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Data-Free and Data-Efficient Physics-Informed Neural Network Approaches to Solve the Buckley–Leverett Problem

Author

Listed:
  • Waleed Diab

    (Department of Petroleum Engineering, Khalifa University of Science and Technology, Abu Dhabi 127788, United Arab Emirates)

  • Omar Chaabi

    (Department of Petroleum Engineering, Khalifa University of Science and Technology, Abu Dhabi 127788, United Arab Emirates)

  • Wenjuan Zhang

    (Department of Petroleum Engineering, Khalifa University of Science and Technology, Abu Dhabi 127788, United Arab Emirates)

  • Muhammad Arif

    (Department of Petroleum Engineering, Khalifa University of Science and Technology, Abu Dhabi 127788, United Arab Emirates)

  • Shayma Alkobaisi

    (College of Information Technology, UAE University, Al Ain 15551, United Arab Emirates)

  • Mohammed Al Kobaisi

    (Department of Petroleum Engineering, Khalifa University of Science and Technology, Abu Dhabi 127788, United Arab Emirates)

Abstract

Physics-informed neural networks (PINNs) are an emerging technology in the scientific computing domain. Contrary to data-driven methods, PINNs have been shown to be able to approximate and generalize well a wide range of partial differential equations (PDEs) by imbedding the underlying physical laws describing the PDE. PINNs, however, can struggle with the modeling of hyperbolic conservation laws that develop shocks, and a classic example of this is the Buckley–Leverett problem for fluid flow in porous media. In this work, we explore specialized neural network architectures for modeling the Buckley–Leverett shock front. We present extensions of the standard multilayer perceptron (MLP) that are inspired by the attention mechanism. The attention-based model was, compared to the multilayer perceptron model, and the results show that the attention-based architecture is more robust in solving the hyperbolic Buckley–Leverett problem, more data-efficient, and more accurate. Moreover, by utilizing distance functions, we can obtain truly data-free solutions to the Buckley–Leverett problem. In this approach, the initial and boundary conditions (I/BCs) are imposed in a hard manner as opposed to a soft manner, where labeled data are provided on the I/BCs. This allows us to use a substantially smaller NN to approximate the solution to the PDE.

Suggested Citation

  • Waleed Diab & Omar Chaabi & Wenjuan Zhang & Muhammad Arif & Shayma Alkobaisi & Mohammed Al Kobaisi, 2022. "Data-Free and Data-Efficient Physics-Informed Neural Network Approaches to Solve the Buckley–Leverett Problem," Energies, MDPI, vol. 15(21), pages 1-13, October.
    • Handle: RePEc:gam:jeners:v:15:y:2022:i:21:p:7864-:d:951204
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    References listed on IDEAS

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    1. Wenjuan Zhang & Mohammed Al Kobaisi, 2022. "On the Monotonicity and Positivity of Physics-Informed Neural Networks for Highly Anisotropic Diffusion Equations," Energies, MDPI, vol. 15(18), pages 1-18, September.
    2. Abreu, Eduardo & Vieira, Jardel, 2017. "Computing numerical solutions of the pseudo-parabolic Buckley–Leverett equation with dynamic capillary pressure," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 137(C), pages 29-48.
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