Overfitting by design: neural network density functionals for water

In density functional theory, simpler exchange-correlation (XC) approximations such as the local density approximation (LDA) are favored for computational speed but rely on limited information, leading to a trade-off between accuracy and generality. Machine-learned XC approximations have seen a lot of interest to address this problem. Here, we train a neural network LDA using a differentiable Kohn-Sham solver, imparting system-specific expertise for water and sacrificing generality for accuracy. Our model achieves 1 kcal / mol errors on gold standard coupled cluster ionization and atomization energies, and improves predictions of spectral lines, electron density distribution, and equilibrium geometry from as few as eight configurations used for training. We proceed to perform transfer learning and obtain results comparable to higher-rung PBE and B3LYP functionals on the WATER27 subset of the GMTKN55 database, even when only a single two-molecule binding energy is used in the transfer process. We furthermore assess performance of this model on a variety of other prediction tasks, including forces, dipole moments, and energetic ordering of isomers. Overall, we demonstrate that simple specialist functionals can be trained from little data, and that transfer learning can aid in this process. By enhancing predictions while maintaining low training costs, we make training of specialist functionals on different systems more accessible. Our approach of training a modified XC density functional approximation (DFA) furthermore allows for a highly interpretable result, as the neural network directly corresponds to a correction of the XC energy per electron.