Machine Learning-Guided Multi-Objective Design of Holmium-Reduced Rare-Earth Permanent Magnets

Heavy rare-earth elements are added to neodymium–iron–boron (Nd₂Fe₁₄B) permanent magnets to keep them magnetically hard at high operating temperatures. Holmium (Ho) is one of these additives, and it sits near the top of every published supply-risk ranking: it is mined almost entirely as a minor by-product, from a small number of countries, and it is almost never recycled. This study asks a direct question: how much holmium does a rare-earth magnet actually need, and what is lost by removing it? We built an end-to-end computational pipeline that combines real density functional theory (DFT) data, a physics-based magnetic correction, and multi-objective evolutionary search. We retrieved 346 rare-earth–transition-metal–boron compounds from the Materials Project database and kept 319 thermodynamically stable or near-stable phases. Composition-based (Magpie) descriptors were used to train Random Forest, gradient-boosted tree (XGBoost) and neural-network surrogate models, which reached coefficients of determination of R² = 0.985 for formation energy and R² = 0.993 for magnetization on held-out test data. Because standard high-throughput DFT freezes the rare-earth 4f electrons into the core, the raw database values cannot distinguish light from heavy rare earths; we corrected this with a two-sublattice molecular-field model that restores ferromagnetic coupling for light rare earths and ferrimagnetic coupling for heavy ones. A six-component supply-chain criticality index was then embedded directly as an optimisation objective, alongside magnetization, formation energy and a literature-derived anisotropy-field proxy, and the design space was searched with the NSGA-II genetic algorithm. Three-objective optimisation converged on cerium-rich (Nd,Ce)₂Fe₁₄B compositions that retain roughly 88–90 % of the corrected magnetization of pure Nd₂Fe₁₄B while cutting the criticality score by about 15 % relative to Nd and 35 % relative to Ho, independently reproducing the experimentally validated cerium-substitution strategy reported in the literature. Adding anisotropy as a fourth objective exposed a trade-off that magnetization-only optimisation cannot see: across the 200-point four-objective Pareto front, criticality and anisotropy are almost perfectly correlated (r = 0.98), so high-coercivity designs necessarily carry supply risk. Critically, that risk is carried by terbium and dysprosium, not holmium. Holmium appeared above 1 % of the rare-earth sublattice in only 14 of 200 Pareto-optimal solutions, because praseodymium offers a higher anisotropy field (8.7 T versus 7.4 T) at a lower criticality score (0.645 versus 0.831) and a favourable magnetic coupling sign — holmium is therefore dominated on all three axes simultaneously. The practical conclusion is that holmium reduction is a comparatively low-cost supply-chain intervention, whereas terbium and dysprosium reduction is not. Keywords: rare-earth permanent magnets; holmium substitution; materials informatics; multi-objective optimisation; NSGA-II; supply-chain criticality; Nd₂Fe₁₄B; machine-learning surrogate models