Energy-Based Error Bound of Physics-Informed Neural Network Solutions in Elasticity
Publication: Journal of Engineering Mechanics
Volume 148, Issue 8
Abstract
An energy-based a posteriori error bound is proposed for the physics-informed neural network solutions of elasticity problems. An admissible displacement-stress solution pair is obtained from a mixed form of physics-informed neural networks, and the proposed error bound is formulated as the constitutive relation error defined by the solution pair. Such an error estimator provides an upper bound of the global error of neural network discretization. The bounding property, as well as the asymptotic behavior of the physics-informed neural network solutions, are studied in a demonstration example.
Get full access to this article
View all available purchase options and get full access to this article.
Data Availability Statement
All data, models, or code that support the findings of this study are available from the corresponding author upon reasonable request.
References
Ainsworth, M., and J. T. Oden. 2011. Vol. 37 of A posteriori error estimation in finite element analysis. New York: Wiley.
Amodei, D., et al. 2016. “Deep speech 2: End-to-end speech recognition in English and mandarin.” In Vol. 48 of Proc., 33rd Int. Conf. on Machine Learning, 173–182. Maastricht, Netherlands: ML Research Press.
Babuška, I., and W. C. Rheinboldt. 1978. “A-posteriori error estimates for the finite element method.” Int. J. Numer. Methods Eng. 12 (10): 1597–1615. https://doi.org/10.1002/nme.1620121010.
Bank, R. E., and A. Weiser. 1985. “Some a posteriori error estimators for elliptic partial differential equations.” Math. Comput. 44 (170): 283–301. https://doi.org/10.1090/S0025-5718-1985-0777265-X.
Baydin, A. G., B. A. Pearlmutter, A. A. Radul, and J. M. Siskind. 2018. “Automatic differentiation in machine learning: A survey.” J. Mach. Learn. Res. 18: 1–43.
Bekele, Y. W. 2020. “Deep learning for one-dimensional consolidation.” Preprint, submitted April 20, 2020. http://arxiv.org/abs2004.11689.
Deng, J., W. Dong, R. Socher, L.-J. Li, K. Li, and L. Fei-Fei. 2009. “Imagenet: A large-scale hierarchical image database.” In Proc., 2009 IEEE Conf. on Computer Vision and Pattern Recognition, 248–255. New York: IEEE.
Deuflhard, P., P. Leinen, and H. Yserentant. 1989. “Concepts of an adaptive hierarchical finite element code.” Impact Comput. Sci. Eng. 1 (1): 3–35. https://doi.org/10.1016/0899-8248(89)90018-9.
Gallimard, L. 2009. “A constitutive relation error estimator based on traction-free recovery of the equilibrated stress.” Int. J. Numer. Methods Eng. 78 (4): 460–482. https://doi.org/10.1002/nme.2496.
Goodfellow, I., Y. Bengio, A. Courville, and Y. Bengio. 2016. Vol. 1. of Deep learning. Cambridge, MA: MIT Press.
Graves, A., A.-R. Mohamed, and G. Hinton. 2013. “Speech recognition with deep recurrent neural networks.” In Proc., 2013 IEEE Int. Conf. on Acoustics, Speech and Signal Processing, 6645–6649. New York: IEEE.
Grigorescu, S., B. Trasnea, T. Cocias, and G. Macesanu. 2020. “A survey of deep learning techniques for autonomous driving.” J. Field Rob. 37 (3): 362–386. https://doi.org/10.1002/rob.21918.
Guo, M., W. Han, and H. Zhong. 2016. “Legendre-Fenchel duality and a generalized constitutive relation error.” Preprint, submitted November 17, 2016. http://arxiv.org/1611.05589.
Ha, J.-W., H. Pyo, and J. Kim. 2016. “Large-scale item categorization in e-commerce using multiple recurrent neural networks.” In Proc., 22nd ACM SIGKDD Int. Conf. on Knowledge Discovery and Data Mining, 107–115. New York: Association for Computing Machinery.
Haghighat, E., and R. Juanes. 2020. “Sciann: A keras wrapper for scientific computations and physics-informed deep learning using artificial neural networks.” Preprint, submitted May 11, 2020. http://arxiv.org/2005.08803.
Haghighat, E., M. Raissi, A. Moure, H. Gomez, and R. Juanes. 2020. “A deep learning framework for solution and discovery in solid mechanics.” Preprint, submitted February 14, 2020. http://arxiv.org/2003.02751.
Jin, X., S. Cai, H. Li, and G. E. Karniadakis. 2020. “Nsfnets (Navier-Stokes flow nets): Physics-informed neural networks for the incompressible navier-stokes equations.” Preprint, submitted March 13, 2020. http://arxiv.org/2003.06496.
Kadeethum, T., T. M. Jørgensen, and H. M. Nick. 2020. “Physics-informed neural networks for solving nonlinear diffusivity and Biot’s equations.” PLoS One 15 (5): e0232683. https://doi.org/10.1371/journal.pone.0232683.
Kharazmi, E., Z. Zhang, and G. E. Karniadakis. 2019. “Variational physics-informed neural networks for solving partial differential equations.” Preprint, submitted November 27, 2019. http://arxiv.org/1912.00873.
Kharazmi, E., Z. Zhang, and G. E. Karniadakis. 2020. “HP-VPINNs: Variational physics-informed neural networks with domain decomposition.” Preprint, submitted March 11, 2020. http://arxiv.org/2003.05385.
Krizhevsky, A., I. Sutskever, and G. E. Hinton. 2012. “Imagenet classification with deep convolutional neural networks.” Adv. Neural Inf. Process. Syst. 25: 1097–1105.
Ladevèze, P., and L. Chamoin. 2016. “The constitutive relation error method: A general verification tool.” In Verifying calculations-forty years on, 59–94. New York: Springer.
Ladevèze, P., and D. Leguillon. 1983. “Error estimate procedure in the finite element method and applications.” SIAM J. Numer. Anal. 20 (3): 485–509.
Ladevèze, P., and J. P. Pelle. 2005. Mastering calculations in linear and nonlinear mechanics. New York: Springer.
LeCun, Y., Y. Bengio, and G. Hinton. 2015. “Deep learning.” Nature 521 (7553): 436–444. https://doi.org/10.1038/nature14539.
Mao, Z., A. D. Jagtap, and G. E. Karniadakis. 2020. “Physics-informed neural networks for high-speed flows.” Comput. Methods Appl. Mech. Eng. 360 (Mar): 112789. https://doi.org/10.1016/j.cma.2019.112789.
Pang, G., L. Lu, and G. E. Karniadakis. 2019. “FPINNs: Fractional physics-informed neural networks.” SIAM J. Sci. Comput. 41 (4): A2603–A2626. https://doi.org/10.1137/18M1229845.
Pled, F., L. Chamoin, and P. Ladevèze. 2011. “On the techniques for constructing admissible stress fields in model verification: Performances on engineering examples.” Int. J. Numer. Methods Eng. 88 (5): 409–441. https://doi.org/10.1002/nme.3180.
Raissi, M., and G. E. Karniadakis. 2018. “Hidden physics models: Machine learning of nonlinear partial differential equations.” J. Comput. Phys. 357 (Mar): 125–141. https://doi.org/10.1016/j.jcp.2017.11.039.
Raissi, M., P. Perdikaris, and G. E. Karniadakis. 2019. “Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations.” J. Comput. Phys. 378 (Feb): 686–707. https://doi.org/10.1016/j.jcp.2018.10.045.
Raissi, M., A. Yazdani, and G. E. Karniadakis. 2020. “Hidden fluid mechanics: Learning velocity and pressure fields from flow visualizations.” Science 367 (6481): 1026–1030. https://doi.org/10.1126/science.aaw4741.
Rao, C., H. Sun, and Y. Liu. 2020. “Physics informed deep learning for computational elastodynamics without labeled data.” Preprint, submitted June 10, 2020. http://arxiv.org/2006.08472.
Sallab, A. E., M. Abdou, E. Perot, and S. Yogamani. 2017. “Deep reinforcement learning framework for autonomous driving.” Electron. Imaging 29 (19): 70–76. https://doi.org/10.2352/ISSN.2470-1173.2017.19.AVM-023.
Shankar, D., S. Narumanchi, H. Ananya, P. Kompalli, and K. Chaudhury. 2017. “Deep learning based large scale visual recommendation and search for e-commerce.” Preprint, submitted March 7, 2017. http://arxiv.org/1703.02344.
Shin, Y., J. Darbon, and G. E. Karniadakis. 2020. “On the convergence and generalization of physics informed neural networks.” Preprint, submitted April 20, 2020. http://arxiv.org/2004.01806.
Yang, X., S. Zafar, J.-X. Wang, and H. Xiao. 2019. “Predictive large-eddy-simulation wall modeling via physics-informed neural networks.” Phys. Rev. Fluids 4 (3): 034602. https://doi.org/10.1103/PhysRevFluids.4.034602.
Yang, Y., and P. Perdikaris. 2018. “Physics-informed deep generative models.” Preprint, submitted December 9, 2018. http://arxiv.org/1812.03511.
Zhao, X., K. Shirvan, R. K. Salko, and F. Guo. 2020. “On the prediction of critical heat flux using a physics-informed machine learning-aided framework.” Appl. Therm. Eng. 164 (Jan): 114540. https://doi.org/10.1016/j.applthermaleng.2019.114540.
Zienkiewicz, O. C., and J. Z. Zhu. 1987. “A simple error estimator and adaptive procedure for practical engineering analysis.” Int. J. Numer. Methods Eng. 24 (2): 337–357. https://doi.org/10.1002/nme.1620240206.
Information & Authors
Information
Published In
Journal of Engineering Mechanics
Volume 148 • Issue 8 • August 2022
Copyright
© 2022 American Society of Civil Engineers.
History
Received: Oct 22, 2021
Accepted: Mar 15, 2022
Published online: May 23, 2022
Published in print: Aug 1, 2022
Discussion open until: Oct 23, 2022
Authors
Metrics & Citations
Metrics
Citations
Download citation
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.
Cited by
- Yong Shang, Fei Wang, Randomized Neural Networks with Petrov–Galerkin Methods for Solving Linear Elasticity and Navier–Stokes Equations, Journal of Engineering Mechanics, 10.1061/JENMDT.EMENG-7463, 150, 4, (2024).