Fig. 3: Integrated PolymorphGen-MLPKD framework enabling efficient transfer of cross-scale accuracy.

a Snapshots of the configuration distribution within Ir’s training dataset compare structures derived from ab initio molecular dynamics (AIMD) with those added by PolymorphGen. b Comparison of the root mean squared errors (RMSE) for energy, force, and virial on configurations strictly excluded from both training sets, between machine learning potentials (MLPs) trained on AIMD-derived and PolymorphGen-derived configurations. c Top: Distribution of datasets on the entropy-symmetry landscape. Polymorph datasets covering stable, metastable, and disordered states enable fair training and testing of MLPs. Bottom: RMSEs of the NequIP, DPA, DPMD, and MACE models compared with density functional theory (DFT) calculations. Both training and testing of the models follow this concept. d Vacancy formation energy of Al⁵⁴, Ir⁵⁴, Mo⁵⁵, and Zr⁵⁶ calculated by message passing neural network (MPNN) models, compared with DFT calculations. e Elastic constants (C₁₁, C₁₂, C₄₄, C₁₃, C₃₃) and bulk modulus (B), shear modulus (G), and Young’s modulus (E) of Al⁵⁷, Ir^58,59,60, Mo⁶¹, and Zr^62,63 calculated using MPNN models compared with experimental data. Inset: Enlarged view of the selected region. f The force prediction RMSEs for different numbers of training configurations demonstrate model loss convergence. Inset: Radial distribution functions of liquid-Ir computed using MPNN trained by 77 and 3259 configurations, compared with DFT results. g Impact of configurational coverage bias on MLP prediction performance. Incomplete sampling by the biased data (inset) manifests as an under-sampled high-error region in the energy prediction error distribution (centre), which is quantified by consistently higher RMSEs across energy, force, and virial predictions compared to the model trained on uniformly sampled data (around). h The error distributions in energy, force, and virial for DNN, DNN model trained via knowledge distillation (DNN-KD), and MPNN models are compared with DFT calculations. The box chart with normal distribution curve depicts the error distribution using five statistics: minimum, lower quartile, median, upper quartile, and maximum. i Phonon dispersion spectra of DFT, DNN, DNN-KD, and MPNN models. Source data are provided as a Source Data file.