We propose a novel approach called Self-Learning Hybrid Monte Carlo (SLHMC)
which is a general method to make use of machine learning potentials to
accelerate the statistical sampling of first-principles
density-functional-theory (DFT) simulations. The trajectories are generated on
an approximate machine learning (ML) potential energy surface. The trajectories
are then accepted or rejected by the Metropolis algorithm based on DFT
energies. In this way the statistical ensemble is sampled exactly at the DFT
level for a given thermodynamic condition. Meanwhile the ML potential is
improved on the fly by training to enhance the sampling, whereby the training
data set, which is sampled from the exact ensemble, is created automatically.
Using the examples of $\alpha$-quartz crystal SiO$_2^{}$ and phonon-mediated
unconventional superconductor YNi$_2^{}$B$_2^{}$C systems, we show that SLHMC
with artificial neural networks (ANN) is capable of very efficient sampling,
while at the same time enabling the optimization of the ANN potential to within
meV/atom accuracy. The ANN potential thus obtained is transferable to ANN
molecular dynamics simulations to explore dynamics as well as thermodynamics.
This makes the SLHMC approach widely applicable for studies on materials in
physics and chemistry.