# Stacked Generative Machine Learning Models for Fast Approximations of Steady-State Navier-Stokes Equations

arXiv: https://arxiv.org/abs/2112.06419 · HTML: https://mlsanity.com/paper/2112.06419/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 17 of its 44 references that are in the map.
- First posted on arXiv 13 Dec 2021.

## Builds on (most cited first, 17 of 17)

- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification](https://mlsanity.com/paper/1502.01852/) (2015)
- [DeepXDE: A deep learning library for solving differential equations](https://mlsanity.com/paper/1907.04502/) (2019)
- [StackGAN: Text to Photo-realistic Image Synthesis with Stacked Generative Adversarial Networks](https://mlsanity.com/paper/1612.03242/) (2016)
- [StackGAN++: Realistic Image Synthesis with Stacked Generative Adversarial Networks](https://mlsanity.com/paper/1710.10916/) (2017)
- [Deep Learning Methods for Reynolds-Averaged Navier-Stokes Simulations of Airfoil Flows](https://mlsanity.com/paper/1810.08217/) (2018)
- [Machine Learning of Linear Differential Equations using Gaussian Processes](https://mlsanity.com/paper/1701.02440/) (2017)
- [Accelerating Eulerian Fluid Simulation With Convolutional Networks](https://mlsanity.com/paper/1607.03597/) (2016)
- [Deep Learning of Vortex Induced Vibrations](https://mlsanity.com/paper/1808.08952/) (2018)
- [Small Data Challenges in Big Data Era: A Survey of Recent Progress on Unsupervised and Semi-Supervised Methods](https://mlsanity.com/paper/1903.11260/) (2019)
- [Physics-informed deep learning for incompressible laminar flows](https://mlsanity.com/paper/2002.10558/) (2020)
- [Deep Learning the Physics of Transport Phenomena](https://mlsanity.com/paper/1709.02432/) (2017)
- [Weakly-Supervised Deep Learning of Heat Transport via Physics Informed Loss](https://mlsanity.com/paper/1807.11374/) (2018)
- [Learning Discrepancy Models From Experimental Data](https://mlsanity.com/paper/1909.08574/) (2019)
- [A Multi-Pass GAN for Fluid Flow Super-Resolution](https://mlsanity.com/paper/1906.01689/) (2019)
- [A Hybrid Frequency-domain/Image-domain Deep Network for Magnetic Resonance Image Reconstruction](https://mlsanity.com/paper/1810.12473/) (2018)
- [A data-driven proxy to Stoke's flow in porous media](https://mlsanity.com/paper/1905.06327/) (2019)

## Abstract (arXiv)

Computational fluid dynamics (CFD) simulations are broadly applied in engineering and physics. A standard description of fluid dynamics requires solving the Navier-Stokes (N-S) equations in different flow regimes. However, applications of CFD simulations are computationally-limited by the availability, speed, and parallelism of high-performance computing. To improve computational efficiency, machine learning techniques have been used to create accelerated data-driven approximations for CFD. A majority of such approaches rely on large labeled CFD datasets that are expensive to obtain at the scale necessary to build robust data-driven models. We develop a weakly-supervised approach to solve the steady-state N-S equations under various boundary conditions, using a multi-channel input with boundary and geometric conditions. We achieve state-of-the-art results without any labeled simulation data, but using a custom data-driven and physics-informed loss function by using and small-scale solutions to prime the model to solve the N-S equations. To improve the resolution and predictability, we train stacked models of increasing complexity generating the numerical solutions for N-S equations. Without expensive computations, our model achieves high predictability with a variety of obstacles and boundary conditions. Given its high flexibility, the model can generate a solution on a 64 x 64 domain within 5 ms on a regular desktop computer which is 1000 times faster than a regular CFD solver. Translation of interactive CFD simulation on local consumer computing hardware enables new applications in real-time predictions on the internet of things devices where data transfer is prohibitive and can increase the scale, speed, and computational cost of boundary-value fluid problems.