# Solving the Discretised Multiphase Flow Equations with Interface Capturing on Structured Grids Using Machine Learning Libraries

arXiv: https://arxiv.org/abs/2401.06755 · HTML: https://mlsanity.com/paper/2401.06755/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 89 references that are in the map.
- First posted on arXiv 12 Jan 2024.

## Builds on (most cited first, 12 of 12)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [PDE-Net: Learning PDEs from Data](https://mlsanity.com/paper/1710.09668/) (2017)
- [Automatic differentiation in ML: Where we are and where we should be going](https://mlsanity.com/paper/1810.11530/) (2018)
- [Inferring incompressible two-phase flow fields from the interface motion using physics-informed neural networks](https://mlsanity.com/paper/2101.09833/) (2021)
- [Solving the Discretised Neutron Diffusion Equations using Neural Networks](https://mlsanity.com/paper/2301.09939/) (2023)
- [Solving the Discretised Boltzmann Transport Equations using Neural Networks: Applications in Neutron Transport](https://mlsanity.com/paper/2301.09991/) (2023)
- [Benchmarking OpenCL, OpenACC, OpenMP, and CUDA: programming productivity, performance, and energy consumption](https://mlsanity.com/paper/1704.05316/) (2017)
- [A Solver for Massively Parallel Direct Numerical Simulation of Three-Dimensional Multiphase Flows](https://mlsanity.com/paper/1410.8568/) (2014)
- [An Interface-Tracking Technique for Multiphase Flow with Soluble Surfactant](https://mlsanity.com/paper/1702.02478/) (2017)
- [Development of an Equation-based Parallelization Method for Multiphase Particle-in-Cell Simulations](https://mlsanity.com/paper/2211.15605/) (2022)

## Built on by (most cited first, 5 of 5)

- [Calibrated Physics-Informed Uncertainty Quantification](https://mlsanity.com/paper/2502.04406/) (2025)
- [Neural Physics: Using AI Libraries to Develop Physics-Based Solvers for Incompressible Computational Fluid Dynamics](https://mlsanity.com/paper/2402.17913/) (2024)
- [Learning Physical Operators using Neural Operators](https://mlsanity.com/paper/2602.23113/) (2026)
- [Parameter estimation for land-surface models using Neural Physics](https://mlsanity.com/paper/2505.02979/) (2025)
- [Quantum Neural Physics: Solving Partial Differential Equations on Quantum Simulators using Quantum Convolutional Neural Networks](https://mlsanity.com/paper/2603.24196/) (2026)

## Abstract (arXiv)

This paper solves the discretised multiphase flow equations using tools and methods from machine-learning libraries. The idea comes from the observation that convolutional layers can be used to express a discretisation as a neural network whose weights are determined by the numerical method, rather than by training, and hence, we refer to this approach as Neural Networks for PDEs (NN4PDEs). To solve the discretised multiphase flow equations, a multigrid solver is implemented through a convolutional neural network with a U-Net architecture. Immiscible two-phase flow is modelled by the 3D incompressible Navier-Stokes equations with surface tension and advection of a volume fraction field, which describes the interface between the fluids. A new compressive algebraic volume-of-fluids method is introduced, based on a residual formulation using Petrov-Galerkin for accuracy and designed with NN4PDEs in mind. High-order finite-element based schemes are chosen to model a collapsing water column and a rising bubble. Results compare well with experimental data and other numerical results from the literature, demonstrating that, for the first time, finite element discretisations of multiphase flows can be solved using an approach based on (untrained) convolutional neural networks. A benefit of expressing numerical discretisations as neural networks is that the code can run, without modification, on CPUs, GPUs or the latest accelerators designed especially to run AI codes.