Physics-Informed Deep Learning for Computational Elastodynamics without Labeled Data
Publication: Journal of Engineering Mechanics
Volume 147, Issue 8
Abstract
Numerical methods such as finite element have been flourishing in the past decades for modeling solid mechanics problems via solving governing partial differential equations (PDEs). A salient aspect that distinguishes these numerical methods is how they approximate the physical fields of interest. Physics-informed deep learning (PIDL) is a novel approach developed in recent years for modeling PDE solutions and shows promise to solve computational mechanics problems without using any labeled data (e.g., measurement data is unavailable). The philosophy behind it is to approximate the quantity of interest (e.g., PDE solution variables) by a deep neural network (DNN) and embed the physical law to regularize the network. To this end, training the network is equivalent to minimization of a well-designed loss function that contains the residuals of the governing PDEs as well as initial/boundary conditions (I/BCs). In this paper, we present a physics-informed neural network (PINN) with mixed-variable output to model elastodynamics problems without resort to the labeled data, in which the I/BCs are forcibly imposed. In particular, both the displacement and stress components are taken as the DNN output, inspired by the hybrid finite-element analysis, which largely improves the accuracy and the trainability of the network. Since the conventional PINN framework augments all the residual loss components in a soft manner with Lagrange multipliers, the weakly imposed I/BCs may not be well satisfied especially when complex I/BCs are present. To overcome this issue, a composite scheme of DNNs is established based on multiple single DNNs such that the I/BCs can be satisfied forcibly in a forcible manner. The proposed PINN framework is demonstrated on several numerical elasticity examples with different I/BCs, including both static and dynamic problems as well as wave propagation in truncated domains. Results show the promise of PINN in the context of computational mechanics applications.
Get full access to this article
View all available purchase options and get full access to this article.
Data Availability Statement
The data, model, and the source codes generated or used during the study are available in a repository online in accordance with the original owner’s data retention policies (https://github.com/Raocp/PINN-elastodynamics).
Acknowledgments
The authors would like to acknowledge the support by the TIER 1 Grant program at Northeastern University.
References
Abadi, M., et al. 2016. “Tensorflow: A system for large-scale machine learning.” In Proc., 12th USENIX Symp. on Operating Systems Design and Implementation, 265–283. Berkeley: USENIX.
Baydin, A. G., B. A. Pearlmutter, A. A. Radul, and J. M. Siskind. 2017. “Automatic differentiation in machine learning: A survey.” J. Mach. Learn. Res. 18 (1): 5595–5637.
Berenger, J.-P. 1994. “A perfectly matched layer for the absorption of electromagnetic waves.” J. Comput. Phys. 114 (2): 185–200. https://doi.org/10.1006/jcph.1994.1159.
Berg, J., and K. Nyström. 2019. “Data-driven discovery of PDES in complex datasets.” J. Comput. Phys. 384 (May): 239–252. https://doi.org/10.1016/j.jcp.2019.01.036.
Both, G.-J., S. Choudhury, P. Sens, and R. Kusters. 2019. “DeepMoD: Deep learning for model discovery in noisy data.” Preprint, submitted April 20, 2019. http://arxiv.org/abs/1904.09406.
Chen, Y., L. Lu, G. E. Karniadakis, and L. Dal Negro. 2020a. “Physics-informed neural networks for inverse problems in nano-optics and metamaterials.” Opt. Express 28 (8): 11618–11633. https://doi.org/10.1364/OE.384875.
Chen, Z., Y. Liu, and H. Sun. 2020b. “Deep learning of physical laws from scarce data.” Preprint, submitted May 5, 2020. http://arxiv.org/abs/2005.03448.
Cybenko, G. 1989. “Approximation by superpositions of a sigmoidal function.” Math. Control Signals Syst. 2 (4): 303–314. https://doi.org/10.1007/BF02551274.
Evans, L. C. 2010. Vol. 19 of Partial differential equations. Providence, RI: American Mathematical Society.
Fang, Z., and J. Zhan. 2019. “Deep physical informed neural networks for metamaterial design.” IEEE Access 8 (Dec): 24506–24513. https://doi.org/10.1109/ACCESS.2019.2963375.
Gao, H., L. Sun, and J.-X. Wang. 2020a. “PhyGeoNet: Physics-informed geometry-adaptive convolutional neural networks for solving parametric PDEs on irregular domain.” Preprint, submitted January 17, 2021. http://arxiv.org/abs/2004.13145.
Gao, H., L. Sun, and J.-X. Wang. 2020b. “Super-resolution and denoising of fluid flow using physics-informed convolutional neural networks without high-resolution labels.” Preprint, submitted November 16, 2020. http://arxiv.org/abs/2011.02364.
Glorot, X., and Y. Bengio. 2010. “Understanding the difficulty of training deep feedforward neural networks.” In Proc., 13th Int. Conf. on Artificial Intelligence and Statistics, 249–256. Princeton, NJ: Society for Artificial Intelligence and Statistics.
Gopalakrishnan, S., A. Chakraborty, and D. R. Mahapatra. 2007. Spectral finite element method: wave propagation, diagnostics and control in anisotropic and inhomogeneous structures. New York: Springer.
Goswami, S., C. Anitescu, S. Chakraborty, and T. Rabczuk. 2020a. “Transfer learning enhanced physics informed neural network for phase-field modeling of fracture.” Theor. Appl. Fract. Mech. 106 (Apr): 102447. https://doi.org/10.1016/j.tafmec.2019.102447.
Goswami, S., C. Anitescu, and T. Rabczuk. 2020b. “Adaptive fourth-order phase field analysis using deep energy minimization.” Theor. Appl. Fract. Mech. 107 (Jun): 102527. https://doi.org/10.1016/j.tafmec.2020.102527.
Han, Z., and S. De Rahul. 2019. “A deep learning-based hybrid approach for the solution of multiphysics problems in electrosurgery.” Comput. Methods Appl. Mech. Eng. 357 (Dec): 112603. https://doi.org/10.1016/j.cma.2019.112603.
He, Q., D. Barajas-Solano, G. Tartakovsky, and A. M. Tartakovsky. 2020. “Physics-informed neural networks for multiphysics data assimilation with application to subsurface transport.” Adv. Water Resour. 141 (Jul): 103610. https://doi.org/10.1016/j.advwatres.2020.103610.
He, Q., and J.-S. Chen. 2020. “A physics-constrained data-driven approach based on locally convex reconstruction for noisy database.” Comput. Methods Appl. Mech. Eng. 363 (May): 112791. https://doi.org/10.1016/j.cma.2019.112791.
Hughes, T. J., J. A. Cottrell, and Y. Bazilevs. 2005. “Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement.” Comput. Methods Appl. Mech. Eng. 194 (39–41): 4135–4195. https://doi.org/10.1016/j.cma.2004.10.008.
Jagtap, A. D., and G. E. Karniadakis. 2020. “Extended physics-informed neural networks (XPINNS): A generalized space-time domain decomposition based deep learning framework for nonlinear partial differential equations.” Commun. Comput. Phys. 28 (5): 2002–2041. https://doi.org/10.4208/cicp.OA-2020-0164.
Jagtap, A. D., E. Kharazmi, and G. E. Karniadakis. 2020. “Conservative physics-informed neural networks on discrete domains for conservation laws: Applications to forward and inverse problems.” Comput. Methods Appl. Mech. Eng. 365 (Jun): 113028. https://doi.org/10.1016/j.cma.2020.113028.
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/abs/2003.06496.
Jung, J., C. Jeong, and E. Taciroglu. 2013. “Identification of a scatterer embedded in elastic heterogeneous media using dynamic XFEM.” Comput. Methods Appl. Mech. Eng. 259 (Jun): 50–63. https://doi.org/10.1016/j.cma.2013.03.001.
Jung, J., and E. Taciroglu. 2014. “Modeling and identification of an arbitrarily shaped scatterer using dynamic XFEM with cubic splines.” Comput. Methods Appl. Mech. Eng. 278 (Aug): 101–118. https://doi.org/10.1016/j.cma.2014.05.001.
Karumuri, S., R. Tripathy, I. Bilionis, and J. Panchal. 2020. “Simulator-free solution of high-dimensional stochastic elliptic partial differential equations using deep neural networks.” J. Comput. Phys. 404 (Mar): 109120. https://doi.org/10.1016/j.jcp.2019.109120.
Kharazmi, E., Z. Zhang, and G. Karniadakis. 2019. “Variational physics-informed neural networks for solving partial differential equations.” Preprint, submitted November 27, 2020. http://arxiv.org/abs/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/abs/2003.05385.
Kingma, D. P., and J. Ba. 2014. “Adam: A method for stochastic optimization.” Preprint, submitted December 22, 2014. http://arxiv.org/abs/1412.6980.
Kissas, G., Y. Yang, E. Hwuang, W. R. Witschey, J. A. Detre, and P. Perdikaris. 2020. “Machine learning in cardiovascular flows modeling: Predicting arterial blood pressure from non-invasive 4D flow MRI data using physics-informed neural networks.” Comput. Methods Appl. Mech. Eng. 358 (Jan): 112623. https://doi.org/10.1016/j.cma.2019.112623.
Lagaris, I. E., A. Likas, and D. I. Fotiadis. 1998. “Artificial neural networks for solving ordinary and partial differential equations.” IEEE Trans. Neural Netw. 9 (5): 987–1000. https://doi.org/10.1109/72.712178.
Lagaris, I. E., A. C. Likas, and D. G. Papageorgiou. 2000. “Neural-network methods for boundary value problems with irregular boundaries.” IEEE Trans. Neural Netw. 11 (5): 1041–1049. https://doi.org/10.1109/72.870037.
Lee, H., and I. S. Kang. 1990. “Neural algorithm for solving differential equations.” J. Comput. Phys. 91 (1): 110–131. https://doi.org/10.1016/0021-9991(90)90007-N.
Liu, D., and Y. Wang. 2019. “Multi-fidelity physics-constrained neural network and its application in materials modeling.” J. Mech. Des. 141 (12): 141. https://doi.org/10.1115/1.4044400.
Liu, Y., V. Filonova, N. Hu, Z. Yuan, J. Fish, Z. Yuan, and T. Belytschko. 2014. “A regularized phenomenological multiscale damage model.” Int. J. Numer. Methods Eng. 99 (12): 867–887. https://doi.org/10.1002/nme.4705.
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.
Márquez-Neila, P., M. Salzmann, and P. Fua. 2017. “Imposing hard constraints on deep networks: Promises and limitations.” Preprint, submitted June 7, 2017. http://arxiv.org/abs/1706.02025.
McKay, M. D., R. J. Beckman, and W. J. Conover. 1979. “Comparison of three methods for selecting values of input variables in the analysis of output from a computer code.” Technometrics 21 (2): 239–245. https://doi.org/10.2307/1268522.
Paszke, A., S. Gross, S. Chintala, G. Chanan, E. Yang, Z. DeVito, Z. Lin, A. Desmaison, L. Antiga, and A. Lerer. 2017. “Automatic differentiation in Pytorch.” In Proc., 31st Conf. on Neural Information Processing Systems. Red Hook, NY: Curran Associates.
Psichogios, D. C., and L. H. Ungar. 1992. “A hybrid neural network-first principles approach to process modeling.” AIChE J. 38 (10): 1499–1511. https://doi.org/10.1002/aic.690381003.
Raissi, M. 2018. “Deep hidden physics models: Deep learning of nonlinear partial differential equations.” J. Mach. Learn. Res. 19 (1): 932–955.
Raissi, M., P. Perdikaris, and G. E. Karniadakis. 2019a. “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., Z. Wang, M. S. Triantafyllou, and G. E. Karniadakis. 2019b. “Deep learning of vortex-induced vibrations.” J. Fluid Mech. 861 (Feb): 119–137. https://doi.org/10.1017/jfm.2018.872.
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., and Y. Liu. 2020. “Three-dimensional convolutional neural network (3D-CNN) for heterogeneous material homogenization.” Comput. Mater. Sci. 184 (Nov): 109850. https://doi.org/10.1016/j.commatsci.2020.109850.
Rao, C., H. Sun, and Y. Liu. 2020. “Physics-informed deep learning for incompressible laminar flows.” Theor. Appl. Mech. Lett. 10 (3): 207–212. https://doi.org/10.1016/j.taml.2020.01.039.
Sahli Costabal, F., Y. Yang, P. Perdikaris, D. E. Hurtado, and E. Kuhl. 2020. “Physics-informed neural networks for cardiac activation mapping.” Front. Phys. 8 (Feb): 42. https://doi.org/10.3389/fphy.2020.00042.
Samaniego, E., C. Anitescu, S. Goswami, V. Nguyen-Thanh, H. Guo, K. Hamdia, X. Zhuang, and T. Rabczuk. 2020. “An energy approach to the solution of partial differential equations in computational mechanics via machine learning: Concepts, implementation and applications.” Comput. Methods Appl. Mech. Eng. 362 (Apr): 112790. https://doi.org/10.1016/j.cma.2019.112790.
Shukla, K., P. C. Di Leoni, J. Blackshire, D. Sparkman, and G. E. Karniadakis. 2020. “Physics-informed neural network for ultrasound nondestructive quantification of surface breaking cracks.” Preprint, submitted May 7, 2020. http://arxiv.org/abs/2005.03596.
Sirignano, J., and K. Spiliopoulos. 2018. “DGM: A deep learning algorithm for solving partial differential equations.” J. Comput. Phys. 375: 1339–1364. https://doi.org/10.1016/j.jcp.2018.08.029.
Smith, M. 2009. ABAQUS/standard user’s manual, version 6.9. Dassault Systèmes Simulia Corporation.
Sun, H., H. Waisman, and R. Betti. 2013. “Nondestructive identification of multiple flaws using XFEM and a topologically adapting artificial bee colony algorithm.” Int. J. Numer. Methods Eng. 95 (10): 871–900. https://doi.org/10.1002/nme.4529.
Sun, H., H. Waisman, and R. Betti. 2014. “A multiscale flaw detection algorithm based on XFEM.” Int. J. Numer. Methods Eng. 100 (7): 477–503. https://doi.org/10.1002/nme.4741.
Sun, H., H. Waisman, and R. Betti. 2016. “A sweeping window method for detection of flaws using an explicit dynamic XFEM and absorbing boundary layers.” Int. J. Numer. Methods Eng. 105 (13): 1014–1040. https://doi.org/10.1002/nme.5006.
Sun, L., H. Gao, S. Pan, and J.-X. Wang. 2020. “Surrogate modeling for fluid flows based on physics-constrained deep learning without simulation data.” Comput. Methods Appl. Mech. Eng. 361 (Apr): 112732. https://doi.org/10.1016/j.cma.2019.112732.
Sun, L., and J.-X. Wang. 2020. “Physics-constrained Bayesian neural network for fluid flow reconstruction with sparse and noisy data.” Theor. Appl. Mech. Lett. 10 (3): 161–169. https://doi.org/10.1016/j.taml.2020.01.031.
Tripathy, R. K., and I. Bilionis. 2018. “Deep UQ: Learning deep neural network surrogate models for high dimensional uncertainty quantification.” J. Comput. Phys. 375 (Dec): 565–588. https://doi.org/10.1016/j.jcp.2018.08.036.
Vlassis, N., R. Ma, and W. Sun. 2020. “Geometric deep learning for computational mechanics Part I: Anisotropic hyperelasticity.” Preprint, submitted October 29, 2020. http://arxiv.org/abs/2001.04292.
Wang, K., and W. Sun. 2018. “A multiscale multi-permeability poroplasticity model linked by recursive homogenizations and deep learning.” Comput. Methods Appl. Mech. Eng. 334 (Jun): 337–380. https://doi.org/10.1016/j.cma.2018.01.036.
Wang, K., and W. Sun. 2019. “Meta-modeling game for deriving theory-consistent, microstructure-based traction–separation laws via deep reinforcement learning.” Comput. Methods Appl. Mech. Eng. 346 (Apr): 216–241. https://doi.org/10.1016/j.cma.2018.11.026.
Wang, S., Y. Teng, and P. Perdikaris. 2020. “Understanding and mitigating gradient pathologies in physics-informed neural networks.” Preprint, submitted January 15, 2020. http://arxiv.org/abs/2001.04536.
Weinan, E., and B. Yu. 2018. “The deep ritz method: A deep learning-based numerical algorithm for solving variational problems.” Commun. Math. Stat. 6 (1): 1–12. https://doi.org/10.1007/s40304-018-0127-z.
Yan, G., H. Sun, and H. Waisman. 2015. “A guided Bayesian inference approach for detection of multiple flaws in structures using the extended finite element method.” Comput. Struct. 152 (May): 27–44. https://doi.org/10.1016/j.compstruc.2015.02.010.
Yang, L., X. Meng, and G. E. Karniadakis. 2020. “B-PINNs: Bayesian physics-informed neural networks for forward and inverse PDE problems with noisy data.” Preprint, submitted December 18, 2020. http://arxiv.org/abs/2003.06097.
Yang, Y., and P. Perdikaris. 2019. “Adversarial uncertainty quantification in physics-informed neural networks.” J. Comput. Phys. 394 (Oct): 136–152. https://doi.org/10.1016/j.jcp.2019.05.027.
Zhang, D., L. Lu, L. Guo, and G. E. Karniadakis. 2019. “Quantifying total uncertainty in physics-informed neural networks for solving forward and inverse stochastic problems.” J. Comput. Phys. 397 (Nov): 108850. https://doi.org/10.1016/j.jcp.2019.07.048.
Zhang, R., Y. Liu, and H. Sun. 2020a. “Physics-guided convolutional neural network (PhyCNN) for data-driven seismic response modeling.” Eng. Struct. 215 (Jul): 110704. https://doi.org/10.1016/j.engstruct.2020.110704.
Zhang, R., Y. Liu, and H. Sun. 2020b. “Physics-informed multi-LSTM networks for metamodeling of nonlinear structures.” Comput. Methods Appl. Mech. Eng. 369 (Sep): 113226. https://doi.org/10.1016/j.cma.2020.113226.
Zhu, C., R. H. Byrd, P. Lu, and J. Nocedal. 1997. “Algorithm 778: L-BFGS-B: Fortran subroutines for large-scale bound-constrained optimization.” ACM Trans. Math. Software (TOMS) 23 (4): 550–560. https://doi.org/10.1145/279232.279236.
Zhu, Y., and N. Zabaras. 2018. “Bayesian deep convolutional encoder–decoder networks for surrogate modeling and uncertainty quantification.” J. Comput. Phys. 366 (Aug): 415–447. https://doi.org/10.1016/j.jcp.2018.04.018.
Zhu, Y., N. Zabaras, P.-S. Koutsourelakis, and P. Perdikaris. 2019. “Physics-constrained deep learning for high-dimensional surrogate modeling and uncertainty quantification without labeled data.” J. Comput. Phys. 394 (Oct): 56–81. https://doi.org/10.1016/j.jcp.2019.05.024.
Information & Authors
Information
Published In
Journal of Engineering Mechanics
Volume 147 • Issue 8 • August 2021
Copyright
© 2021 American Society of Civil Engineers.
History
Received: Nov 24, 2020
Accepted: Feb 23, 2021
Published online: May 21, 2021
Published in print: Aug 1, 2021
Discussion open until: Oct 21, 2021
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.