# Uncertainty Quantification for Transport in Porous media using Parameterized Physics Informed neural Networks

arXiv: https://arxiv.org/abs/2205.12730 · HTML: https://mlsanity.com/paper/2205.12730/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 38 references that are in the map.
- First posted on arXiv 19 May 2022.

## Builds on (most cited first, 9 of 9)

- [Generative Adversarial Networks](https://mlsanity.com/paper/2203.00667/) (2022)
- [Auto-Encoding Variational Bayes](https://mlsanity.com/paper/1312.6114/) (2013)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Implicit Neural Representations with Periodic Activation Functions](https://mlsanity.com/paper/2006.09661/) (2020)
- [Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domains](https://mlsanity.com/paper/2006.10739/) (2020)
- [On the Spectral Bias of Neural Networks](https://mlsanity.com/paper/1806.08734/) (2018)
- [On Wasserstein Two Sample Testing and Related Families of Nonparametric Tests](https://mlsanity.com/paper/1509.02237/) (2015)
- [A Deep Neural Network Surrogate for High-Dimensional Random Partial Differential Equations](https://mlsanity.com/paper/1806.02957/) (2018)
- [Physics Informed Deep Learning for Transport in Porous Media. Buckley Leverett Problem](https://mlsanity.com/paper/2001.05172/) (2020)

## Built on by (most cited first, 2 of 2)

- [A Critical Review of Physics-Informed Machine Learning Applications in Subsurface Energy Systems](https://mlsanity.com/paper/2308.04457/) (2023)
- [Extended Physics Informed Neural Network for Hyperbolic Two-Phase Flow in Porous Media](https://mlsanity.com/paper/2511.13734/) (2025)

## Abstract (arXiv)

We present a Parametrization of the Physics Informed Neural Network (P-PINN) approach to tackle the problem of uncertainty quantification in reservoir engineering problems. We demonstrate the approach with the immiscible two phase flow displacement (Buckley-Leverett problem) in heterogeneous porous medium. The reservoir properties (porosity, permeability) are treated as random variables. The distribution of these properties can affect dynamic properties such as the fluids saturation, front propagation speed or breakthrough time. We explore and use to our advantage the ability of networks to interpolate complex high dimensional functions. We observe that the additional dimensions resulting from a stochastic treatment of the partial differential equations tend to produce smoother solutions on quantities of interest (distributions parameters) which is shown to improve the performance of PINNS. We show that provided a proper parameterization of the uncertainty space, PINN can produce solutions that match closely both the ensemble realizations and the stochastic moments. We demonstrate applications for both homogeneous and heterogeneous fields of properties. We are able to solve problems that can be challenging for classical methods. This approach gives rise to trained models that are both more robust to variations in the input space and can compete in performance with traditional stochastic sampling methods.