Ant Colony Optimization for Density Functionals in Strongly Correlated Systems

Abstract

The Ant Colony Optimization (ACO) algorithm is a nature-inspired metaheuristic method used for optimization problems. Although not a machine learning method per se, ACO is often employed alongside machine learning models to enhance performance through optimization. We adapt an ACO algorithm to optimize the so-called FVC density functional for the ground-state energy of strongly correlated systems. We find the parameter configurations that maximize optimization efficiency, while reducing the mean relative error ($MRE$) of the ACO functional. We then analyze the algorithm's performance across different dimensionalities ($1D-5D$), which are related to the number of parameters to be optimized within the FVC functional. Our results indicate that $15$ ants with a pheromone evaporation rate superior to $0.2$ are sufficient to minimize the $MRE$ for a vast regime of parameters of the strongly-correlated system -- interaction, particle density and spin magnetization. While the optimizations $1D$, $2D$, and $4D$ yield $1.5\%< MRE< 2.7\%$, the $3D$ and $5D$ optimizations lower the $MRE$ to $\sim0.8\%$, reflecting a $67\%$ error reduction compared to the original FVC functional ($MRE = 2.4\%$). As simulation time grows almost linearly with dimension, our results highlight the potential of ant colony algorithms for density-functional problems, combining effectiveness with low computational cost.