Interaction embedding via factorized linear models for multi-element machine learning interatomic potentials

Linear machine-learning interatomic potentials, such as the Ultra-Fast Force Field (UF3), deliver near-density functional theory accuracy at low computational cost. Their number of parameters, however, grows combinatorially with the number of chemical species, limiting their use on multi-element systems. We address this scaling problem with pseudo-interaction UF3 (PI-UF3), a factorized linear model in which the Z(k) real k-body potentials are expressed as linear combinations of a small number P(k) ≪ Z(k) of shared pseudo-interactions. Learned element embeddings are standard in nonlinear potentials; inside a linear model, the embedding becomes a rank constraint on the coefficient matrix of each body order, and the resulting bilinear problem is trained with alternating least squares, where each block update is one closed-form linear solve. We benchmark PI-UF3 on a five-element refractory high-entropy alloy (HEA) and a subset of MD22, training on forces, with restarts and cross-validation. On the chemically homogeneous HEA, compression acts as implicit regularization: with three-body terms included, PI-UF3's test force root-mean-squared error drops below the unfactorized UF3 baseline with 3% of the baseline's parameters on the predefined split and 7% on every cross-validation fold, and the maximally compressed model recovers a chemically meaningful ordering of homonuclear interaction strengths consistent with experimental bulk melting temperatures. On the chemically heterogeneous MD22 subset, where UF3 is not expressive enough to overfit, PI-UF3 approaches the baseline from above and reaches parity with it at about two thirds of the parameter count. Compression thus regularizes only where the unfactorized model overfits, while a parameter reduction is available in both regimes.