Hostname: page-component-7dd5485656-2pp2p Total loading time: 0 Render date: 2025-11-02T20:50:46.985Z Has data issue: false hasContentIssue false
  • English
  • Français

Solving parametric PDE problems with artificial neural networks

Part of: Connections between Deep learning and Partial Differential Equations

Published online by Cambridge University Press:  01 July 2020

YUEHAW KHOO Open the ORCID record for YUEHAW KHOO [Opens in a new window] ,
JIANFENG LU  and
LEXING YING
YUEHAW KHOO
Affiliation:
Department of Statistics, University of Chicago, IL 60615, USA, email: ykhoo@uchicago.edu
JIANFENG LU
Affiliation:
Department of Mathematics, Department of Chemistry and Department of Physics, Duke University, Durham, NC 27708, USA, email: jianfeng@math.duke.edu
LEXING YING
Affiliation:
Department of Mathematics and ICME, Stanford University, Stanford, CA 94305, USA, email: lexing@stanford.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

The curse of dimensionality is commonly encountered in numerical partial differential equations (PDE), especially when uncertainties have to be modelled into the equations as random coefficients. However, very often the variability of physical quantities derived from PDE can be captured by a few features on the space of the coefficient fields. Based on such observation, we propose using neural network to parameterise the physical quantity of interest as a function of input coefficients. The representability of such quantity using a neural network can be justified by viewing the neural network as performing time evolution to find the solutions to the PDE. We further demonstrate the simplicity and accuracy of the approach through notable examples of PDEs in engineering and physics.

Keywords

Neural-networkparametric PDEuncertainty quantification

Information

Type
Papers
Information
European Journal of Applied Mathematics , Volume 32 , Special Issue 3: Connections between Deep learning and Partial Differential Equations , June 2021 , pp. 421 - 435
Check for updates
Copyright
© The Author(s), 2020. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

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, J., Levenberg, M., Mane, 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., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y. & Zheng, X. (2016) Tensorflow: Large-scale machine learning on heterogeneous distributed systems. arXiv preprint arXiv:1603.04467.Google Scholar
Carleo, G. & Troyer, M. (2017) Solving the quantum many-body problem with artificial neural networks. Science 355(6325), 602–606.CrossRefGoogle ScholarPubMed
Cheng, M., Hou, T. Y., Yan, M. & Zhang, Z. (2013) A data-driven stochastic method for elliptic PDEs with random coefficients. SIAM/ASA J. Uncertainty Quant. 1(1), 452–493.CrossRefGoogle Scholar
Chollet, F. (2017) Keras (2015). http://keras.io.Google Scholar
Dozat, T. (2016) Incorporating Nesterov momentum into ADAM. In: Proceedings of the ICLR Workshop.Google Scholar
Han, J., Jentzen, A. & Weinan, E. (2017) Overcoming the curse of dimensionality: solving high-dimensional partial differential equations using deep learning. arXiv preprint arXiv:1707.02568.Google Scholar
He, K., Zhang, X., Ren, S. & Sun, J. (2016) Deep residual learning for image recognition. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 770–778.CrossRefGoogle Scholar
Hinton, G. E. & Salakhutdinov, R. R. (2006) Reducing the dimensionality of data with neural networks. Science 313(5786), 504–507.CrossRefGoogle ScholarPubMed
Khoo, Y., Lu, J. & Ying, L. (2018) Solving for high dimensional committor functions using artificial neural networks. arXiv preprint arXiv:1802.10275.Google Scholar
Lagaris, I. E., Likas, A. & Fotiadis, D. I. (1998) Artificial neural networks for solving ordinary and partial differential equations. IEEE Trans. Neural Networks 9(5), 987–1000.CrossRefGoogle ScholarPubMed
LeCun, Y., Bengio, Y. & Hinton, G. (2015) Deep learning. Nature 521(7553), 436–444.CrossRefGoogle ScholarPubMed
Long, Z., Lu, Y., Ma, X. & Dong, B. (2017) PDE-net: learning PDEs from data. arXiv preprint arXiv:1710.09668.Google Scholar
Matthies, H. G. & Keese, A. (2005) Galerkin methods for linear and nonlinear elliptic stochastic partial differential equations. Comput. Methods Appl. Mecha. Eng. 194(12), 1295–1331.CrossRefGoogle Scholar
Rudd, K. & Ferrari, S. (2015) A constrained integration (CINT) approach to solving partial differential equations using artificial neural networks. Neurocomputing 155, 277–285.CrossRefGoogle Scholar
Schmidhuber, J. (2015) Deep learning in neural networks: an overview. Neural Networks 61, 85–117.CrossRefGoogle ScholarPubMed
Stefanou, G. (2009) The stochastic finite element method: past, present and future. Comput. Methods Appl. Mecha. Eng. 198(9), 1031–1051.CrossRefGoogle Scholar
Torlai, G. & Melko, R. G. (2016) Learning thermodynamics with Boltzmann machines. Phys. Rev. B 94(16), 165134.CrossRefGoogle Scholar
Wiener, N. (1938) The homogeneous chaos. Am. J. Math. 60(4), 897–936.CrossRefGoogle Scholar
Xiu, D. & Karniadakis, G. E. (2002) The Wiener–Askey polynomial chaos for stochastic differential equations. SIAM J. Sci. Comput. 24(2), 619–644.CrossRefGoogle Scholar
Xu, J. & Zikatanov, L. (2017) Algebraic multigrid methods. Acta Numerica 26, 591–721.CrossRefGoogle Scholar