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Article

Physics-Informed Neural Networks with Periodic Activation Functions for Solute Transport in Heterogeneous Porous Media

by
Salah A. Faroughi
1,*,
Ramin Soltanmohammadi
1,
Pingki Datta
1,
Seyed Kourosh Mahjour
1 and
Shirko Faroughi
2
1
Geo-Intelligence Laboratory, Ingram School of Engineering, Texas State University, San Marcos, TX 78666, USA
2
Department of Mechanical Engineering, School of Engineering, Urmia University of Technology, Urmia 57561-51818, Iran
*
Author to whom correspondence should be addressed.
Mathematics 2024, 12(1), 63; https://doi.org/10.3390/math12010063
Submission received: 9 November 2023 / Revised: 28 November 2023 / Accepted: 30 November 2023 / Published: 24 December 2023
(This article belongs to the Special Issue Advances in Computational Fluid Dynamics)

Abstract

Simulating solute transport in heterogeneous porous media poses computational challenges due to the high-resolution meshing required for traditional solvers. To overcome these challenges, this study explores a mesh-free method based on deep learning to accelerate solute transport simulation. We employ Physics-informed Neural Networks (PiNN) with a periodic activation function to solve solute transport problems in both homogeneous and heterogeneous porous media governed by the advection-dispersion equation. Unlike traditional neural networks that rely on large training datasets, PiNNs use strong-form mathematical models to constrain the network in the training phase and simultaneously solve for multiple dependent or independent field variables, such as pressure and solute concentration fields. To demonstrate the effectiveness of using PiNNs with a periodic activation function to resolve solute transport in porous media, we construct PiNNs using two activation functions, sin and tanh, for seven case studies, including 1D and 2D scenarios. The accuracy of the PiNNs’ predictions is then evaluated using absolute point error and mean square error metrics and compared to the ground truth solutions obtained analytically or numerically. Our results demonstrate that the PiNN with sin activation function, compared to tanh activation function, is up to two orders of magnitude more accurate and up to two times faster to train, especially in heterogeneous porous media. Moreover, PiNN’s simultaneous predictions of pressure and concentration fields can reduce computational expenses in terms of inference time by three orders of magnitude compared to FEM simulations for two-dimensional cases.
Keywords: physics-informed neural networks; solute transport; heterogeneous porous media; advection-dispersion equation; deep learning; scientific computing physics-informed neural networks; solute transport; heterogeneous porous media; advection-dispersion equation; deep learning; scientific computing

Share and Cite

MDPI and ACS Style

Faroughi, S.A.; Soltanmohammadi, R.; Datta, P.; Mahjour, S.K.; Faroughi, S. Physics-Informed Neural Networks with Periodic Activation Functions for Solute Transport in Heterogeneous Porous Media. Mathematics 2024, 12, 63. https://doi.org/10.3390/math12010063

AMA Style

Faroughi SA, Soltanmohammadi R, Datta P, Mahjour SK, Faroughi S. Physics-Informed Neural Networks with Periodic Activation Functions for Solute Transport in Heterogeneous Porous Media. Mathematics. 2024; 12(1):63. https://doi.org/10.3390/math12010063

Chicago/Turabian Style

Faroughi, Salah A., Ramin Soltanmohammadi, Pingki Datta, Seyed Kourosh Mahjour, and Shirko Faroughi. 2024. "Physics-Informed Neural Networks with Periodic Activation Functions for Solute Transport in Heterogeneous Porous Media" Mathematics 12, no. 1: 63. https://doi.org/10.3390/math12010063

APA Style

Faroughi, S. A., Soltanmohammadi, R., Datta, P., Mahjour, S. K., & Faroughi, S. (2024). Physics-Informed Neural Networks with Periodic Activation Functions for Solute Transport in Heterogeneous Porous Media. Mathematics, 12(1), 63. https://doi.org/10.3390/math12010063

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