用机器学习力场提升量子效应模拟效率，精准预测材料热性质。

Accounting for nuclear quantum effects (NQEs) can significantly alter material properties at finite temperatures. Atomic modeling using the path-integral molecular dynamics (PIMD) method can fully account for such effects, but requires computationally efficient and accurate models of interatomic interactions. Empirical potentials are fast but may lack sufficient accuracy, whereas quantum-mechanical calculations are highly accurate but computationally expensive. Machine-learned interatomic potentials offer a solution to this challenge, providing near-quantum-mechanical accuracy while maintaining high computational efficiency compared to density functional theory (DFT) calculations. In this context, an interface was developed to integrate moment tensor potentials (MTPs) from the MLIP-2 software package into PIMD calculations using the i-PI software package. This interface was then applied to active learning of potentials and to investigate the influence of NQEs on material properties, namely the temperature dependence of lattice parameters and thermal expansion coefficients, as well as radial distribution functions, for lithium hydride (LiH) and silicon (Si) systems. The results were compared with experimental data, quasi-harmonic approximation calculations, and predictions from the universal machine learning force field MatterSim. These comparisons demonstrated the high accuracy and effectiveness of the MTP-PIMD approach.