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Physics informed neural networks for continuum micromechanics. (English) Zbl 07526114

Summary: Recently, physics informed neural networks have successfully been applied to a broad variety of problems in applied mathematics and engineering. The principle idea is the usage of a neural network as a global ansatz function for partial differential equations. Due to the global approximation, physics informed neural networks have difficulties in displaying localized effects and strong nonlinear solution fields by optimization. In this work we consider nonlinear stress and displacement fields invoked by material inhomogeneities with sharp phase interfaces. This constitutes a challenging problem for a method relying on a global ansatz. To overcome convergence issues, adaptive training strategies and domain decomposition are studied. It is shown, that the domain decomposition approach is capable to accurately resolve nonlinear stress, displacement and energy fields in heterogeneous microstructures obtained from real-world \(\mu\)CT-scans.

MSC:

74-XX Mechanics of deformable solids
82-XX Statistical mechanics, structure of matter
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[1] David Müzel, S.; Bonhin, E. P.; Guimarães, N. M.; Guidi, E. S., Application of the finite element method in the analysis of composite materials: A review, Polymers, 12, 4, 818 (2020)
[2] LeCun, Y.; Bengio, Y.; Hinton, G., Deep learning, Nature, 521, 7553, 436-444 (2015)
[3] Lantz, V.; Abiri, N.; Carlsson, G.; Pistol, M.-E., Deep learning for inverse problems in quantum mechanics, Int. J. Quantum Chem., 121, 9, Article e26599 pp. (2021)
[4] Min, S.; Lee, B.; Yoon, S., Deep learning in bioinformatics, Brief. Bioinform., 18, 5, 851-869 (2017)
[5] Piccialli, F.; Di Somma, V.; Giampaolo, F.; Cuomo, S.; Fortino, G., A survey on deep learning in medicine: Why, how and when?, Inf. Fusion, 66, 111-137 (2021)
[6] Lu, L.; Jin, P.; Pang, G.; Zhang, Z.; Karniadakis, G. E., Learning nonlinear operators via DeepONet based on the universal approximation theorem of operators, Nat. Mach. Intell., 3, 3, 218-229 (2021)
[7] Bock, F. E.; Aydin, R. C.; Cyron, C. J.; Huber, N.; Kalidindi, S. R.; Klusemann, B., A review of the application of machine learning and data mining approaches in continuum materials mechanics, Front. Mater., 6, 110 (2019)
[8] Kutz, J. N., Deep learning in fluid dynamics, J. Fluid Mech., 814, 1-4 (2017)
[9] Hsu, Y.-C.; Yu, C.-H.; Buehler, M. J., Using deep learning to predict fracture patterns in crystalline solids, Matter, 3, 1, 197-211 (2020)
[10] Henkes, A.; Caylak, I.; Mahnken, R., A deep learning driven pseudospectral PCE based FFT homogenization algorithm for complex microstructures, Comput. Methods Appl. Mech. Engrg., 385, Article 114070 pp. (2021)
[11] Lagaris, I. E.; Likas, A.; Fotiadis, D. I., Artificial neural networks for solving ordinary and partial differential equations, IEEE Trans. Neural Netw., 9, 5, 987-1000 (1998)
[12] Raissi, M.; Perdikaris, P.; Karniadakis, G. E., Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations, J. Comput. Phys., 378, 686-707 (2019)
[13] Karniadakis, G. E.; Kevrekidis, I. G.; Lu, L.; Perdikaris, P.; Wang, S.; Yang, L., Physics-informed machine learning, Nature Rev. Phys., 1-19 (2021)
[14] Rackauckas, C.; Ma, Y.; Martensen, J.; Warner, C.; Zubov, K.; Supekar, R.; Skinner, D.; Ramadhan, A.; Edelman, A., Universal differential equations for scientific machine learning (2020), arXiv preprint arXiv:2001.04385
[15] Misyris, G. S.; Venzke, A.; Chatzivasileiadis, S., Physics-informed neural networks for power systems, (2020 IEEE Power & Energy Society General Meeting (PESGM) (2020), IEEE), 1-5
[16] Ji, W.; Qiu, W.; Shi, Z.; Pan, S.; Deng, S., Stiff-pinn: Physics-informed neural network for stiff chemical kinetics (2020), arXiv preprint arXiv:2011.04520
[17] Raissi, M.; Yazdani, A.; Karniadakis, G. E., Hidden fluid mechanics: Learning velocity and pressure fields from flow visualizations, Science, 367, 6481, 1026-1030 (2020)
[18] Zhu, Q.; Liu, Z.; Yan, J., Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks, Comput. Mech., 67, 2, 619-635 (2021)
[19] Wessels, H.; Weißenfels, C.; Wriggers, P., The neural particle method-an updated Lagrangian physics informed neural network for computational fluid dynamics, Comput. Methods Appl. Mech. Engrg., 368, Article 113127 pp. (2020)
[20] Cai, S.; Mao, Z.; Wang, Z.; Yin, M.; Karniadakis, G. E., Physics-informed neural networks (PINNs) for fluid mechanics: A review (2021), arXiv preprint arXiv:2105.09506
[21] Niaki, S. A.; Haghighat, E.; Campbell, T.; Poursartip, A.; Vaziri, R., Physics-informed neural network for modelling the thermochemical curing process of composite-tool systems during manufacture, Comput. Methods Appl. Mech. Engrg., 384, Article 113959 pp. (2021)
[22] Zhang, E.; Yin, M.; Karniadakis, G. E., Physics-informed neural networks for nonhomogeneous material identification in elasticity imaging (2020), arXiv preprint arXiv:2009.04525
[23] Cai, S.; Wang, Z.; Wang, S.; Perdikaris, P.; Karniadakis, G. E., Physics-informed neural networks for heat transfer problems, J. Heat Transfer, 143, 6, Article 060801 pp. (2021)
[24] Vahab, M.; Haghighat, E.; Khaleghi, M.; Khalili, N., A physics informed neural network approach to solution and identification of biharmonic equations of elasticity (2021), arXiv preprint arXiv:2108.07243
[25] Haghighat, E.; Raissi, M.; Moure, A.; Gomez, H.; Juanes, R., A physics-informed deep learning framework for inversion and surrogate modeling in solid mechanics, Comput. Methods Appl. Mech. Engrg., 379, Article 113741 pp. (2021)
[26] Rao, C.; Sun, H.; Liu, Y., Physics-informed deep learning for incompressible laminar flows, Theor. Appl. Mech. Lett., 10, 3, 207-212 (2020)
[27] Samaniego, E.; Anitescu, C.; Goswami, S.; Nguyen-Thanh, V. M.; Guo, H.; Hamdia, K.; Zhuang, X.; Rabczuk, T., An energy approach to the solution of partial differential equations in computational mechanics via machine learning: Concepts, implementation and applications, Comput. Methods Appl. Mech. Engrg., 362, Article 112790 pp. (2020)
[28] Kollmannsberger, S.; D’Angella, D.; Jokeit, M.; Herrmann, L., Deep energy method, (Deep Learning in Computational Mechanics (2021), Springer), 85-91
[29] Guo, M.; Haghighat, E., An energy-based error bound of physics-informed neural network solutions in elasticity (2020), arXiv preprint arXiv:2010.09088
[30] Chen, C.-T.; Gu, G. X., Learning hidden elasticity with deep neural networks, Proc. Natl. Acad. Sci., 118, 31 (2021)
[31] Shin, Y.; Darbon, J.; Karniadakis, G. E., On the convergence and generalization of physics informed neural networks (2020), ArXiv E-Prints, arXiv-2004
[32] Nguyen-Thanh, V. M.; Zhuang, X.; Rabczuk, T., A deep energy method for finite deformation hyperelasticity, Eur. J. Mech. A Solids, 80, Article 103874 pp. (2020)
[33] Li, S.; Wang, G., Introduction to Micromechanics and Nanomechanics (2008), World Scientific Publishing Company
[34] Hornik, K.; Stinchcombe, M.; White, H., Multilayer feedforward networks are universal approximators, Neural Netw., 2, 5, 359-366 (1989)
[35] Bishop, C. M., Pattern Recognition and Machine Learning (2006), springer
[36] Goodfellow, I.; Bengio, Y.; Courville, A.; Bengio, Y., Deep learning, Vol. 1 (2016), MIT press Cambridge
[37] Aggarwal, C. C., Neural networks and deep learning, Vol. 10 (2018), Springer
[38] Géron, A., Hands-on Machine Learning with Scikit-Learn, Keras, and TensorFlow: Concepts, Tools, and Techniques to Build Intelligent Systems (2019), O’Reilly Media
[39] Chollet, F., Deep Learning with Python, Vol. 361 (2018), Manning New York
[40] Ramachandran, P.; Zoph, B.; Le, Q. V., Searching for activation functions (2017), arXiv preprint arXiv:1710.05941
[41] Berg, J.; Nyström, K., A unified deep artificial neural network approach to partial differential equations in complex geometries, Neurocomputing, 317, 28-41 (2018)
[42] Abadi, M.; Agarwal, A.; Barham, P.; Brevdo, E.; Chen, Z.; Citro, C.; Corrado, G. S.; Davis, A.; Dean, J.; Devin, M.; Ghemawat, S.; Goodfellow, I.; Harp, A.; Irving, G.; Isard, M.; Jia, Y.; Jozefowicz, R.; Kaiser, L.; Kudlur, M.; Levenberg, J.; Mané, D.; Monga, R.; Moore, S.; Murray, D.; Olah, C.; Schuster, M.; Shlens, J.; Steiner, B.; Sutskever, I.; Talwar, K.; Tucker, P.; Vanhoucke, V.; Vasudevan, V.; Viégas, F.; Vinyals, O.; Warden, P.; Wattenberg, M.; Wicke, M.; Yu, Y.; Zheng, X., TensorFlow: Large-scale machine learning on heterogeneous systems (2015), Software available from tensorflow.org, URL https://www.tensorflow.org/
[43] Sun, L.; Gao, H.; Pan, S.; Wang, J.-X., Surrogate modeling for fluid flows based on physics-constrained deep learning without simulation data, Comput. Methods Appl. Mech. Engrg., 361, Article 112732 pp. (2020)
[44] Wang, S.; Teng, Y.; Perdikaris, P., Understanding and mitigating gradient pathologies in physics-informed neural networks (2020), arXiv preprint arXiv:2001.04536
[45] Wang, S.; Yu, X.; Perdikaris, P., When and why pinns fail to train: A neural tangent kernel perspective (2020), arXiv preprint arXiv:2007.14527
[46] Fletcher, R., Practical methods of optimization, Vol. 80, 4 (1987), ohn wiley & sons: ohn wiley & sons New York
[47] Wight, C. L.; Zhao, J., Solving allen-cahn and cahn-hilliard equations using the adaptive physics informed neural networks (2020), arXiv preprint arXiv:2007.04542
[48] C. Szegedy, V. Vanhoucke, S. Ioffe, J. Shlens, Z. Wojna, Rethinking the inception architecture for computer vision, in: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, 2016, pp. 2818-2826.
[49] Jagtap, A. D.; Kharazmi, E.; Karniadakis, G. E., Conservative physics-informed neural networks on discrete domains for conservation laws: Applications to forward and inverse problems, Comput. Methods Appl. Mech. Engrg., 365, Article 113028 pp. (2020)
[50] Chen, Y.; Lu, L.; Karniadakis, G. E.; Dal Negro, L., Physics-informed neural networks for inverse problems in nano-optics and metamaterials, Opt. Express, 28, 8, 11618-11633 (2020)
[51] Anton, D.; Wessels, H., Identification of material parameters from full-field displacement data using physics-informed neural networks (2021), Researchgate Preprint
[52] Fuhg, J. N.; Bouklas, N., On physics-informed data-driven isotropic and anisotropic constitutive models through probabilistic machine learning and space-filling sampling (2021), arXiv preprint arXiv:2109.11028
[53] Shukla, K.; Jagtap, A. D.; Karniadakis, G. E., Parallel physics-informed neural networks via domain decomposition (2021), arXiv preprint arXiv:2104.10013
[54] Jagtap, A. D.; Karniadakis, G. E., 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 (2020)
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