I am looking for a reference explaining how to solve the Navier-Stokes equations numerically using machine learning algorithms .
Thank you in advance for your help .



Navier-Stokes equations and machine learning
One of the earliest papers is [Application of machine learning algorithms to flow modeling and optimization](https://web.stanford.edu/group/ctr/ResBriefs99/petros.pdf) (1999).



> 
> A model reduction can be accomplished by projecting the Navier-Stokes
>  equations on a properly selected lower dimensional phase subspace. A
>  reasonable choice for a “proper” selection criterion for the base of
>  this manifold is the maximization of the energy content of the
>  projection. This operation is called Proper Orthogonal Decomposition
>  (POD), or Linear Principal Components Analysis (PCA).
> 
> 
> The linear POD is an approximation of the flow vector $v$ by a finite
>  expansion of orthonormal functions $\phi\_n$ such that: $v = V + 
> > \sum\_{i=1}^n a\_n(t)\phi\_n(x)$, where $V$ is the time averaged flow,
>  $\phi\_n$ is the set of the first $n$ eigenvectors of the covariance
>  matrix $C = E[(v\_i −V )(v\_j −V )]$; when this representation for $v$ is
>  substituted in the Navier Stokes equations, the original PDE model is
>  transformed in an ODE model, composed by n equations. The POD can be
>  expressed as a multi-layer feed-forward neural network.
> 
> 
> 


For more recent work, see 


* [Convolutional
Neural Networks for Steady Flow Approximation](https://autodeskresearch.com/publications/convolutional-neural-networks-steady-flow-approximation) (2016)
* [Artificial
neural network method for solving the Navier–Stokes equations](https://link.springer.com/article/10.1007/s00521-014-1762-2)
(2014)


