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

Gauge invariant input to neural network for path optimization method