Predicting sulfide precipitation in magma oceans on Earth, Mars and the Moon using machine learning

The sulfur content at sulfide saturation (SCSS) of a silicate melt can regulate the stability of sulfides and, therefore, chalcophile elements' behaviors in planetary magma oceans. Many studies have reported high-pressure experiments to determine SCSS using either linear or exponential regressions to parameterize the thermodynamics of the system. Although these empirical equations describe the effects of different parameters on SCSS, they perform poorly when predicting laboratory measurements. Here, we compiled 542 published analyses of experiments performed on a range of sulfide and silicate compositions at varying P-T conditions (<24 GPa, <2673 K). Using empirical equations, linear regression, Random Forest algorithms, and a hybrid algorithm employing empirical fits to P-T conditions and the Random Forest algorithm for compositions, we developed several SCSS models and compared them to laboratory measurements. The Random Forest and hybrid models (R-2 = 0.82-0.91, mean average error [MAE] < 746 ppmw S, residual mean standard error [RMSE] < 972 ppmw S), significantly outperform previous empirical models (R-2 = 0.28-0.69, MAE = 622-1,170 ppmw S, RMSE = 1,070-1,744 ppmw S), whereas linear regression performs moderately well, i.e., between the classic and machine learning models. We applied our hybrid model to predict SCSS during magma ocean solidification on Earth, Mars, and the Moon, and we compared our model results to expected S contents in the residual magma oceans calculated by mass balance. Our results confirm that during early accretion, sulfides precipitated from magma oceans and into the outer cores of Earth and Mars, but not the Moon. Subsequently, once the respective magma oceans began precipitating minerals with increasingly FeO-rich and SiO2-, Al2O3-, and MgO-depleted compositions, the increasing S concentration in the residual magma was offset by temperature and compositional effects on SCSS, preventing sulfide precipitation during intermediate stages of crystallization. Sulfides precipitated late during magma ocean crystallization, but failed to percolate through the underlying crystalline mantle, significantly contributing to the modern bulk-silicate sulfur abundances of Earth, Mars, and the Moon. Our calculations suggest that late-stage sulfide precipitation occurred at shallow depths of 120-220 km, 40-320 km, and < 10 km in the magma oceans of Earth, Mars, and the Moon, respectively.