Improving efficiency of the path optimization method for a gauge theory

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