Derivative Informed Learning of Exchange-Correlation Functionals
2026-06-02 • Machine Learning
Machine Learning
AI summaryⓘ
The authors explore training faster machine-learned exchange-correlation functionals to imitate a popular but slower hybrid functional, B3LYP. They introduce a new loss function called DI-Loss that not only matches the energy output but also aligns how the energy changes locally, by supervising the energy’s derivatives. This approach significantly improves the prediction accuracy of the learned functionals compared to just using energy and density data. Additionally, the learned functionals speed up self-consistent calculations and enhance predictions of excited states when applied in time-dependent density functional theory.
machine-learned exchange-correlation functionalhybrid functionalB3LYPdensity functional theoryderivative-informed lossself-consistent field (SCF)excited statestime-dependent DFTmean absolute error (MAE)density matrices
Authors
Eike S. Eberhard, Luca A. Thiede, Abdul Aldossary, Andreas Burger, Nicholas Gao, Vignesh Bhethanabotla, Alán Aspuru-Guzik, Stephan Günnemann
Abstract
Machine-learned (ML) exchange-correlation (XC) functionals aim to replace human-designed density functional approximations by learning directly from reference data, but they still do not consistently outperform traditional $\mathcal{O}(N^4)$-scaling hybrid functionals. We study a hybrid-distillation setting in which $\mathcal{O}(N^3)$-scaling ML-XC functionals are trained to reproduce B3LYP/def2-SVP targets. We introduce Derivative Informed XC-Loss (DI-Loss), a loss that incorporates additional information from the reference hybrid functional by supervising first and second derivatives of the energy on the Grassmannian of admissible density matrices. Rather than only matching the self-consistent fixed point, DI-Loss aligns the local first- and second-order response of the learned functional with that of the target functional. Across four evaluated architectures, DI-Loss consistently improves the main energy metrics. Averaged uniformly across architectures, the total-energy MAE decreases by 66% relative to energy and density supervision alone. The density-sensitive mean-field energy metric $E_ρ$ improves from $1.2$ to $0.8$ mEh on average, while dipole and $\mathcal{L}_2$ density errors do not improve uniformly. We further show that densities from the distilled functionals reduce hybrid-functional SCF iterations by up to 50%. In downstream TDDFT calculations, Hessian supervision improves excited-state predictions, with XCdiff reducing the mean excitation-energy MAE by 19 - 35%.