# Extended Physics Informed Neural Network for Hyperbolic Two-Phase Flow in Porous Media

arXiv: https://arxiv.org/abs/2511.13734 · HTML: https://mlsanity.com/paper/2511.13734/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 22 of its 64 references that are in the map.
- First posted on arXiv 5 Nov 2025.

## Builds on (most cited first, 22 of 22)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Sequence to Sequence Learning with Neural Networks](https://mlsanity.com/paper/1409.3215/) (2014)
- [Solving high-dimensional partial differential equations using deep learning](https://mlsanity.com/paper/1707.02568/) (2017)
- [Physics-Constrained Deep Learning for High-dimensional Surrogate Modeling and Uncertainty Quantification without Labeled Data](https://mlsanity.com/paper/1901.06314/) (2019)
- [hp-VPINNs: Variational Physics-Informed Neural Networks With Domain Decomposition](https://mlsanity.com/paper/2003.05385/) (2020)
- [Variational Physics-Informed Neural Networks For Solving Partial Differential Equations](https://mlsanity.com/paper/1912.00873/) (2019)
- [Accelerating Eulerian Fluid Simulation With Convolutional Networks](https://mlsanity.com/paper/1607.03597/) (2016)
- [Adversarial Uncertainty Quantification in Physics-Informed Neural Networks](https://mlsanity.com/paper/1811.04026/) (2018)
- [Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating PDEs](https://mlsanity.com/paper/2006.16144/) (2020)
- [Physics informed deep learning for computational elastodynamics without labeled data](https://mlsanity.com/paper/2006.08472/) (2020)
- [Deep Learning of Vortex Induced Vibrations](https://mlsanity.com/paper/1808.08952/) (2018)
- [From PINNs to PIKANs: Recent Advances in Physics-Informed Machine Learning](https://mlsanity.com/paper/2410.13228/) (2024)
- [Learning in Modal Space: Solving Time-Dependent Stochastic PDEs Using Physics-Informed Neural Networks](https://mlsanity.com/paper/1905.01205/) (2019)
- [D3M: A deep domain decomposition method for partial differential equations](https://mlsanity.com/paper/1909.12236/) (2019)
- [On the convergence of physics informed neural networks for linear second-order elliptic and parabolic type PDEs](https://mlsanity.com/paper/2004.01806/) (2020)
- [Physics-Informed Neural Networks with Adaptive Localized Artificial Viscosity](https://mlsanity.com/paper/2203.08802/) (2022)
- [Physics-informed semantic inpainting: Application to geostatistical modeling](https://mlsanity.com/paper/1909.09459/) (2019)
- [Physics Informed Deep Learning for Transport in Porous Media. Buckley Leverett Problem](https://mlsanity.com/paper/2001.05172/) (2020)
- [PINNs for the Solution of the Hyperbolic Buckley-Leverett Problem with a Non-convex Flux Function](https://mlsanity.com/paper/2112.14826/) (2021)
- [Uncertainty Quantification for Transport in Porous media using Parameterized Physics Informed neural Networks](https://mlsanity.com/paper/2205.12730/) (2022)
- [Learning Generic Solutions for Multiphase Transport in Porous Media via the Flux Functions Operator](https://mlsanity.com/paper/2307.01354/) (2023)

## Abstract (arXiv)

The accurate solution of nonlinear hyperbolic partial differential equations (PDEs) remains challenging due to steep gradients, discontinuities, and multiscale structures that make conventional solvers computationally demanding. Physics-Informed Neural Networks (PINNs) embed the governing equations into the learning process, enabling mesh-free solution of PDEs, yet they often struggle to capture steep gradients, discontinuities, and complex nonlinear wave interactions. To address these limitations, we employ the Extended Physics-Informed Neural Network (XPINN) framework to solve the nonlinear Buckley-Leverett equation with a nonconvex flux, modeling immiscible two-phase flow in porous media. The computational domain is dynamically decomposed in space and time into evolving pre-shock and post-shock subdomains, allowing localized subnetworks to efficiently learn distinct flow behaviors, with coupling enforced via the Rankine-Hugoniot jump condition to ensure physically consistent flux continuity. We compare XPINN with standard PINNs and its variants, including PINN with artificial viscosity, PINN with Welge construction, and PINN with the Oleinik entropy condition, and across all cases, XPINN consistently outperforms the other methods, accurately resolving sharp fronts and capturing the correct physical behavior. Importantly, XPINN achieves this using the simpler Adam optimizer, whereas some PINN variants require more complex or higher-order strategies such as L-BFGS to reach comparable accuracy, demonstrating that XPINN is a robust and scalable approach for challenging hyperbolic PDEs without artificial diffusion or entropy corrections. The code is available at github.com/saifkhanengr/XPINN-for-Buckley-Leverett.