Local atomic energy regularization with posterior information optimization in machine-learning interatomic potentials

Decomposition of global properties (e.g., total energy) into local entities, as implemented in machine-learning- and neural-network-driven materials modeling, inherently introduces uncertainty due to the physically undefined nature of local properties. The validity of local decomposition constitutes an essential but still debated consideration in current studies. To address this ambiguity, we introduce a probabilistic atomic energy regularization (AER) framework designed to constrain local energy projections in machine learning interatomic potentials (MLIPs). The essence of AER is to regulate local-environment-determined atomic properties by incorporating statistically significant long-range correlations, thereby establishing a probability-based framework that selectively filters local decomposition incompatible with established correlation constraints. Beyond significantly improving decomposition robustness in regular solids, AER demonstrates excellent physical scalability as evidenced by the thermal transport simulations for $\ensuremath{\beta}\text{\ensuremath{-}}\mathrm{C}{\mathrm{u}}_{1.95}\mathrm{Se}$ with highly fluctuating local environment and higher-order nonlinearity. Our findings not only underscore the critical role of prior beliefs in MLIP development but also offer a generalized framework for resolving property decomposition controversies across data-driven machine-learning models.