BMach: a Bayesian machine for optimizing Hubbard U parameters in DFT+U with machine learning

Accurately determining the effective Hubbard parameter $(U_{eff})$ in Density Functional Theory plus U (DFT+U) remains a significant challenge, often relying on empirical methods or linear response theory, which frequently fail to predict accurate material properties. This study introduces BMach, an advanced Bayesian optimization algorithm that refines $U_{eff}$ by incorporating electronic properties, such as band gaps and eigenvalues, alongside structural properties like lattice parameters. Implemented within the Quantum Espresso platform, BMach demonstrates superior accuracy and reduced computational cost compared to traditional methods. The BMach-optimized $U_{eff}$ values yield electronic properties that align closely with experimental and high-level theoretical results, providing a robust framework for high-throughput materials discovery and detailed electronic property characterization across diverse material systems.