# PINNs for the Solution of the Hyperbolic Buckley-Leverett Problem with a Non-convex Flux Function

arXiv: https://arxiv.org/abs/2112.14826 · HTML: https://mlsanity.com/paper/2112.14826/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 37 references that are in the map.
- First posted on arXiv 29 Dec 2021.

## Builds on (most cited first, 10 of 10)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [DeepXDE: A deep learning library for solving differential equations](https://mlsanity.com/paper/1907.04502/) (2019)
- [Deep Hidden Physics Models: Deep Learning of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1801.06637/) (2018)
- [Ab-Initio Solution of the Many-Electron Schr"odinger Equation with Deep Neural Networks](https://mlsanity.com/paper/1909.02487/) (2019)
- [Physics-informed neural network for ultrasound nondestructive quantification of surface breaking cracks](https://mlsanity.com/paper/2005.03596/) (2020)
- [NVIDIA SimNet^{TM}: an AI-accelerated multi-physics simulation framework](https://mlsanity.com/paper/2012.07938/) (2020)
- [A proof that artificial neural networks overcome the curse of dimensionality in the numerical approximation of Black-Scholes partial differential equations](https://mlsanity.com/paper/1809.02362/) (2018)
- [Deep Learning of Turbulent Scalar Mixing](https://mlsanity.com/paper/1811.07095/) (2018)
- [A Physics-Constrained Deep Learning Model for Simulating Multiphase Flow in 3D Heterogeneous Porous Media](https://mlsanity.com/paper/2105.09467/) (2021)

## Built on by (most cited first, 3 of 3)

- [Conservative Physics-Informed Neural Networks for Non-Conservative Hyperbolic Conservation Laws Near Critical States](https://mlsanity.com/paper/2305.12817/) (2023)
- [Coupled Integral PINN for Discontinuity](https://mlsanity.com/paper/2411.11276/) (2024)
- [Extended Physics Informed Neural Network for Hyperbolic Two-Phase Flow in Porous Media](https://mlsanity.com/paper/2511.13734/) (2025)

## Abstract (arXiv)

The displacement of two immiscible fluids is a common problem in fluid flow in porous media. Such a problem can be posed as a partial differential equation (PDE) in what is commonly referred to as a Buckley-Leverett (B-L) problem. The B-L problem is a non-linear hyperbolic conservation law that is known to be notoriously difficult to solve using traditional numerical methods. Here, we address the forward hyperbolic B-L problem with a nonconvex flux function using physics-informed neural networks (PINNs). The contributions of this paper are twofold. First, we present a PINN approach to solve the hyperbolic B-L problem by embedding the Oleinik entropy condition into the neural network residual. We do not use a diffusion term (artificial viscosity) in the residual-loss, but we rely on the strong form of the PDE. Second, we use the Adam optimizer with residual-based adaptive refinement (RAR) algorithm to achieve an ultra-low loss without weighting. Our solution method can accurately capture the shock-front and produce an accurate overall solution. We report a L2 validation error of 2 x 10-2 and a L2 loss of 1x 10-6. The proposed method does not require any additional regularization or weighting of losses to obtain such accurate solution.