Application of a neural network to the sign problem via the path optimization method

We introduce the feedforward neural network to attack the sign problem via the path optimization method. The variables of integration are complexified and the integration path is optimized in the complexified space by minimizing the cost function, which reflects the seriousness of the sign problem. For the preparation and optimization of the integral path in multi-dimensional systems, we utilize the feedforward neural network. We examine the validity and usefulness of the method in the 2D complex $\lambda \phi^4$ theory at finite chemical potential as an example of the quantum field theory with the sign problem. We show that the average phase factor is significantly enhanced after the optimization and then we can safely perform the hybrid Monte Carlo method.