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

Solute transport in porous media is relevant to a wide range of applications in hydrogeology, geothermal energy, underground CO2 storage, and a variety of chemical engineering systems. Due to the complexity of solute transport in heterogeneous porous media, traditional solvers require high resolution meshing and are therefore expensive computationally. This study explores the application of a mesh-free method based on deep learning to accelerate the simulation of solute transport. We employ Physics-informed Neural Networks (PiNN) to solve solute transport problems in homogeneous and heterogeneous porous media governed by the advection-dispersion equation. Unlike traditional neural networks that learn from large training datasets, PiNNs only leverage the strong form mathematical models to simultaneously solve for multiple dependent or independent field variables (e.g., pressure and solute concentration fields). In this study, we construct PiNN using a periodic activation function to better represent the complex physical signals (i.e., pressure) and their derivatives (i.e., velocity). Several case studies are designed with the intention of investigating the proposed PiNN's capability to handle different degrees of complexity. A manual hyperparameter tuning method is used to find the best PiNN architecture for each test case. Point-wise error and mean square error (MSE) measures are employed to assess the performance of PiNNs' predictions against the ground truth solutions obtained analytically or numerically using the finite element method. Our findings show that the predictions of PiNN are in good agreement with the ground truth solutions while reducing computational complexity and cost by, at least, three orders of magnitude.