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A framework for data-driven solution and parameter estimation of PDEs using conditional generative adversarial networks

Nature Computational Science volume 1, pages 819–829 (2021)Cite this article

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A preprint version of the article is available at arXiv.

Abstract

Here we employ and adapt the image-to-image translation concept based on conditional generative adversarial networks (cGAN) for learning a forward and an inverse solution operator of partial differential equations (PDEs). We focus on steady-state solutions of coupled hydromechanical processes in heterogeneous porous media and present the parameterization of the spatially heterogeneous coefficients, which is exceedingly difficult using standard reduced-order modeling techniques. We show that our framework provides a speed-up of at least 2,000 times compared to a finite-element solver and achieves a relative root-mean-square error (r.m.s.e.) of less than 2% for forward modeling. For inverse modeling, the framework estimates the heterogeneous coefficients, given an input of pressure and/or displacement fields, with a relative r.m.s.e. of less than 7%, even for cases where the input data are incomplete and contaminated by noise. The framework also provides a speed-up of 120,000 times compared to a Gaussian prior-based inverse modeling approach while also delivering more accurate results.

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Fig. 1: Illustration of strongly heterogeneous input and output fields.
Fig. 2: The results of three test cases using the W model.
Fig. 3: Relative r.m.s.e. of the test set using the W model.
Fig. 4: Results of equation (4) for the W model using different numbers of training samples.
Fig. 5: The results of three test cases of the W model and pyPCGA.
Fig. 6: The results of three test cases of the W model and pyPCGA.

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Data availability

We provide the training, validation and testing data or the scripts used to generate them at 10.24433/CO.6650973.v161. We also discuss these data characteristics in Supplementary Section 4.1. Source data are provided with this paper.

Code availability

The scripts used to produce these results as well as training data are available at 10.24433/CO.6650973.v161. The finite-element source codes used to generate the training, validation and testing data are available at https://github.com/teeratornk/jcp_YJCPH_110030_git, and a tutorial for the multiphenics package is available at https://github.com/multiphenics/multiphenics/tree/master/tutorials/09_multiphysics_examples. Scripts of the pyPCGA used in the paper are available at https://github.com/jonghyunharrylee/pyPCGA/tree/master/examples/fenics_hydromechanics.

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Acknowledgements

D.O. acknowledges support from Los Alamos National Laboratory’s Laboratory Directed Research and Development Early Career Award (20200575ECR). H.V. is grateful for funding support from the US Department of Energy (DOE) Basic Energy Sciences (LANLE3W1). N.B. acknowledges start-up funds from Cornell University. Y.C. acknowledges LDRD funds (21-FS-042) from Lawrence Livermore National Laboratory. Lawrence Livermore National Laboratory is operated by Lawrence Livermore National Security, LLC, for the US DOE, National Nuclear Security Administration, under contract no. DE-AC52-07NA27344 (LLNL-JRNL-823007).

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Authors and Affiliations

  1. Sibley School of Mechanical and Aerospace, EngineeringCornell University, Ithaca, NY, USA

    Teeratorn Kadeethum, Jan Niklas Fuhg & Nikolaos Bouklas

  2. Computational Earth Science, Los Alamos National Laboratory, Los Alamos, NM, USA

    Daniel O’Malley & Hari S. Viswanathan

  3. Center for Applied Scientific Computing, Lawrence Livermore National Laboratory, Livermore, CA, USA

    Youngsoo Choi

  4. Civil & Environmental Engineering/Water Resources Research Center, University of Hawai’i at Manoa, Honolulu, HI, USA

    Jonghyun Lee

  5. Center for Applied Mathematics, Cornell University, Ithaca, NY, USA

    Nikolaos Bouklas

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  1. Teeratorn Kadeethum
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Contributions

T.K. contributed conceptualization, formal analysis, software and validation, wrote the original draft, and reviewed and edited the manuscript. D.O. contributed conceptualization, formal analysis, supervision and validation, and reviewed and edited the manuscript. J.N.F. contributed software and validation, and reviewed and edited the manuscript. Y.C. contributed conceptualization, formal analysis, supervision and validation, and reviewed and edited the manuscript. J.L. contributed software, formal analysis and supervision, and reviewed and edited the manuscript. H.S.V. contributed conceptualization and supervision, and reviewed and edited the manuscript. N.B. contributed conceptualization, formal analysis, funding acquisition and supervision, and reviewed and edited the manuscript.

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Correspondence to Nikolaos Bouklas.

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Editor recognition statement Handling editor: Fernando Chirigati, in collaboration with the Nature Computational Science team. Nature Computational Science thanks the anonymous reviewers for their contribution to the peer review of this work.

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Supplementary sections 1–5, Figs. 1–28 and Tables 1–4.

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Kadeethum, T., O’Malley, D., Fuhg, J.N. et al. A framework for data-driven solution and parameter estimation of PDEs using conditional generative adversarial networks. Nat Comput Sci 1, 819–829 (2021). https://doi.org/10.1038/s43588-021-00171-3

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