  The accurate prediction of solid-solid structural phase transitions at finite
temperature is a challenging task, since the dynamics is so slow that direct
simulations of the phase transitions by first-principles (FP) methods are
typically not possible. Here, we study the $\alpha$-$\beta$ phase transition of
Zr at ambient pressure by means of on-the-fly machine-learned force fields.
These are automatically generated during FP molecular dynamics (MD) simulations
without the need of human intervention, while retaining almost FP accuracy. Our
MD simulations successfully reproduce the first-order displacive nature of the
phase transition, which is manifested by an abrupt jump of the volume and a
cooperative displacement of atoms at the phase transition temperature. The
phase transition is further identified by the simulated x-ray powder
diffraction, and the predicted phase transition temperature is in reasonable
agreement with experiment. Furthermore, we show that using a singular value
decomposition and pseudo inversion of the design matrix generally improves the
machine-learned force field compared to the usual inversion of the squared
matrix in the regularized Bayesian regression.
