Inferring material properties of the lower mantle minerals using Mixture Density Networks

Abstract

Interpretation of information available from seismic data in terms of temperature and composition requires an understanding of the physical properties of minerals, in particular, the elastic properties of candidate Earth minerals at the relevant (here, lower mantle) pressure and temperature. A common practice for the bulk elastic properties is to measure volume at a range of pressures and temperatures using experiments or computational methods. These datasets are then typically fit to a pre-determined functional form, or equation of state to allow computation of elastic properties at any other pressure or temperature. However, errors, both random and systematic, limitations in the number of data and choice of pressure marker and scale, as well as different functional forms of equations of state, all contribute to the uncertainties in mineral seismic properties. In an attempt to present a more comprehensive view of these uncertainties, we use neural-network based techniques to infer the relationship among: pressure, temperature, volume, bulk modulus, and thermal expansivity of MgO. We illustrate our approach on experimental data, but an extension to ab initio data is straightforward. The type of neural network used is called a Mixture Density Network (MDN) which is a combination of a conventional feed-forward neural network and a mixture model that consists of Gaussian functions. MDNs are capable of approximating arbitrary probability density functions, which allows us to compute the uncertainties in the predicted equations of state. Since the networks interpolate locally between input samples, pressure-volume-temperature relations are implicitly learned from data without imposing any explicit thermodynamic assumptions or ad-hoc relationships. We use the partial derivatives of the mapping between inputs (pressure and temperature) and output (volume) to compute the isothermal bulk modulus and thermal expansivity. Flexibility of the MDNs allows us to investigate the uncertainty due to certain data in one region of pressure-temperature space without influencing the posterior probability density everywhere. In general, we find that the elastic properties of MgO are well-constrained by experimental data. However, our study highlights regions in which sparse or inconsistent data lead to poorly constrained elastic properties, namely: at low pressure and high temperature (<25 GPa and >1500 K), and temperatures above 2700 K. While the former conditions are likely not important for the Earth's lower mantle, they are relevant in other planetary bodies such as the Moon and Mars. Comparison with conventional equation of state forms shows that assuming a certain functional form of the pressure-volume-temperature relationship leads to potential bias in uncertainty quantification, because the uncertainties are then specific to the underlying form. In combination with data sets of other lower mantle minerals, this technique should improve uncertainty quantification in interpretations of seismic data.