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Physics-Informed Neural Networks

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Deep Learning in Computational Mechanics

Part of the book series: Studies in Computational Intelligence ((SCI,volume 977))

Abstract

Physics-informed neural networks (PINNs) are used for problems where data are scarce. The underlying physics is enforced via the governing differential equation, including the residual in the cost function. PINNs can be used for both solving and discovering differential equations. In this chapter, PINNs are illustrated with three one-dimensional examples. The first example shows how the displacement of a static bar can be computed. The temperature evolution in a one-dimensional spatial domain is determined using the non-linear heat equation, using both a continuous and a discrete approach. Finally, the data-driven identification is illustrated with the static bar, where the cross-sectional stiffness is estimated from the displacement.

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References

  1. Dimitris C. Psichogios and Lyle H. Ungar. “A hybrid neural network-first principles approach to process modeling”. In: AIChE J. 38.10 (Oct. 1992), pp. 1499–1511. ISSN: 0001-1541, 1547-5905. DOI https://doi.org/10.1002/aic.690381003 (visited on 07/02/2020).

  2. I.E. Lagaris, A. Likas, and D.I. Fotiadis. “Artificial neural networks for solving ordinary and partial differential equations”. In: IEEE Trans. Neural Netw. 9.5 (Sept. 1998), pp. 987–1000. ISSN: 10459227. DOI https://doi.org/10.1109/72.712178 (visited on 01/08/2020).

  3. Risi Kondor. “N-body Networks: a Covariant Hierarchical Neural Network Architecture for Learning Atomic Potentials”. In: arXiv:1803.01588 [cs] (Mar. 5, 2018) (visited on 07/15/2020).

  4. Matthew Hirn, Stéphane Mallat, and Nicolas Poilvert. “Wavelet Scattering Regression of Quantum Chemical Energies”. In: Multiscale Model. Simul. 15.2 (Jan. 2017), pp. 827–863. ISSN: 1540-3459, 1540-3467. DOI https://doi.org/10.1137/16M1075454. arXiv:1605.04654 (visited on 07/15/2020).

  5. Stéphane Mallat. “Understanding deep convolutional networks”. In: Phil. Trans. R. Soc. A 374.2065 (Apr. 13, 2016), p. 20150203. ISSN: 1364-503X, 1471-2962. DOI https://doi.org/10.1098/rsta.2015.0203 (visited on 07/15/2020).

  6. M. Raissi, P. Perdikaris, and G.E. Karniadakis. “Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations”. In: Journal of Computational Physics 378 (Feb. 2019), pp. 686–707. ISSN: 00219991. DOI https://doi.org/10.1016/j.jcp.2018.10.045. URL: https://linkinghub.elsevier.com/retrieve/pii/S0021999118307125 (visited on 01/08/2020).

  7. Maziar Raissi. maziarraissi/PINNs. original-date: 2018-01-21T04:04:32Z. July 25, 2020. URL: https://github.com/maziarraissi/PINNs (visited on 07/27/2020).

  8. Steven Brunton, Bernd Noack, and Petros Koumoutsakos. “Machine Learning for Fluid Mechanics”. In: Annu. Rev. Fluid Mech. 52.1 (Jan. 5, 2020), pp. 477–508. ISSN: 0066-4189, 1545-4479. DOI https://doi.org/10.1146/annurev-fluid-010719-060214. arXiv: 1905.11075 (visited on 06/26/2020).

  9. Michael Frank, Dimitris Drikakis, and Vassilis Charissis. “Machine-Learning Methods for Computational Science and Engineering”. In: Computation 8.1 (Mar. 3, 2020), p. 15. ISSN: 2079-3197. DOI https://doi.org/10.3390/computation8010015. URL: https://www.mdpi.com/2079-3197/8/1/15 (visited on 07/02/2020).

  10. Esteban Samaniego et al. “An Energy Approach to the Solution of Partial Differential Equations in Computational Mechanics via Machine Learning: Concepts, Implementation and Applications”. In: arXiv:1908.10407 [cs, math, stat] (Sept. 2, 2019) (visited on 01/08/2020).

  11. Dong C. Liu and Jorge Nocedal. “On the limited memory BFGS method for large scale optimization”. In: Mathematical Programming 45.1 (Aug. 1989), pp. 503–528. ISSN: 0025-5610, 1436-4646. DOI https://doi.org/10.1007/BF01589116 (visited on 07/13/2020).

  12. S. Kollmannsberger et al. “A hierarchical computational model for moving thermal loads and phase changes with applications to selective laser melting”. In: Computers & Mathematics with Applications 75.5 (Mar. 2018), pp. 1483–1497. ISSN: 08981221. DOI https://doi.org/10.1016/j.camwa.2017.11.014. URL: https://linkinghub.elsevier.com/retrieve/pii/S0898122117307289 (visited on 07/21/2020).

  13. Stefan Kollmannsberger et al. “Accurate Prediction of Melt Pool Shapes in Laser Powder Bed Fusion by the Non-Linear Temperature Equation Including Phase Changes: Model validity: isotropic versus anisotropic conductivity to capture AM Benchmark Test AMB2018-02”. In: Integr Mater Manuf Innov 8.2 (June 2019), pp. 167–177. ISSN: 2193-9764, 2193-9772. DOI https://doi.org/10.1007/s40192-019-00132-9 (visited on 07/20/2020).

  14. Patrick J. Roache. “Code Verification by the Method of Manufactured Solutions”. In: J. Fluids Eng 124.1 (Mar. 1, 2002). Publisher: American Society of Mechanical Engineers Digital Collection, pp. 4–10. ISSN: 0098-2202. DOI https://doi.org/10.1115/1.1436090. URL: https://asmedigitalcollection.asme.org/fluidsengineering/article/124/1/4/462791/Code-Verification-by-the-Method-of-Manufactured (visited on 07/23/2020).

  15. Michael Stein. “Large Sample Properties of Simulations Using Latin Hypercube Sampling”. In: Technometrics 29.2 (May 1987), pp. 143–151. ISSN: 0040-1706, 1537-2723. DOI https://doi.org/10.1080/00401706.1987.10488205 (visited on 07/13/2020).

  16. Diederik P. Kingma and Jimmy Ba. “Adam: A Method for Stochastic Optimization”. In: arXiv:1412.6980 [cs] (Jan. 29, 2017) (visited on 07/30/2020).

  17. Mohammad Amin Nabian and Hadi Meidani. “A Deep Neural Network Surrogate for High-Dimensional Random Partial Differential Equations”. In: Probabilistic Engineering Mechanics 57 (July 2019), pp. 14–25. ISSN: 02668920. DOI https://doi.org/10.1016/j.probengmech.2019.05.001. arXiv:1806.02957 (visited on 02/21/2020).

  18. Arieh Iserles. A First Course in the Numerical Analysis of Differential Equations. Google-Books-ID: 3acgAwAAQBAJ. Cambridge University Press, Nov. 27, 2008. 481 pp. ISBN: 978-1-139-47376-7.

    Google Scholar 

  19. Alexandre M. Tartakovsky et al. “Learning Parameters and Constitutive Relationships with Physics Informed Deep Neural Networks”. In: (Aug. 2018). URL: https://arxiv.org/pdf/1808.03398.pdf.

  20. Maziar Raissi, Paris Perdikaris, and George Em Karniadakis. “Inferring solutions of differential equations using noisy multi-fidelity data”. In: Journal of Computational Physics 335 (Apr. 2017), pp. 736–746. ISSN: 00219991. DOI https://doi.org/10.1016/j.jcp.2017.01.060. arXiv:1607.04805 (visited on 07/16/2020).

  21. Maziar Raissi, Paris Perdikaris, and George Em Karniadakis. “Machine learning of linear differential equations using Gaussian processes”. In: Journal of Computational Physics 348 (Nov. 1, 2017), pp. 683–693. ISSN: 0021-9991. DOI https://doi.org/10.1016/j.jcp.2017.07.050. URL: http://www.sciencedirect.com/science/article/pii/S0021999117305582 (visited on 07/16/2020).

  22. Maziar Raissi and George Em Karniadakis. “Hidden Physics Models: Machine Learning of Nonlinear Partial Differential Equations”. In: Journal of Computational Physics 357 (Mar. 2018), pp. 125–141. ISSN: 00219991. DOI https://doi.org/10.1016/j.jcp.2017.11.039. arXiv:1708.00588 (visited on 07/16/2020).

  23. Maziar Raissi, Paris Perdikaris, and George Em Karniadakis. “Numerical Gaussian Processes for Time-dependent and Non-linear Partial Differential Equations”. In: arXiv:1703.10230 [cs, math, stat] (Mar. 29, 2017) (visited on 07/16/2020).

  24. Guofei Pang, Lu Lu, and George Em Karniadakis. “fPINNs: Fractional Physics-Informed Neural Networks”. In: arXiv:1811.08967 [physics] (Nov. 19, 2018) (visited on 07/16/2020).

  25. Maziar Raissi, Alireza Yazdani, and George Em Karniadakis. “Hidden Fluid Mechanics: A Navier-Stokes Informed Deep Learning Framework for Assimilating Flow Visualization Data”. In: arXiv:1808.04327 [physics, stat] (Aug. 13, 2018) (visited on 04/09/2020).

  26. E. Kharazmi, Z. Zhang, and G. E. Karniadakis. “Variational Physics- Informed Neural Networks For Solving Partial Differential Equations”. In: arXiv:1912.00873 [physics, stat] (Nov. 27, 2019) (visited on 07/16/2020).

  27. Christian Beck, Weinan E, and Arnulf Jentzen. “Machine learning approximation algorithms for high-dimensional fully nonlinear partial differential equations and second-order backward stochastic differential equations”. In: J Nonlinear Sci 29.4 (Aug. 2019), pp. 1563–1619. ISSN: 0938-8974, 1432-1467. DOI https://doi.org/10.1007/s00332-018-9525-3. arXiv:1709.05963 (visited on 07/16/2020).

  28. Weinan E and Bing Yu. “The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems”. In: arXiv:1710.00211 [cs, stat] (Sept. 30, 2017) (visited on 01/14/2020).

  29. Kaiming He et al. “Deep Residual Learning for Image Recognition”. In: arXiv:1512.03385 [cs] (Dec. 10, 2015) (visited on 07/16/2020).

  30. Justin Sirignano and Konstantinos Spiliopoulos. “DGM: A deep learning algorithm for solving partial differential equations”. In: Journal of Computational Physics 375 (Dec. 2018), pp. 1339–1364. ISSN: 00219991. DOI https://doi.org/10.1016/j.jcp.2018.08.029. arXiv:1708.07469 (visited on 01/08/2020).

  31. Samuel H. Rudy et al. “Data-driven discovery of partial differential equations”. In: Sci. Adv. 3.4 (Apr. 2017), e1602614. ISSN: 2375-2548. DOI https://doi.org/10.1126/sciadv.1602614 (visited on 01/08/2020).

  32. Yinhao Zhu et al. “Physics-Constrained Deep Learning for High-dimensional Surrogate Modeling and Uncertainty Quantification without Labeled Data”. In: Journal of Computational Physics 394 (Oct. 2019), pp. 56–81. ISSN: 00219991. DOI https://doi.org/10.1016/j.jcp.2019.05.024. arXiv:1901.06314 (visited on 07/06/2020).

  33. Nicholas Geneva and Nicholas Zabaras. “Modeling the Dynamics of PDE Systems with Physics-Constrained Deep Auto-Regressive Networks”. In: Journal of Computational Physics 403 (Feb. 2020), p. 109056. ISSN: 00219991. DOI https://doi.org/10.1016/j.jcp.2019.109056. arXiv:1906.05747 (visited on 11/09/2020).

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Authors and Affiliations

  1. School of Engineering and Design, Chair of Computational Modeling and Simulation, TU Munich, München, Germany

    Stefan Kollmannsberger & Davide D’Angella

  2. Department of Health Sciences and Technology, Institute for Biomechanics, ETH Zürich, Zürich, Switzerland

    Moritz Jokeit

  3. Bavarian Graduate School of Computational Engineering, TU Munich, München, Germany

    Leon Herrmann

Authors
  1. Stefan Kollmannsberger
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  2. Davide D’Angella
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  3. Moritz Jokeit
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  4. Leon Herrmann
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Corresponding author

Correspondence to Stefan Kollmannsberger .

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Kollmannsberger, S., D’Angella, D., Jokeit, M., Herrmann, L. (2021). Physics-Informed Neural Networks. In: Deep Learning in Computational Mechanics. Studies in Computational Intelligence, vol 977. Springer, Cham. https://doi.org/10.1007/978-3-030-76587-3_5

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