☆ 4.7 Article

Physics-informed graph neural Galerkin networks: A unified framework for solving PDE-governed forward and inverse problems

COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING (2022)

Journal

COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING

Volume 390, Issue -, Pages -

Publisher

ELSEVIER SCIENCE SA

DOI: 10.1016/j.cma.2021.114502

Keywords

Partial differential equations; Inverse problem; Physics-informed machine learning; Graph convolutional neural networks; Mechanics

Funding

National Science Foundation, United States of America [CMMI-1934300, OAC-2047127]
Air Force Office of Scientific Research (AFOSR) , United States of America [FA9550-20-1-0236]
College of Engineering at University of Notre Dame, United States of America

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Abstract

Despite the promise of PINNs in solving forward and inverse problems, there are technical challenges that hinder complex and realistic applications. To address these challenges, a discrete PINN framework based on GCN and PDE is proposed to solve PDEs in both forward and inverse settings, offering improved scalability, boundary enforcement, and handling of irregular geometries.

Despite the great promise of the physics-informed neural networks (PINNs) in solving forward and inverse problems, several technical challenges are present as roadblocks for more complex and realistic applications. First, most existing PINNs are based on point-wise formulation with fully-connected networks to learn continuous functions, which suffer from poor scalability and hard boundary enforcement. Second, the infinite search space over-complicates the non-convex optimization for network training. Third, although the convolutional neural network (CNN)-based discrete learning can significantly improve training efficiency, CNNs struggle to handle irregular geometries with unstructured meshes. To properly address these challenges, we present a novel discrete PINN framework based on graph convolutional network (GCN) and variational structure of PDE to solve forward and inverse partial differential equations (PDEs) in a unified manner. The use of a piecewise polynomial basis can reduce the dimension of search space and facilitate training and convergence. Without the need of tuning penalty parameters in classic PINNs, the proposed method can strictly impose boundary conditions and assimilate sparse data in both forward and inverse settings. The flexibility of GCNs is leveraged for irregular geometries with unstructured meshes. The effectiveness and merit of the proposed method are demonstrated over a variety of forward and inverse computational mechanics problems governed by both linear and nonlinear PDEs.(c) 2021 Elsevier B.V. All rights reserved.

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Surrogate-Based Physics-Informed Neural Networks for Elliptic Partial Differential Equations

Peng Zhi, Yuching Wu, Cheng Qi, Tao Zhu, Xiao Wu, Hongyu Wu

Summary: The study investigates the role of deep learning in computational mechanics. A convolutional neural network technique based on modified loss function is proposed as a surrogate of the finite element method. Physics-informed neural networks are developed to solve boundary value problems based on elliptic partial differential equations. The proposed method is applied for the first time to solve boundary value problems with elliptic partial differential equations. The results show that the proposed surrogate-based approach is in good agreement with the conventional FEM and modification of the loss function improves the prediction accuracy of the neural network. The study demonstrates the potential of the deep learning approach as a significant surrogate model.

MATHEMATICS (2023)