Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Physics-integrated machine learning: embedding a neural network in the
  Navier-Stokes equations. Part I   In this paper the physics- (or PDE-) integrated machine learning (ML)
framework is investigated. The Navier-Stokes (NS) equations are solved using
Tensorflow library for Python via Chorin's projection method. The methodology
for the solution is provided, which is compared with a classical solution
implemented in Fortran. This solution is integrated with a neural network (NN).
Such integration allows one to train a NN embedded in the NS equations without
having the target (labeled training) data for the direct outputs from the NN;
instead, the NN is trained on the field data (quantities of interest), which
are the solutions for the NS equations. To demonstrate the performance of the
framework, a case study is formulated: the 2D lid-driven cavity with
non-constant velocity-dependent dynamic viscosity is considered. A NN is
trained to predict the dynamic viscosity from the velocity fields. The
performance of the physics-integrated ML is compared with classical ML
framework, when a NN is directly trained on the available data (fields of the
dynamic viscosity). Both frameworks showed similar accuracy; however, despite
its complexity and computational cost, the physics-integrated ML offers
principal advantages, namely: (i) the target outputs (labeled training data)
for a NN might be unknown and can be recovered using 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. The
advantage (i) is demonstrated in this paper, while the advantages (ii) and
(iii) are the subjects for future work. Such integration of PDEs with ML opens
a door for a tighter data-knowledge connection, which may potentially influence
the further development of the physics-based modelling with ML for data-driven
thermal fluid models.

6
The categories of this paper are: astro-ph
0
The categories of this paper are: hep-ph
9
The categories of this paper are: nlin
30
The categories of this paper are: q-alg
32
The categories of this paper are: atom-ph
16
The categories of this paper are: stat
25
The categories of this paper are: patt-sol
18
The categories of this paper are: econ
4
The categories of this paper are: cond-mat
36
The categories of this paper are: cmp-lg
14
The categories of this paper are: math-ph
24
The categories of this paper are: dg-ga
1
The categories of this paper are: cs
12
The categories of this paper are: nucl-th
13
The categories of this paper are: quant-ph
11
The categories of this paper are: hep-lat