Revisiting the machine-learning density functional for the one-dimensional Hubbard model with random external potential
We revisit the machine-learning (ML) approach to the universal density functional $F[\mathbf{n}]$ of the one-dimensional Hubbard model with a site-dependent random potential $\mathbf{v}=\{v_{i}\}$. We generate exact ground-state data via exact diagonalization for a periodic chain with $L=8$ in the paramagnetic sector $(N_\uparrow,N_\downarrow)=(2,2)$, with site electron densities $n_{i} = n_{i\uparrow}=n_{i\downarrow}$. The resulting density-potential dataset is analyzed. Using principal component analysis of the joint feature space $(\mathbf n,\mathbf v)$, we identify the intrinsic low-dimensional structure of the data. Then, we restricted the study of the dataset with an energy-based filtering criterion to concentrate the data around weakly perturbed energy values with zero potential. A compact one-dimensional convolutional neural network is trained to learn the universal functional considering the lattice periodicity through unilateral wrapping and enforce the lattice symmetries by data augmentation (translations and mirror reflections), achieving near-exact predictions of $F[\mathbf n]$. Finally, we address the fact that accurate functional values do not necessarily imply accurate functional derivatives. By augmenting training with a variational consistency term that constrains the Euler-Lagrange relation between $\partial F/\partial n_i$ and the gauge-fixed potential we reconstruct the external potentials from automatic differentiation. These results clarify the roles of dataset geometry, symmetry, gauge fixing, and derivative-based constraints in learning physically consistent density functionals.