Predicting Sulfide Precipitation in Magma Oceans on Earth, Mars, and the Moon Using Machine Learning

Abstract

The sulfur content at sulfide saturation (SCSS) in silicate melts plays a pivotal role in governing the behavior of chalcophile elements in planetary magma oceans. Numerous high-pressure experiments have been conducted to determine SCSS, employing various regression methods to capture the thermodynamic characteristics of the system. However, existing empirical equations have shown limited predictive accuracy when applied to laboratory measurements. In this study, we have compiled and analyzed 542 experimental datasets encompassing diverse sulfide and silicate compositions under varying pressure-temperature (P-T) conditions (up to 24 GPa and 2673 K). Employing empirical equations, linear regression, Random Forest algorithms, and a novel hybrid approach combining empirical fits for P-T conditions with Random Forest modeling for compositions, we have developed multiple SCSS models. These models have been rigorously compared with laboratory measurements. Our findings reveal that the Random Forest and hybrid models exhibit exceptional predictive performance (R2 = 0.82–0.91, mean average error [MAE] < 746 ppmw S, residual mean standard error [RMSE] < 972 ppmw S) in comparison to previous empirical models (R2 = 0.28–0.69, MAE = 622–1,170 ppmw S, RMSE = 1,070–1,744 ppmw S). Linear regression falls in between the performance of classical and machine learning models. Furthermore, we have applied our hybrid model to predict SCSS during the solidification of magma oceans on Earth, Mars, and the Moon. A comparison of our model results with expected sulfur contents in residual magma oceans, calculated through mass balance, offers valuable insights. Our analysis confirms that sulfides precipitated during the early accretion phases of Earth and Mars, but not on the Moon. Subsequently, evolving compositions of magma oceans offset increasing sulfur concentrations, preventing sulfide precipitation during intermediate stages of crystallization. Late-stage sulfide precipitation, contributing significantly to the bulk-silicate sulfur abundances of Earth, Mars, and the Moon, occurred at shallow depths (120–220 km, 40–320 km, and <10 km, respectively) within their respective magma oceans. This study sheds light on predicting SCSS under a range of conditions, advancing our understanding of chalcophile element behavior in planetary magma oceans.