We start from the polynomic interatomic potentials introduced by Wojde{\l} etal. [J. Phys. Condens. Matt. 25, 305401(2013)] and take advantage of one oftheir key features -- namely, the linear dependence of the energy on thepotential's adjustable parameters -- to devise a scheme for the construction offirst-principles-based ({\em second-principles}) models for large-scalelattice-dynamical simulations. Our method presents the following convenientfeatures. The parameters of the model are computed in a very fast and efficientway, as it is possible to recast the fit to a training set of first-principlesdata into a simple matrix diagonalization problem. Our method selectsautomatically the interactions that are most relevant to reproduce thetraining-set data, by choosing from a pool that includes virtually all possiblecoupling terms, and produces a family of models of increasing complexity andaccuracy. We work with practical and convenient cross-validation criterialinked to the physical properties that will be relevant in future simulationsbased on the new model, and which greatly facilitate the task of identifying apotential that is simultaneously simple (thus computationally light), veryaccurate, and predictive. We also discuss practical ways to guarantee that ourenergy models are bounded from below, with a minimal impact on their accuracy.Finally, we demonstrate our scheme with an application to ferroelasticperovskite SrTiO$_{3}$, which features many non-trivial lattice-dynamicalfeatures (e.g., a phase transition driven by soft phonons, competing structuralinstabilities, highly anharmonic dynamics) and provides a very demanding test.
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