  We propose a general framework for finding the ground state of many-body
fermionic systems by using feed-forward neural networks. The anticommutation
relation for fermions is usually implemented to a variational wave function by
the Slater determinant (or Pfaffian), which is a computational bottleneck
because of the numerical cost of $O(N^3)$ for $N$ particles. We bypass this
bottleneck by explicitly calculating the sign changes associated with particle
exchanges in real space and using fully connected neural networks for
optimizing the rest parts of the wave function. This reduces the computational
cost to $O(N^2)$ or less. We show that the accuracy of the approximation can be
improved by optimizing the "variance" of the energy simultaneously with the
energy itself. We also find that a reweighting method in Monte Carlo sampling
can stabilize the calculation. These improvements can be applied to other
approaches based on variational Monte Carlo methods. Moreover, we show that the
accuracy can be further improved by using the symmetry of the system, the
representative states, and an additional neural network implementing a
generalized Gutzwiller-Jastrow factor. We demonstrate the efficiency of the
method by applying it to a two-dimensional Hubbard model.
