  We propose a general way to construct an effective Hamiltonian in the
Self-learning Monte Carlo method (SLMC), which speeds up Monte Carlo
simulations by training an effective model to propose uncorrelated
configurations in the Markov chain. Its applications are, however, limited.
This is because it is not obvious to find the explicit form of the effective
Hamiltonians. Particularly, it is difficult to make trainable effective
Hamiltonians including many-body interactions. In order to overcome this
critical difficulty, we introduce the Behler-Parrinello neural networks (BPNNs)
as "effective Hamiltonian'' without any prior knowledge, which is used to
construct the potential-energy surfaces in interacting many particle systems
for molecular dynamics. We combine SLMC with BPNN by focusing on a divisibility
of Hamiltonian and propose how to construct the element-wise configurations. We
apply it to quantum impurity models. We observed significant improvement of the
acceptance ratio from 0.01 (the effective Hamiltonian with the explicit form)
to 0.76 (BPNN). This drastic improvement implies that the BPNN effective
Hamiltonian includes many body interaction, which is omitted in the effective
Hamiltonian with the explicit forms. The BPNNs make SLMC more promising.
