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

By Han Gao, Matthew J. Zahr, Jian-Xun Wang. Computer Methods in Applied Mechanics and Engineering, 2021

310 citations recorded by the source corpus.

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.

Subfields: Computational Mechanics, Electronic, Optical and Magnetic Materials, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/a4697474a885787e4312765be5af29a8b6d908c0

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computational Mechanics

1. C. Ooi
2. Yew-Soon Ong
3. Pao-Hsiung Chiu
4. My Ha Dao
5. Jian Cheng Wong

In Statistical and Nonlinear Physics

1. C. Ooi
2. Yew-Soon Ong
3. Pao-Hsiung Chiu
4. My Ha Dao
5. Jian Cheng Wong

Citations in the corpus

• CAN-PINN: A Fast Physics-Informed Neural Network Based on Coupled-Automatic-Numerical Differentiation Method (2021, methodology): “To naturally handle the complex geometries, Gao et al. [51] further proposed the Physics-informed graph neural Galerkin networks, which utilized nodal continuous Galerkin method.”