical derivatives such as finite difference and Fourier differentiation [27, 51].”