  We investigate the sign problem in field theories by using the path
optimization method with use of the neural network. For theories with the sign
problem, integral in the complexified variable space is a promising approach to
obtain a finite (non-zero) average phase factor. In the path optimization
method, the imaginary part of variables are given as functions of the real
part, $y_i=y_i(\{x\})$, and are optimized to enhance the average phase factor.
The feedforward neural network can be used to give and to optimize functions
with many variables. The combined framework, the path optimization with use of
the neural network, is applied to the complex $\phi^4$ theory at finite
density, the 0+1 dimensional QCD at finite density, and the Polyakov loop
extended Nambu-Jona-Lasinio (PNJL) model, all of which have the sign problem.
In these cases, the average phase factor is found to be enhanced significantly.
In the complex $\phi^4$ theory, it is demonstrated that the number density is
calculated at a high precision. On the optimized path, the imaginary part is
found to have strong correlation with the real part on the temporal nearest
neighbor site. In the 0+1 dimensional QCD, we compare the results in two
different treatments of the link variable: optimization after the diagonal
gauge fixing and optimization without the diagonal gauge fixing. These two
methods show consistent eigenvalue distribution of the link variables. In the
PNJL model with homogeneous field ansatz, finite volume results approach the
mean field results as expected, and the phase transition behavior can be
described.
