Pre-training Neural Networks to Learn the Exchange–Correlation Functional in Density Functional Theory

We investigate strategies to train neural networks to reproduce known exchange–correlation (XC) functionals in Density Functional Theory as a first step toward learning the exact, unknown XC functional. We focus on the pre-training stage, where the networks learn XC enhancement factors directly from the density, its gradient, and Laplacian. We find uniformly sampled training grids struggle to reach chemical accuracy, while representative data from benchmark sets yield markedly better results. However, even in this case, the functional-driven errors remain above chemical accuracy, underscoring the intrinsic difficulty of the learning problem. We further show that explicitly incorporating physical constraints into the neural network architecture improves transferability and performance—often surpassing the reference functional itself—demonstrating the critical role of such constraints. This pre-training provides a physically grounded starting point for the full training stage, where both the functional parameters and the electronic density are optimized self-consistently. This “double optimization problem,” in which the functional must yield the exact ground-state density as its self-consistent fixed point, constitutes the primary bottleneck of XC functional optimization.