One of the earliest papers is Application of machine learning algorithms to flow modeling and optimization (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 (2016).

One of the earliest papers is Application of machine learning algorithms to flow modeling and optimization (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 (2016)
• Artificial neural network method for solving the Navier–Stokes equations (2014)