# Learning Generic Solutions for Multiphase Transport in Porous Media via the Flux Functions Operator

arXiv: https://arxiv.org/abs/2307.01354 · HTML: https://mlsanity.com/paper/2307.01354/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 64 references that are in the map.
- First posted on arXiv 3 Jul 2023.

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

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [Neural Operator: Graph Kernel Network for Partial Differential Equations](https://mlsanity.com/paper/2003.03485/) (2020)
- [When and why PINNs fail to train: A neural tangent kernel perspective](https://mlsanity.com/paper/2007.14527/) (2020)
- [Learning the solution operator of parametric partial differential equations with physics-informed DeepOnets](https://mlsanity.com/paper/2103.10974/) (2021)
- [Machine Learning for Fluid Mechanics](https://mlsanity.com/paper/1905.11075/) (2019)
- [Multipole Graph Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2006.09535/) (2020)
- [Tackling Climate Change with Machine Learning](https://mlsanity.com/paper/1906.05433/) (2019)
- [Transfer learning enhanced physics informed neural network for phase-field modeling of fracture](https://mlsanity.com/paper/1907.02531/) (2019)
- [Physics-Informed Deep Neural Operator Networks](https://mlsanity.com/paper/2207.05748/) (2022)
- [Learning Operators with Coupled Attention](https://mlsanity.com/paper/2201.01032/) (2022)
- [Improved architectures and training algorithms for deep operator networks](https://mlsanity.com/paper/2110.01654/) (2021)
- [Physics informed WNO](https://mlsanity.com/paper/2302.05925/) (2023)
- [Physics-informed Neural Networks for Solving Nonlinear Diffusivity and Biot's equations](https://mlsanity.com/paper/2002.08235/) (2020)
- [FC-PINO: High Precision Physics-Informed Neural Operators via Fourier Continuation](https://mlsanity.com/paper/2211.15960/) (2022)
- [Physics-informed neural network solution of thermo-hydro-mechanical (THM) processes in porous media](https://mlsanity.com/paper/2203.01514/) (2022)
- [Applications of physics-informed scientific machine learning in subsurface science: A survey](https://mlsanity.com/paper/2104.04764/) (2021)
- [The FluidFlower International Benchmark Study: Process, Modeling Results, and Comparison to Experimental Data](https://mlsanity.com/paper/2302.10986/) (2023)

## 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)

Traditional numerical schemes for simulating fluid flow and transport in porous media can be computationally expensive. Advances in machine learning for scientific computing have the potential to help speed up the simulation time in many scientific and engineering fields. DeepONet has recently emerged as a powerful tool for accelerating the solution of partial differential equations (PDEs) by learning operators (mapping between function spaces) of PDEs. In this work, we learn the mapping between the space of flux functions of the Buckley-Leverett PDE and the space of solutions (saturations). We use Physics-Informed DeepONets (PI-DeepONets) to achieve this mapping without any paired input-output observations, except for a set of given initial or boundary conditions; ergo, eliminating the expensive data generation process. By leveraging the underlying physical laws via soft penalty constraints during model training, in a manner similar to Physics-Informed Neural Networks (PINNs), and a unique deep neural network architecture, the proposed PI-DeepONet model can predict the solution accurately given any type of flux function (concave, convex, or non-convex) while achieving up to four orders of magnitude improvements in speed over traditional numerical solvers. Moreover, the trained PI-DeepONet model demonstrates excellent generalization qualities, rendering it a promising tool for accelerating the solution of transport problems in porous media.