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

arXiv: https://arxiv.org/abs/2212.08965 · HTML: https://mlsanity.com/paper/2212.08965/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 74 references that are in the map.
- First posted on arXiv 17 Dec 2022.

## Builds on (most cited first, 19 of 19)

- [Implicit Neural Representations with Periodic Activation Functions](https://mlsanity.com/paper/2006.09661/) (2020)
- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [DeepXDE: A deep learning library for solving differential equations](https://mlsanity.com/paper/1907.04502/) (2019)
- [Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's next](https://mlsanity.com/paper/2201.05624/) (2022)
- [Solving high-dimensional partial differential equations using deep learning](https://mlsanity.com/paper/1707.02568/) (2017)
- [Physics-informed neural networks (PINNs) for fluid mechanics: A review](https://mlsanity.com/paper/2105.09506/) (2021)
- [Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1711.10561/) (2017)
- [Gradient-enhanced physics-informed neural networks for forward and inverse PDE problems](https://mlsanity.com/paper/2111.02801/) (2021)
- [A unified deep artificial neural network approach to partial differential equations in complex geometries](https://mlsanity.com/paper/1711.06464/) (2017)
- [CAN-PINN: A Fast Physics-Informed Neural Network Based on Coupled-Automatic-Numerical Differentiation Method](https://mlsanity.com/paper/2110.15832/) (2021)
- [Physics-Informed Neural Networks for Multiphysics Data Assimilation with Application to Subsurface Transport](https://mlsanity.com/paper/1912.02968/) (2019)
- [Hidden Fluid Mechanics: A Navier-Stokes Informed Deep Learning Framework for Assimilating Flow Visualization Data](https://mlsanity.com/paper/1808.04327/) (2018)
- [How important are activation functions in regression and classification? A survey, performance comparison, and future directions](https://mlsanity.com/paper/2209.02681/) (2022)
- [Physics-informed neural network simulation of multiphase poroelasticity using stress-split sequential training](https://mlsanity.com/paper/2110.03049/) (2021)
- [Physics-Guided, Physics-Informed, and Physics-Encoded Neural Networks in Scientific Computing](https://mlsanity.com/paper/2211.07377/) (2022)
- [Automatic differentiation in ML: Where we are and where we should be going](https://mlsanity.com/paper/1810.11530/) (2018)
- [Hyper-parameter tuning of physics-informed neural networks: Application to Helmholtz problems](https://mlsanity.com/paper/2205.06704/) (2022)
- [Physics-Informed Neural Network Method for Solving One-Dimensional Advection Equation Using PyTorch](https://mlsanity.com/paper/2103.09662/) (2021)

## Built on by (most cited first, 8 of 8)

- [EPi-cKANs: Elasto-Plasticity Informed Kolmogorov-Arnold Networks Using Chebyshev Polynomials](https://mlsanity.com/paper/2410.10897/) (2024)
- [Quantum advantage for learning shallow neural networks with natural data distributions](https://mlsanity.com/paper/2503.20879/) (2025)
- [NeuroPMD: Neural Fields for Density Estimation on Product Manifolds](https://mlsanity.com/paper/2501.02994/) (2025)
- [Exploring Efficient Quantification of Modeling Uncertainties with Differentiable Physics-Informed Machine Learning Architectures](https://mlsanity.com/paper/2506.18247/) (2025)
- [Physics-Informed Chebyshev Polynomial Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2602.01737/) (2026)
- [Architectural Strategies for the optimization of Physics-Informed Neural Networks](https://mlsanity.com/paper/2402.02711/) (2024)
- [Scientific machine learning in Hydrology: a unified perspective](https://mlsanity.com/paper/2506.06308/) (2025)
- [Improving physics-informed neural network extrapolation via transfer learning and adaptive activation functions](https://mlsanity.com/paper/2507.12659/) (2025)

## Abstract (arXiv)

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.