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We discuss the sign problem in the Polyakov loop extended Nambu--Jona-Lasiniomodel with repulsive vector-type interaction by using the path optimizationmethod. In this model, both of the Polyakov loop and the vector-typeinteraction cause the model sign problem, and several prescriptions have beenutilized even in the mean field treatment. In the path optimization method,integration variables are complexified and the integration path (manifold) isoptimized to evade the sign problem, or equivalently to enhance the averagephase factor. Within the homogeneous field ansatz, the path is optimized byusing the feedforward neural network. We find that the assumptions adopted inprevious works, $\mathrm{Re}\,A_8 \simeq 0$ and $\mathrm{Re}\,\omega \simeq 0$,can be justified from the Monte-Carlo configurations sampled on the optimizedpath. We also derive the Euler-Lagrange equation for the optimal path tosatisfy. The two optimized paths, the solution of the Euler-Lagrange equationand the variationally optimized path, agree with each other in the region withlarge statistical weight.

Evading the model sign problem in the PNJL model with repulsive vector-type interaction via path optimization