# Surrogate Modeling of Fluid Dynamics with a Multigrid Inspired Neural Network Architecture

arXiv: https://arxiv.org/abs/2105.03854 · HTML: https://mlsanity.com/paper/2105.03854/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 60 references that are in the map.
- First posted on arXiv 9 May 2021.

## Builds on (most cited first, 13 of 13)

- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [V-Net: Fully Convolutional Neural Networks for Volumetric Medical Image Segmentation](https://mlsanity.com/paper/1606.04797/) (2016)
- [An Empirical Evaluation of Generic Convolutional and Recurrent Networks for Sequence Modeling](https://mlsanity.com/paper/1803.01271/) (2018)
- [Physics-Constrained Deep Learning for High-dimensional Surrogate Modeling and Uncertainty Quantification without Labeled Data](https://mlsanity.com/paper/1901.06314/) (2019)
- [Theory-guided Data Science: A New Paradigm for Scientific Discovery from Data](https://mlsanity.com/paper/1612.08544/) (2016)
- [Deep Learning Methods for Reynolds-Averaged Navier-Stokes Simulations of Airfoil Flows](https://mlsanity.com/paper/1810.08217/) (2018)
- [Prediction of Aerodynamic Flow Fields Using Convolutional Neural Networks](https://mlsanity.com/paper/1905.13166/) (2019)
- [Physics-Guided Machine Learning for Scientific Discovery: An Application in Simulating Lake Temperature Profiles](https://mlsanity.com/paper/2001.11086/) (2020)
- [Application of Convolutional Neural Network to Predict Airfoil Lift Coefficient](https://mlsanity.com/paper/1712.10082/) (2017)
- [U-net architectures for fast prediction of incompressible laminar flows](https://mlsanity.com/paper/1910.13532/) (2019)
- [A study on using image based machine learning methods to develop the surrogate models of stamp forming simulations](https://mlsanity.com/paper/2010.03370/) (2020)
- [Multilevel Artificial Neural Network Training for Spatially Correlated Learning](https://mlsanity.com/paper/1806.05703/) (2018)
- [Multigrid-in-Channels Architectures for Wide Convolutional Neural Networks](https://mlsanity.com/paper/2006.06799/) (2020)

## Built on by (most cited first, 9 of 9)

- [Solving the Discretised Neutron Diffusion Equations using Neural Networks](https://mlsanity.com/paper/2301.09939/) (2023)
- [Graph Neural Network Based Surrogate Model of Physics Simulations for Geometry Design](https://mlsanity.com/paper/2302.00557/) (2023)
- [Robustness of Physics-Informed Neural Networks to Noise in Sensor Data](https://mlsanity.com/paper/2211.12042/) (2022)
- [FastFlow: AI for Fast Urban Wind Velocity Prediction](https://mlsanity.com/paper/2211.12035/) (2022)
- [Image-based Artificial Intelligence empowered surrogate model and shape morpher for real-time blank shape optimisation in the hot stamping process](https://mlsanity.com/paper/2212.05885/) (2022)
- [ConfEviSurrogate: A Conformalized Evidential Surrogate Model for Uncertainty Quantification](https://mlsanity.com/paper/2504.02919/) (2025)
- [Identification of Physical Processes and Unknown Parameters of 3D Groundwater Contaminant Problems via Theory-guided U-net](https://mlsanity.com/paper/2205.00134/) (2022)
- [Surrogate models for Rock-Fluid Interaction: A Grid-Size-Invariant Approach](https://mlsanity.com/paper/2602.22188/) (2026)
- [Acceleration of an algebraic multigrid pressure solver using graph neural networks](https://mlsanity.com/paper/2606.19251/) (2026)

## Abstract (arXiv)

Algebraic or geometric multigrid methods are commonly used in numerical solvers as they are a multi-resolution method able to handle problems with multiple scales. In this work, we propose a modification to the commonly-used U-Net neural network architecture that is inspired by the principles of multigrid methods, referred to here as U-Net-MG. We then demonstrate that this proposed U-Net-MG architecture can successfully reduce the test prediction errors relative to the conventional U-Net architecture when modeling a set of fluid dynamic problems. In total, we demonstrate an improvement in the prediction of velocity and pressure fields for the canonical fluid dynamics cases of flow past a stationary cylinder, flow past 2 cylinders in out-of-phase motion, and flow past an oscillating airfoil in both the propulsion and energy harvesting modes. In general, while both the U-Net and U-Net-MG models can model the systems well with test RMSEs of less than 1%, the use of the U-Net-MG architecture can further reduce RMSEs by between 20% and 70%.