Machine learning for accuracy in density functional approximations

TL;DR

The paper surveys the burgeoning use of machine learning to enhance density functional approximations, categorizing approaches into ML-based exchange-correlation functionals and post-DFT corrections. It compiles benchmark datasets and ground-truth targets (thermochemistry, molecular structures, transition-metal surfaces, and charge densities) to train and evaluate ML models, including explicit-form functionals and neural-network XC models such as pcNN and DM21. It discusses key successes, transferability challenges, and the need for physical constraints and robust benchmarks to ensure reliable performance across chemistries and solids. The work underscores both the potential for achieving chemical accuracy with ML-augmented DFAs and the practical hurdles in extending these models to extended systems and diverse material classes, calling for careful data curation, validation, and methodological advances.

Abstract

Machine learning techniques have found their way into computational chemistry as indispensable tools to accelerate atomistic simulations and materials design. In addition, machine learning approaches hold the potential to boost the predictive power of computationally efficient electronic structure methods, such as density functional theory, to chemical accuracy and to correct for fundamental errors in density functional approaches. Here, recent progress in applying machine learning to improve the accuracy of density functional and related approximations is reviewed. Promises and challenges in devising machine learning models transferable between different chemistries and materials classes are discussed with the help of examples applying promising models to systems far outside their training sets.

Figures (12)

• Figure 1: Overview of ML approaches to increasing the accuracy of electronic structure predictions based on electronic or atomic structural features. Machine-learned XC functionals are trained on high-accuracy benchmark data to improve upon the predictive power of existing DFAs. Post-DFT and $\Delta$-ML methods provide improved energetics on fixed DFT charge densities, and other ML approaches supplement the Kohn-Sham Hamiltonian with Hubbard and dispersion terms.
• Figure 2: Schematic of a neural network-based XC functional. Local features of the charge density $\rho$ at position r and, depending on the XC functional type, kinetic ($\tau$) or EXX energy densities are inputs to the neural network yielding the XC energy $E_{\rm XC}(\textbf{r})$. With the help of backpropagation, the gradient of the XC energy with respect to the inputs can be obtained, from which the effective one-body KS potential is computed.
• Figure 3: Comparison of the XC enhancement of the PBE,PBE SCAN,SCAN and neural network-based XC functionals without (NN) and with physical constraints (pcNN) for different values of Wigner-Seitz radius $r_s$, reduced density gradient $s$, KS kinetic energy density $\tau$ (relative to Thomas-Fermi kinetic energy density $\tau_{\rm unif}$), and relative spin polarization $\zeta$. Two top rows: XC enhancement over local exchange $\epsilon_{\rm X}^{\rm unif}$, bottom row: pcNN XC enhancement over SCAN XC functional. Reproduced from Ref. [$\!\!\!$pcNN], [border = 0, color = 0 0 1]https://doi.org/10.1103/PhysRevResearch.4.013106DOI: 10.1103/PhysRevResearch.4.013106 under the terms of the Creative Commons Attribution 4.0 International License. Copyright 2022, the Authors. Published by the American Physical Society.
• Figure 4: Reaction enthalpies of lowest energetic C$_7$H$_{10}$O$_2$ isomers with respect to the most stable isomer 7-oxabicyclooctan-7-one (depicted in inset). Gaussian-4-Møller-Plesset-2 benchmarks are shown by black bars and B3LYP DFT results by red bars. Blue bars show $\Delta$-ML predictions based on atomic-structural kernels significantly improving over the B3LYP results to within chemical accuracy $<1$ kcal/mol. Reproduced with permission from Ramakrishnan et al.,Ramakrishnan2015[border = 0, color = 0 0 1]https://doi.org/10.1021/acs.jctc.5b00099DOI: 10.1021/acs.jctc.5b00099. Copyright 2015 American Chemical Society.
• Figure 5: Comparison of water molecule charge density differences between coupled cluster theory and PBE (contour plots to the left) and between the NeuralXC functional and PBE (right). Reproduced from Ref. [$\!\!\!$Dick2020], [border = 0, color = 0 0 1]https://doi.org/10.1038/s41467-020-17265-7DOI: 10.1038/s41467-020-17265-7 under the terms of the Creative Commons Attribution 4.0 International License. Copyright 2020, the Authors. Published by Springer Nature.