Extended Data Fig. 4: The calculation of effective Hamiltonian parameters of the PTS_x=0.25 by the active learning approach.

a, Bayesian error. The dashed line in denotes the threshold to perform first-principles calculations. b, Potential energy. The blue and orange lines in denote the potential energy predicted by the effective Hamiltonian model during the fitting process and using the parameters obtained after the fitting, respectively. The open circles in denote the potential energy obtained from the first-principles calculations. c, Local dipolar mode. The active learning is performed on 2 × 2 × 2 supercell of the perovskite unit-cell (40 atoms). The Bayesian error (a) exhibits a sharp drop at the beginning of fitting, with first principles calculations called frequently. After about 5000 steps, the Bayesian error threshold (a) remains basically unchanged, and first-principles calculations are less frequently called, indicating that the fitting is approaching convergence. The potential energy values predicted by the effective Hamiltonian are close to those obtained from first principles calculations (b). The local dipolar mode (c) shows one nonzero average value during the whole fitting process, characterizing a macroscopic tetrahedral ferroelectric phase.