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

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 n...

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Bibliographic Details
Published in:Computer methods in applied mechanics and engineering 2022-02, Vol.390, p.114502, Article 114502
Main Authors: Gao, Han, Zahr, Matthew J., Wang, Jian-Xun
Format: Article
Language:English
Subjects:
Artificial neural networks
Boundary conditions
Computational geometry
Continuity (mathematics)
Convexity
Graph convolutional neural networks
Inverse problem
Inverse problems
Mechanics
Neural networks
Optimization
Partial differential equations
Physics-informed machine learning
Polynomials
Training
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Summary: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.
ISSN:0045-7825
1879-2138
DOI:10.1016/j.cma.2021.114502