# Performance and accuracy assessments of an incompressible fluid solver coupled with a deep Convolutional Neural Network

arXiv: https://arxiv.org/abs/2109.09363 · HTML: https://mlsanity.com/paper/2109.09363/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 27 of its 104 references that are in the map.
- First posted on arXiv 20 Sep 2021.

## Builds on (most cited first, 27 of 27)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Very Deep Convolutional Networks for Large-Scale Image Recognition](https://mlsanity.com/paper/1409.1556/) (2014)
- [Learning Transferable Architectures for Scalable Image Recognition](https://mlsanity.com/paper/1707.07012/) (2017)
- [Learning Mesh-Based Simulation with Graph Networks](https://mlsanity.com/paper/2010.03409/) (2020)
- [Machine learning accelerated computational fluid dynamics](https://mlsanity.com/paper/2102.01010/) (2021)
- [Machine Learning for Fluid Mechanics](https://mlsanity.com/paper/1905.11075/) (2019)
- [Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1711.10561/) (2017)
- [Deep multi-scale video prediction beyond mean square error](https://mlsanity.com/paper/1511.05440/) (2015)
- [Hidden Physics Models: Machine Learning of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1708.00588/) (2017)
- [Solver-in-the-Loop: Learning from Differentiable Physics to Interact with Iterative PDE-Solvers](https://mlsanity.com/paper/2007.00016/) (2020)
- [Physics Informed Deep Learning (Part II): Data-driven Discovery of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1711.10566/) (2017)
- [Linearly-Recurrent Autoencoder Networks for Learning Dynamics](https://mlsanity.com/paper/1712.01378/) (2017)
- [On Translation Invariance in CNNs: Convolutional Layers can Exploit Absolute Spatial Location](https://mlsanity.com/paper/2003.07064/) (2020)
- [Accelerating Eulerian Fluid Simulation With Convolutional Networks](https://mlsanity.com/paper/1607.03597/) (2016)
- [Deep Fluids: A Generative Network for Parameterized Fluid Simulations](https://mlsanity.com/paper/1806.02071/) (2018)
- [Latent-space Physics: Towards Learning the Temporal Evolution of Fluid Flow](https://mlsanity.com/paper/1802.10123/) (2018)
- [Learning Neural PDE Solvers with Convergence Guarantees](https://mlsanity.com/paper/1906.01200/) (2019)
- [Global field reconstruction from sparse sensors with Voronoi tessellation-assisted deep learning](https://mlsanity.com/paper/2101.00554/) (2021)
- [Estimating Continuous Distributions in Bayesian Classifiers](https://mlsanity.com/paper/1302.4964/) (2013)
- [tempoGAN: A Temporally Coherent, Volumetric GAN for Super-resolution Fluid Flow](https://mlsanity.com/paper/1801.09710/) (2018)
- [Geodesic Convolutional Shape Optimization](https://mlsanity.com/paper/1802.04016/) (2018)
- [Poisson CNN: Convolutional neural networks for the solution of the Poisson equation on a Cartesian mesh](https://mlsanity.com/paper/1910.08613/) (2019)
- [Turbulence Enrichment using Physics-informed Generative Adversarial Networks](https://mlsanity.com/paper/2003.01907/) (2020)
- [Effects of boundary conditions in fully convolutional networks for learning spatio-temporal dynamics](https://mlsanity.com/paper/2106.11160/) (2021)
- [Learning Incompressible Fluid Dynamics from Scratch -- Towards Fast, Differentiable Fluid Models that Generalize](https://mlsanity.com/paper/2006.08762/) (2020)
- [Adaptive Neural Network-Based Approximation to Accelerate Eulerian Fluid Simulation](https://mlsanity.com/paper/2008.11832/) (2020)

## Built on by (most cited first, 8 of 8)

- [Weak baselines and reporting biases lead to overoptimism in machine learning for fluid-related partial differential equations](https://mlsanity.com/paper/2407.07218/) (2024)
- [Using neural networks to solve the 2D Poisson equation for electric field computation in plasma fluid simulations](https://mlsanity.com/paper/2109.13076/) (2021)
- [Invariant preservation in machine learned PDE solvers via error correction](https://mlsanity.com/paper/2303.16110/) (2023)
- [An Implicit GNN Solver for Poisson-like problems](https://mlsanity.com/paper/2302.10891/) (2023)
- [DS-GPS : A Deep Statistical Graph Poisson Solver (for faster CFD simulations)](https://mlsanity.com/paper/2211.11763/) (2022)
- [Multi-Level GNN Preconditioner for Solving Large Scale Problems](https://mlsanity.com/paper/2402.08296/) (2024)
- [Coupling Machine Learning Local Predictions with a Computational Fluid Dynamics Solver to Accelerate Transient Buoyant Plume Simulations](https://mlsanity.com/paper/2409.07175/) (2024)
- [Acceleration of an algebraic multigrid pressure solver using graph neural networks](https://mlsanity.com/paper/2606.19251/) (2026)

## Abstract (arXiv)

The resolution of the Poisson equation is usually one of the most computationally intensive steps for incompressible fluid solvers. Lately, Deep Learning, and especially Convolutional Neural Networks (CNN), has been introduced to solve this equation, leading to significant inference time reduction at the cost of a lack of guarantee on the accuracy of the solution. This drawback might lead to inaccuracies and potentially unstable simulations. It also makes impossible a fair assessment of the CNN speedup, for instance, when changing the network architecture, since evaluated at different error levels. To circumvent this issue, a hybrid strategy is developed, which couples a CNN with a traditional iterative solver to ensure a user-defined accuracy level. The CNN hybrid method is tested on two flow cases, consisting of a variable-density plume with and without obstacles, demostrating remarkable generalization capabilities, ensuring both the accuracy and stability of the simulations. The error distribution of the predictions using several network architectures is further investigated. Results show that the threshold of the hybrid strategy defined as the mean divergence of the velocity field is ensuring a consistent physical behavior of the CNN-based hybrid computational strategy. This strategy allows a systematic evaluation of the CNN performance at the same accuracy level for various network architectures. In particular, the importance of incorporating multiple scales in the network architecture is demonstrated, since improving both the accuracy and the inference performance compared with feedforward CNN architectures, as these networks can provide solutions 1 10-25 faster than traditional iterative solvers.