  We investigate the efficiency of a gauge invariant input to a neural network
for the path optimization method. While the path optimization with a completely
gauge-fixed link-variable input has successfully tamed the sign problem in a
simple gauge theory, the optimization does not work well when the gauge degrees
of freedom remain. We propose to employ a gauge invariant input, such as
plaquette, to overcome this problem. The efficiency of the gauge invariant
input to the neural network is evaluated for the 2-dimensional $U(1)$ gauge
theory with a complex coupling. The average phase factor is significantly
enhanced by the path optimization with the plaquette input, indicating good
control of the sign problem. It opens a possibility that the path optimization
is available to complicated gauge theories, including Quantum Chromodynamics,
in a realistic setup.
