Inversion of the spatially dependent mechanical field based on PIGNNs
Yuchen Bi, Hesheng Tang
Abstract. Deep learning (DL) is a promising approach to predicting physical phenomena, and has been widely researched in the inversion of the spatially dependent mechanical field due to their remarkable fitting abilities. The traditional physics-informed neural networks (PINNs) disregard the locality of physical evolution processes, which makes the model results lack representation ability and effectiveness. Physics-informed graph neural networks (PIGNNs) can adapt to various types of two-dimensional unstructured grids due to their flexible manipulative data, which enhances the interpretability of the model. Therefore, in this work, a physics and data-driven graph neural networks model is constructed for inversion of the spatially dependent parameter. This model combines traditional numerical methods with unstructured data, and embedded physical information into the graph networks model in the form of discrete differentiation, which made the model have better generalization and interpretability. To verify the feasibility of the model, parameter inversion is performed on plane structures with varying spatially dependent mechanical fields. The results demonstrate that PIGNNs can accurately identify the spatial stiffness variations, and its accuracy in inverting spatially dependent stiffness surpasses that of traditional PINNs. This indicates that the model has strong potential for solving inverse problems related to mechanical parameters.
Keywords
Spatially Dependent Mechanical Field, Parametric Inversion, Physics and Data-Driven, PIGNNs
Published online 3/25/2025, 8 pages
Copyright © 2025 by the author(s)
Published under license by Materials Research Forum LLC., Millersville PA, USA
Citation: Yuchen Bi, Hesheng Tang, Inversion of the spatially dependent mechanical field based on PIGNNs, Materials Research Proceedings, Vol. 50, pp 307-314, 2025
DOI: https://doi.org/10.21741/9781644903513-36
The article was published as article 36 of the book Structural Health Monitoring
Content from this work may be used under the terms of the Creative Commons Attribution 3.0 license. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.
References
[1] Zhao X-Y, Sun D, Toh K-C. A Newton-CG Augmented Lagrangian Method for Semidefinite Programming. SIAM J Optim 2010; 20: 1737-65. https://doi.org/10.1137/080718206
[2] Baydin AG, Pearlmutter BA, Radul AA, Siskind JM. Automatic Differentiation in Machine Learning: a Survey. Journal of Machine Learning Research 2018;18:1-43.
[3] Mao Z, Jagtap AD, Karniadakis GE. Physics-informed neural networks for high-speed flows. Computer Methods in Applied Mechanics and Engineering 2020; 360: 112789. https://doi.org/10.1016/j.cma.2019.112789
[4] Sun L, Wang J-X. Physics-constrained bayesian neural network for fluid flow reconstruction with sparse and noisy data. Theoretical and Applied Mechanics Letters 2020; 10: 161-9. https://doi.org/10.1016/j.taml.2020.01.031
[5] Haghighat E, Raissi M, Moure A, Gomez H, Juanes R. A physics-informed deep learning framework for inversion and surrogate modeling in solid mechanics. Computer Methods in Applied Mechanics and Engineering 2021; 379: 113741. https://doi.org/10.1016/j.cma.2021.113741
[6] Guo Q, Zhao Y, Lu C, Luo J. High-dimensional inverse modeling of hydraulic tomography by physics informed neural network (HT-PINN). Journal of Hydrology 2023; 616: 128828. https://doi.org/10.1016/j.jhydrol.2022.128828
[7] Chen Y, Huang D, Zhang D, Zeng J, Wang N, Zhang H, et al. Theory-guided hard constraint projection (HCP): A knowledge-based data-driven scientific machine learning method. Journal of Computational Physics 2021; 445: 110624. https://doi.org/10.1016/j.jcp.2021.110624
[8] Rezaei S, Harandi A, Moeineddin A, Xu B-X, Reese S. A mixed formulation for physics-informed neural networks as a potential solver for engineering problems in heterogeneous domains: Comparison with finite element method. Computer Methods in Applied Mechanics and Engineering 2022;401: 115616. https://doi.org/10.1016/j.cma.2022.115616
[9] Aulakh DJS, Beale SB, Pharoah JG. A generalized framework for unsupervised learning and data recovery in computational fluid dynamics using discretized loss functions. Physics of Fluids 2022; 34: 077111. https://doi.org/10.1063/5.0097480
[10] Jiang L, Wang L, Chu X, Xiao Y, Zhang H. PhyGNNet: Solving spatiotemporal PDEs with Physics-informed Graph Neural Network. Proceedings of the 2023 2nd Asia Conference on Algorithms, Computing and Machine Learning, New York, NY, USA: Association for Computing Machinery; 2023, p. 143-7. https://doi.org/10.1145/3590003.3590029
[11] Gao H, Zahr MJ, Wang J-X. 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; 390: 114502. https://doi.org/10.1016/j.cma.2021.114502
[12] Xiang Z, Peng W, Yao W. RBF-MGN: Solving spatiotemporal PDEs with Physics-informed Graph Neural Network 2022. https://doi.org/10.1145/3590003.3590029
[13] Pfaff T, Fortunato M, Sanchez-Gonzalez A, Battaglia PW. Learning Mesh-Based Simulation with Graph Networks 2021.
[14] Mavriplis D. Revisiting the Least-Squares Procedure for Gradient Reconstruction on Unstructured Meshes. 16th AIAA Computational Fluid Dynamics Conference, American Institute of Aeronautics and Astronautics; n.d.
