In a seminal paper, Behler and Parinello introduced atom-centered symmetry functions (ACSF) and high-dimensional NNPs (HDNNP). The ACSFs encode the local chemical environment as a set of radial and angular symmetry functions within a distance cutoff, which guarantees permutational, rotational, and translational invariance, and enables transferability across different systems. The decomposition of total energy into atomic energies makes the computational scaling linear with system size. Inspired by Behler's HDNNP, a number of modern NNPs have been developed , with a majority of them categorized as models using either predefined descriptors or learnable descriptors.