  The work is a continuation of a paper by Iskhakov A.S. and Dinh N.T.
"Physics-integrated machine learning: embedding a neural network in the
Navier-Stokes equations". Part I // arXiv:2008.10509 (2020) [1]. The proposed
in [1] physics-integrated (or PDE-integrated (partial differential equation))
machine learning (ML) framework is furtherly investigated. The Navier-Stokes
equations are solved using the Tensorflow ML library for Python programming
language via the Chorin's projection method. The Tensorflow solution is
integrated with a deep feedforward neural network (DFNN). Such integration
allows one to train a DFNN embedded in the Navier-Stokes equations without
having the target (labeled training) data for the direct outputs from the DFNN;
instead, the DFNN is trained on the field variables (quantities of interest),
which are solutions for the Navier-Stokes equations (velocity and pressure
fields). To demonstrate performance of the framework, two additional case
studies are formulated: 2D turbulent lid-driven cavities with predicted by a
DFNN (a) turbulent viscosity and (b) derivatives of the Reynolds stresses.
Despite its complexity and computational cost, the proposed physics-integrated
ML shows a potential to develop a "PDE-integrated" closure relations for
turbulent models and offers principal advantages, namely: (i) the target
outputs (labeled training data) for a DFNN might be unknown and can be
recovered using the knowledge base (PDEs); (ii) it is not necessary to extract
and preprocess information (training targets) from big data, instead it can be
extracted by PDEs; (iii) there is no need to employ a physics- or
scale-separation assumptions to build a closure model for PDEs. The advantage
(i) is demonstrated in the Part I paper [1], while the advantage (ii) is the
subject of the current paper.
