PNcsp+: A Periodic Number-Based Crystal Structure Prediction Method Enhanced by Machine Learning.

Crystal structure prediction (CSP) is central to materials discovery, yet its efficiency and interpretability remain limited by the vast configurational space and reliance on costly local optimizations. Although template-based and machine-learning (ML) approaches have improved exploration, many approaches still require large data sets, complex similarity metrics, or opaque generative pipelines. In this work, we introduce PNcsp+, an enhanced and chemically interpretable CSP framework that uses the Mendeleev Periodic Number (PN) as a transparent descriptor of elemental similarity. PNcsp+ expands the original implementation through a larger prototype library, an improved data management strategy, and ML-assisted prototype scoring by combining cutting-edge neural network models such as MACE, M3GNet, and ALIGNN-FF. Despite its simplicity, PNcsp+ reaches state-of-the-art performance. In evaluations on the CSPBench data set─a curated set of 180 benchmark crystal structures for assessing CSP methods─our approach surpasses alternative methods by achieving 86.1% space group accuracy and 85.0% structure matching accuracy within the Top-5 predictions, all without structure relaxations. Moreover, our case study on several hybrid systems, including ammonium and methylammonium cations, demonstrated that molecular components emerge autonomously in the predicted lattices, guided solely by PN-derived similarity relationships. Overall, PNcsp+ shows that fundamental periodic trends, combined with targeted ML-based evaluation, offer an efficient, scalable, and interpretable CSP framework, enabling accelerated discovery across both inorganic and hybrid chemical spaces.