  In this study the effects of Joule heating parameter on entropy generation
and heat transfer in MHD flow inside a channel is investigated by means of
Physics-Informed Neural Networks (PINNs) in form of a parametric analysis in
addition to exploring the solution to the ill-posed problem. All of the
governing equations are reformulated in terms of first order derivatives and
the dimensionless form of the governing equations has been employed to further
lessen the number of parameters and achieve better compatibility with loss
function terms. Dimensionless groups such as Reynolds number, Prandtl number,
Hartmann number and Joule heating parameter have been designated as the input
for the neural network in order to perform the parametric study. Besides
achieving high accuracy for case of parameters confined in the predefined
ranges, the generalization ability of the method is depicted by solving the
problem for the cases where the dimensionless parameters were outside of the
assumed ranges. Moreover, the ability of handling Neumann boundary conditions
is also investigated in the present study despite being neglected prevalently
in the literature concerning PINNs. The effects of Joule heating parameter on
entropy generation are researched using a parametric approach utilizing PINNs
which is another novel aspect of the article at hand. As a concluding remark,
an ill-posed problem is also defined such that the Joule heating parameter is
included in a learning process and the method has been able to determine Joule
heating parameter as a parameter of interest alongside other learnable
parameters of the neural network such as weights and biases.
