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We propose a data-driven approach for constructing machine-learninginteratomic potentials (MLIPs) trained under a regularization with the aim ofavoiding nonphysical heat flux. Specifically, we introduce a regularizationterm for the heat flux into the cost function of MLIPs to be minimized. Sincethe treatment of heat flux using MLIPs with regularization can be decomposedinto elemental contributions or conducted in frequency space, this approach isexpected to be useful for investigating the origin of thermal conductivityobtained from the Green-Kubo formula. However, the strength of regularizationneeds to be appropriately set because it may reduce not only the nonphysicalpart but also the intrinsic heat flux one. To this end, we investigated theconditions for constructing MLIPs that can reproduce the power spectra of heatflux associated with the empirical interatomic potential of Ag$_2$Se, whichconsists of pairwise functions and do not contain a nonphysical heat flux. Theappropriate strength could be estimated from the variation of the magnitude ofregularization term as well as root mean square errors for total potentialenergy, atomic force, and virial stress with respect to the strengths, withoutreference spectrum data. As an application example, we explored the differencesin power spectra between superionic and nonsuperionic conducting phases basedon the heat flux regularization to MLIPs trained with the first-principlescalculation data of Ag$_2$S. Furthermore, our results demonstrate that trainingwith the regularization improves the robustness of MLIPs as well as thereduction of the nonphysical heat flux.

Construction of Machine-Learning Interatomic Potential Under Heat Flux  Regularization and Its Application to Power Spectrum Analysis for Silver  Chalcogenides