  We investigate efficiency of a gauge-covariant neural network and an
approximation of the Jacobian in optimizing the complexified integration path
toward evading the sign problem in lattice field theories. For the construction
of the complexified integration path, we employ the path optimization method.
The $2$-dimensional $\text{U}(1)$ gauge theory with the complex gauge coupling
constant is used as a laboratory to evaluate the efficiency. It is found that
the gauge-covariant neural network, which is composed of the Stout-like
smearing, can enhance the average phase factor, as the gauge-invariant input
does. For the approximation of the Jacobian, we test the most drastic case in
which we perfectly drop the Jacobian during the learning process. It reduces
the numerical cost of the Jacobian calculation from ${\cal O}(N^3)$ to ${\cal
O}(1)$, where $N$ means the number of degrees of freedom of the theory. The
path optimization using this Jacobian approximation still enhances the average
phase factor at expense of a slight increase of the statistical error.
