A nonlocal physics-informed deep learning framework using the peridynamic differential operator. (English) Zbl 07415658

Summary: The Physics-Informed Neural Network (PINN) framework introduced recently incorporates physics into deep learning, and offers a promising avenue for the solution of partial differential equations (PDEs) as well as identification of the equation parameters. The performance of existing PINN approaches, however, may degrade in the presence of sharp gradients, as a result of the inability of the network to capture the solution behavior globally. We posit that this shortcoming may be remedied by introducing long-range (nonlocal) interactions into the network’s input, in addition to the short-range (local) space and time variables. Following this ansatz, here we develop a nonlocal PINN approach using the Peridynamic Differential Operator (PDDO) – a numerical method which incorporates long-range interactions and removes spatial derivatives in the governing equations. Because the PDDO functions can be readily incorporated in the neural network architecture, the nonlocality does not degrade the performance of modern deep-learning algorithms. We apply nonlocal PDDO-PINN to the solution and identification of material parameters in solid mechanics and, specifically, to elastoplastic deformation in a domain subjected to indentation by a rigid punch, for which the mixed displacement-traction boundary condition leads to localized deformation and sharp gradients in the solution. We document the superior behavior of nonlocal PINN with respect to local PINN in both solution accuracy and parameter inference, illustrating its potential for simulation and discovery of partial differential equations whose solution develops sharp gradients.


35-XX Partial differential equations
82-XX Statistical mechanics, structure of matter
Full Text: DOI arXiv


[1] Goodfellow, I.; Bengio, Y.; Courville, A., Deep Learning, 800 (2016), MIT press, URL https://www.deeplearningbook.org
[2] Bishop, C. M., Pattern Recognition and Machine Learning (2006), Springer-Verlag: Springer-Verlag Berlin, Heidelberg, URL https://www.springer.com/gp/book/9780387310732 · Zbl 1107.68072
[3] Krizhevsky, A.; Sutskever, I.; Hinton, G. E., Imagenet classification with deep convolutional neural networks, (Pereira, F.; Burges, C. J.C.; Bottou, L.; Weinberger, K. Q., Advances in Neural Information Processing Systems, NIPS 2012 (2012)), 1097-1105
[4] LeCun, Y.; Bengio, Y.; Hinton, G., Deep learning, Nature, 521, 7553, 436-444 (2015)
[5] Jannach, D.; Zanker, M.; Felfernig, A.; Friedrich, G., Recommender Systems: An Introduction (2010), Cambridge University Press
[6] Zhang, S.; Yao, L.; Sun, A.; Tay, Y., Deep learning based recommender system: A survey and new perspectives, ACM Comput. Surv., 52, 1, 1-38 (2019)
[7] A. Graves, M. Abdel-Rahman, G. Hinton, Speech recognition with deep recurrent neural networks, in: IEEE International Conference on Acoustics, Speech and Signal Processing, 2013, pp. 6645-6649.
[8] Bojarski, M.; Del Testa, D.; Dworakowski, D.; Firner, B.; Flepp, B.; Goyal, P.; Jackel, L. D.; Monfort, M.; Muller, U.; Zhang, J., End to end learning for self-driving cars (2016), arXiv preprint arXiv:1604.07316
[9] Miotto, R.; Wang, F.; Wang, S.; Jiang, X.; Dudley, J. T., Deep learning for healthcare: Review, opportunities and challenges, Brief. Bioinform., 19, 6, 1236-1246 (2018)
[10] Brunton, S. L.; Kutz, J. N., Data-Driven Science and Engineering: Machine Learning, Dynamical Systems, and Control (2019), Cambridge University Press · Zbl 1407.68002
[11] Brenner, M. P.; Eldredge, J. D.; Freund, J. B., Perspective on machine learning for advancing fluid mechanics, Phys. Rev. Fluids, 4, 10, Article 100501 pp. (2019), URL https://link.aps.org/doi/10.1103/PhysRevFluids.4.100501
[12] Ghaboussi, J.; Sidarta, D., New nested adaptive neural networks (NANN) for constitutive modeling, Comput. Geotech., 22, 1, 29-52 (1998)
[13] Kirchdoerfer, T.; Ortiz, M., Data-driven computational mechanics, Comput. Methods Appl. Mech. Engrg., 304, 81-101 (2016) · Zbl 1425.74503
[14] 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, 379, 113741 (2021), URL https://doi.org/10.1016/j.cma.2021.113741 · Zbl 07340350
[15] DeVries, P. M.R.; Viégas, F.; Wattenberg, M.; Meade, B. J., Deep learning of aftershock patterns following large earthquakes, Nature, 560, 7720, 632-634 (2018)
[16] Kong, Q.; Trugman, D. T.; Ross, Z. E.; Bianco, M. J.; Meade, B. J.; Gerstoft, P., Machine learning in seismology: Turning data into insights, Seismol. Res. Lett., 90, 1, 3-14 (2018)
[17] Bergstra, J.; Breuleux, O.; Bastien, F.; Lamblin, P.; Pascanu, R.; Desjardins, G.; Turian, J.; Warde-Farley, D.; Bengio, Y., Theano: A CPU and GPU math expression compiler, (Proceedings of the Python for Scientific Computing Conference, SciPy, vol. 4 (2010)), 3
[18] Abadi, M.; Barham, P.; Chen, J.; Chen, Z.; Davis, A.; Dean, J.; Devin, M.; Ghemawat, S.; Irving, G.; Isard, M.; Kudlur, M.; Levenberg, J.; Monga, R.; Moore, S.; Murray, D. G.; Steiner, B.; Tucker, P.; Vasudevan, V.; Warden, P.; Wicke, M.; Yu, Y.; Zheng, X., TensorFlow: A system for large-scale machine learning, (12th USENIX Symposium on Operating Systems Design and Implementation, OSDI 16 (2016), USENIX Association: USENIX Association Savannah, GA), 265-283, URL https://www.usenix.org/conference/osdi16/technical-sessions/presentation/abadi
[19] Paszke, A.; Gross, S.; Massa, F.; Lerer, A.; Bradbury, J.; Chanan, G.; Killeen, T.; Lin, Z.; Gimelshein, N.; Antiga, L., PyTorch: An imperative style, high-performance deep learning library, (Wallach, H.; Larochelle, H.; Beygelzimer, A.; d’Alché Buc, F.; Fox, E.; Garnett, R., Advances in Neural Information Processing Systems, NIPS 2019 (2019)), 8024-8035
[20] Haghighat, E.; Juanes, R., SciANN: A Keras/TensorFlow wrapper for scientific computations and physics-informed deep learning using artificial neural networks, Computer Methods in Applied Mechanics and Engineering, 373, 113552 (2021), URL https://doi.org/10.1016/j.cma.2020.113552 · Zbl 07337817
[21] Han, J.; Jentzen, A.; E, W., Solving high-dimensional partial differential equations using deep learning, Proc. Natl. Acad. Sci., 115, 34, 8505-8510 (2018), URL https://www.pnas.org/content/115/34/8505 · Zbl 1416.35137
[22] Bar-Sinai, Y.; Hoyer, S.; Hickey, J.; Brenner, M. P., Learning data-driven discretizations for partial differential equations, Proc. Natl. Acad. Sci., 116, 31, 15344-15349 (2019), URL https://www.pnas.org/content/116/31/15344 · Zbl 1431.65195
[23] Rudy, S.; Alla, A.; Brunton, S. L.; Kutz, J. N., Data-driven identification of parametric partial differential equations, SIAM J. Appl. Dyn. Syst., 18, 2, 643-660 (2019), URL https://epubs.siam.org/doi/abs/10.1137/18M1191944 · Zbl 1456.65096
[24] 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) · Zbl 1415.68175
[25] Champion, K.; Lusch, B.; Kutz, J. N.; Brunton, S. L., Data-driven discovery of coordinates and governing equations, Proc. Natl. Acad. Sci., 116, 45, 22445-22451 (2019), URL https://www.pnas.org/content/116/45/22445 · Zbl 1433.68396
[26] Raissi, M.; Yazdani, A.; Karniadakis, G. E., Hidden fluid mechanics: Learning velocity and pressure fields from flow visualizations, Science, 367, 6481, 1026-1030 (2020)
[27] Baydin, A. G.; Pearlmutter, B. A.; Radul, A. A.; Siskind, J. M., Automatic differentiation in machine learning: A survey, J. Mach. Learn. Res., 18, 1, 5595-5637 (2017), URL https://dl.acm.org/doi/abs/10.5555/3122009.3242010
[28] Kharazmi, E.; Zhang, Z.; Karniadakis, G. E., Hp-vpinns: variational physics-informed neural networks with domain decomposition, Computer Methods in Applied Mechanics and Engineering, 374, 113547 (2021), URL https://doi.org/10.1016/j.cma.2020.113547 · Zbl 07338008
[29] Meng, X.; Li, Z.; Zhang, D.; Karniadakis, G. E., PPINN: Parareal physics-informed neural network for time-dependent PDEs, Computer Methods in Applied Mechanics and Engineering, 370, 113250 (2020), URL https://doi.org/10.1016/j.cma.2020.113250 · Zbl 07337119
[30] Madenci, E.; Barut, A.; Futch, M., Peridynamic differential operator and its applications, Comput. Methods Appl. Mech. Engrg., 304, 408-451 (2016) · Zbl 1425.74043
[31] Madenci, E.; Barut, A.; Dorduncu, M., Peridynamic Differential Operator for Numerical Analysis (2019), Springer
[32] Silling, S. A., Reformulation of elasticity theory for discontinuities and long-range forces, J. Mech. Phys. Solids, 48, 1, 175-209 (2000) · Zbl 0970.74030
[33] Silling, S. A.; Askari, E., A meshfree method based on the peridynamic model of solid mechanics, Comput. Struct., 83, 17-18, 1526-1535 (2005)
[34] Silling, S. A.; Epton, M.; Weckner, O.; Xu, J.; Askari, E., Peridynamic states and constitutive modeling, J. Elasticity, 88, 2, 151-184 (2007) · Zbl 1120.74003
[35] Silling, S. A.; Lehoucq, R. B., Convergence of peridynamics to classical elasticity theory, J. Elasticity, 93, 1, 13 (2008) · Zbl 1159.74316
[36] Madenci, E.; Oterkus, E., Peridynamic Theory and Its Applications (2014), Springer · Zbl 1295.74001
[37] Madenci, E.; Dorduncu, M.; Barut, A.; Futch, M., Numerical solution of linear and nonlinear partial differential equations using the peridynamic differential operator, Numer. Methods Partial Differential Equations, 33, 5, 1726-1753 (2017) · Zbl 1375.65124
[38] Chen, T.; Li, M.; Li, Y.; Lin, M.; Wang, N.; Wang, M.; Xiao, T.; Xu, B.; Zhang, C.; Zhang, Z., MXNet: A flexible and efficient machine learning library for heterogeneous distributed systems (2015), arXiv:1512.01274
[39] Duchi, J.; Hazan, E.; Singer, Y., Adaptive subgradient methods for online learning and stochastic optimization, J. Mach. Learn. Res., 12, Jul, 2121-2159 (2011), URL http://jmlr.org/papers/v12/duchi11a.html · Zbl 1280.68164
[40] Kingma, D. P.; Ba, J., Adam: A method for stochastic optimization (2014), arXiv:1412.6980
[41] Chollet, F., Keras (2015), URL https://github.com/fchollet/keras
[42] Simo, J. C.; Hughes, T. J.R., Computational Inelasticity (1998), Springer · Zbl 0934.74003
[43] COMSOL, J. C., COMSOL Multiphysics User’s Guide (2020), COMSOL: COMSOL Stockholm, Sweden
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.