Graph neural network maps PBE formation energies onto the r2SCAN scale, cutting the cross-functional gap to 14.3 meV/atom
Synopsis
Formation energies from different exchange-correlation functionals have not been directly mergeable: in MatPES the same structures differ by 107 meV/atom on average between PBE and r2SCAN. Using more than 380,000 structurally paired PBE–r2SCAN entries, researchers trained a graph neural network to predict the energy residual between the two functionals and correct PBE formation energies onto the r2SCAN scale. With GPTFF as the backbone, the model reaches a mean absolute error of 14.3 meV/atom on the held-out test set, below the 18.2 meV/atom of CHGNet.
Interpretation
Cross-functional formation-energy offsets can be learned by a structure-aware graph neural network, lifting low-cost PBE data to the r2SCAN energy scale without repeating the expensive calculations. Earlier attempts to unify DFT energy scales relied on GGA/GGA+U compatibility schemes, fitted elemental-phase reference energies, coordination-corrected enthalpies and PBE(+U)–r2SCAN mixing, all built on regression or reference-state corrections designed for specific datasets or protocols. Across 380,190 structurally paired MatPES samples, PBE and r2SCAN formation energies differ by 107 meV/atom on average and by up to 1.960 eV/atom; by bonding type the mean difference is 181 meV/atom for ionic compounds and 62, 73 and 86 meV/atom for covalent, metallic and mixed-bonding materials, so the offset tracks chemical environment rather than a global shift.
The corrected energies feed directly into three downstream judgments: phase stability, battery voltage and reaction thermodynamics. Raw PBE energies place r2SCAN-stable phases above the convex hull and underestimate lithium deintercalation voltage plateaus. In six binary phase diagrams (Fe-O, Mn-O, Ti-Cl, Mo-Cl, V-S, Cu-S), the GPTFF correction reproduces the r2SCAN stability classification for six of the eight phases whose PBE and r2SCAN classifications differ, a 75% recovery rate; LiCoO2 and LiMn2O4 voltage profiles track r2SCAN and sit closer to experiment; in the BaO+TiO2 and Li2O+Fe2O3 reactions the most stable product switches from the PBE choice to the one favored by r2SCAN and experiment.
The method treats the formation-energy difference between the two functionals as the learning target and adds the predicted residual back to the original PBE energy. This residual learning reuses established machine-learning interatomic potential architectures instead of introducing a separate correction formula. With GPTFF and CHGNet as backbones, crystal structures are mapped to the per-atom r2SCAN-minus-PBE formation-energy difference, with data split 80:10:10 into training, validation and test; GPTFF reaches a test MAE of 14.3 meV/atom, RMSE of 39.3 meV/atom and R² of 0.9118, versus 18.2 meV/atom, 44.0 meV/atom and 0.8889 for CHGNet.
Perspective
Validation covers structurally paired PBE–r2SCAN inorganic data from MatPES, with downstream tests on six binary phase diagrams, LiCoO2 and LiMn2O4 voltage profiles and four solid-state reaction systems; entries containing Tb, Sm, Th or Dy are excluded for lack of data.
The r2SCAN convex-hull topology is not fully reproduced: FeO2, Mn5O8 and V7S8 are stable in r2SCAN but remain above the corrected hull, while MoCl4, V3S5 and CuS2 are predicted on the corrected hull; the corrected LiMn2O4 profile adds a plateau transition at x = 0.083 and misses the transition at x = 0.25, and the Li2O+Fe2O3 reaction energy still differs from r2SCAN and experiment by about 0.025 eV/atom.