Permutationally Invariant Fourier Series for Accurate and Robust Data-Driven Many-Body Potentials

We present a robust solution to the long-standing challenge of eliminating unphysical energy predictions, or "holes," in machine-learned many-body potentials, which can destabilize simulations when encountering configurations beyond the training set. By leveraging permutationally invariant Fourier series (PIFSs) within the MB-nrg data-driven many-body formalism, we introduce a new approach that significantly enhances the numerical stability of MB-nrg potential energy functions (PEFs) while preserving accuracy and transferability. Unlike conventional strategies that attempt to "plug holes" by expanding training data sets, PIFSs provide a more fundamental and efficient means of ensuring physically meaningful extrapolation across diverse molecular configurations. Using water as a benchmark system, we demonstrate that the MB-pol(PIFS) PEF retains the high accuracy of MB-pol across gas and condensed phases while extending the PEF's stability to a much broader range of thermodynamic conditions. Our results suggest that the PIFS-based MB-nrg many-body formalism provides a general framework for constructing accurate and robust physics-based/machine-learned potentials applicable to a broad range of molecular systems.