# Physics Informed Deep Learning for Transport in Porous Media. Buckley Leverett Problem

arXiv: https://arxiv.org/abs/2001.05172 · HTML: https://mlsanity.com/paper/2001.05172/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 15 Jan 2020.

## Built on by (most cited first, 10 of 10)

- [Physics-informed neural network simulation of multiphase poroelasticity using stress-split sequential training](https://mlsanity.com/paper/2110.03049/) (2021)
- [Physics-Informed Neural Networks with Adaptive Localized Artificial Viscosity](https://mlsanity.com/paper/2203.08802/) (2022)
- [Learning Large-scale Subsurface Simulations with a Hybrid Graph Network Simulator](https://mlsanity.com/paper/2206.07680/) (2022)
- [On the Limitations of Physics-informed Deep Learning: Illustrations Using First Order Hyperbolic Conservation Law-based Traffic Flow Models](https://mlsanity.com/paper/2302.12337/) (2023)
- [Physics-informed attention-based neural network for solving non-linear partial differential equations](https://mlsanity.com/paper/2105.07898/) (2021)
- [Inverse modeling of nonisothermal multiphase poromechanics using physics-informed neural networks](https://mlsanity.com/paper/2209.03276/) (2022)
- [Characteristics-Informed Neural Networks for Forward and Inverse Hyperbolic Problems](https://mlsanity.com/paper/2212.14012/) (2022)
- [Uncertainty Quantification for Transport in Porous media using Parameterized Physics Informed neural Networks](https://mlsanity.com/paper/2205.12730/) (2022)
- [Extended Physics Informed Neural Network for Hyperbolic Two-Phase Flow in Porous Media](https://mlsanity.com/paper/2511.13734/) (2025)
- [Feature-Modulated UFNO for Improved Prediction of Multiphase Flow in Porous Media](https://mlsanity.com/paper/2511.20543/) (2025)

## Abstract (arXiv)

We present a new hybrid physics-based machine-learning approach to reservoir modeling. The methodology relies on a series of deep adversarial neural network architecture with physics-based regularization. The network is used to simulate the dynamic behavior of physical quantities (i.e. saturation) subject to a set of governing laws (e.g. mass conservation) and corresponding boundary and initial conditions. A residual equation is formed from the governing partial-differential equation and used as part of the training. Derivatives of the estimated physical quantities are computed using automatic differentiation algorithms. This allows the model to avoid overfitting, by reducing the variance and permits extrapolation beyond the range of the training data including uncertainty implicitely derived from the distribution output of the generative adversarial networks. The approach is used to simulate a 2 phase immiscible transport problem (Buckley Leverett). From a very limited dataset, the model learns the parameters of the governing equation and is able to provide an accurate physical solution, both in terms of shock and rarefaction. We demonstrate how this method can be applied in the context of a forward simulation for continuous problems. The use of these models for the inverse problem is also presented, where the model simultaneously learns the physical laws and determines key uncertainty subsurface parameters. The proposed methodology is a simple and elegant way to instill physical knowledge to machine-learning algorithms. This alleviates the two most significant shortcomings of machine-learning algorithms: the requirement for large datasets and the reliability of extrapolation. The principles presented in this paper can be generalized in innumerable ways in the future and should lead to a new class of algorithms to solve both forward and inverse physical problems.