Data-Free and Data-Efficient Physics Informed Neural Network Approaches to Solve the Buckley-Leverett Problem
Note: to run the models, upload the nessesary data files to session storage in Colab
This code accompanies the manuscript titled "Data-Free and Data-Efficient Physics Informed Neural Network Approaches to Solve the Buckley-Leverett Problem ", authored by Waleed Diab, Omar Chaabi, Wenjuan Zhang, Muhammad Arif, Shayma Alkobaisi, Mohammed Al Kobaisi.
Physics Informed Neural Networks (PINNs) is 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). This is accomplished by regularizing standard feed-forward neural networks with the underlying physical laws through automatic differentiation. 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. The difficulty can be overcome by embedding the PDE residual with the Oleinik entropy condition. Recent successes in solving the two-phase hyperbolic Buckley-Leverett problem using PINNs are, however, computationally inefficient. In this work, we explore specialized neural network architectures for approximating PDEs. Such methodology has been successfully implemented for image processing via convolutional neural networks for example. We present extensions of the standard Multilayer Perceptron (MLP) that are inspired by the Attention mechanism. The attention mechanism is the building block of transformers that have achieved tremendous success in the domain of machine translation. 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 is provided on the I/BCs. This allows us to use a substantially smaller NN to approximate the solution to the PDE. To the best of our knowledge, this is the first implementation of such an approach to fluid flow in porous media problems.
