Shear properties of mantle minerals are vital for interpreting seismic shear wave speeds and therefore inferring the composition and dynamics of a planetary interior. Shear wave speed and elastic tensor components, from which the shear modulus can be computed, are usually measured in the laboratory mimicking the Earth's (or a planet's) internal pressure and temperature conditions. A functional form that relates the shear modulus to pressure (and temperature) is fitted to the measurements and used to interpolate within and extrapolate beyond the range covered by the data. Assuming a functional form provides prior information, and the constraints on the predicted shear modulus and its uncertainties might depend largely on the assumed prior rather than the data. In the present study, we propose a data-driven approach in which we train a neural network to learn the relationship between the pressure, temperature and shear modulus from the experimental data without prescribing a functional form a priori. We present an application to MgO, but the same approach works for any other mineral if there are sufficient data to train a neural network. At low pressures, the shear modulus of MgO is well-constrained by the data. However, our results show that different experimental results are inconsistent even at room temperature, seen as multiple peaks and diverging trends in probability density functions predicted by the network. Furthermore, although an explicit finite-strain equation mostly agrees with the likelihood predicted by the neural network, there are regions where it diverges from the range given by the networks. In those regions, it is the prior assumption of the form of the equation that provides constraints on the shear modulus regardless of how the Earth behaves (or data behave). In situations where realistic uncertainties are not reported, one can become overconfident when interpreting seismic models based on those defined equations of state. In contrast, the trained neural network provides a reasonable approximation to experimental data and quantifies the uncertainty from experimental errors, interpolation uncertainty, data sparsity and inconsistencies from different experiments.

A comparison of seismic observables with mineral seismic properties predicted by experimental or theoretical methods allows us to infer the structure and composition of the Earth's (or a planetary) interior (e.g.

Seismic properties of minerals and sensitivities of wave speeds to temperature and composition are usually derived from equation-of-state (EOS) modelling. While the volumetric properties are calculated from thermodynamic principles and EOSs (e.g. Birch–Murnaghan or Vinet with Grüneisen models), the shear modulus lacks a thermodynamic expression. Nevertheless, the shear modulus may be cast in a functional form similar to the bulk modulus (e.g.

In

Experimental shear modulus data for MgO are available from various measurement techniques. In this study, we collate data (Fig.

Experimental MgO

With RPR and BS in a single crystal, it is also possible to determine the elastic tensor of a sample (e.g.

As shown in Fig.

Prediction performance of the trained MDN as a function of pressure using the test set. The mean of the posterior pdf's on the shear modulus predicted by the MDN is subtracted from the actual shear modulus values of the test set (i.e. target values) to compute the variation in the shear modulus. The mean variation is shown as circles, and the size of uncertainty (1 standard deviation) is given by grey error bars. One could also represent the same information by using the log-likelihood function given by the MDN instead. The dashed cyan line refers to a perfectly resolved shear modulus. Hence, the closer the data plot to the line, the better the resolving power of the neural network. The differences between target and predicted shear modulus values are mostly located close to the cyan line, although intermediate- and high-pressure predictions are more uncertain and are located away from it. The range of temperature of the test data is given by the colour bar on the right. Note some error bars are smaller than or comparable to the plotting symbol.

We follow a standard approach of training neural networks, whereby training, monitoring and test sets are randomly generated from the total dataset. These sub-sets have similar pressure and temperature distributions. Although the total data contain similar numbers of SC and PC measurements, not all the referenced studies (in Fig.

The mean and the variance (or the standard deviation) of the posterior pdf

Figure

MDN-predicted pdf's for the shear modulus of MgO at every 5 GPa interval

For 300 K and above approximately 50 GPa (Fig.

Note that unless explicitly mentioned, to prepare the plots shown in this paper, we use the network trained with all data (i.e. including

Probability density functions for the shear modulus of MgO along a 0 GPa isobar

Besides ambient temperature, we plot (Fig.

We take the density with uncertainties from our previous study on the volumetric properties of MgO

Mean and standard deviation (1 SD) of the shear modulus and wave speeds of MgO along 300 and 2000 K isotherms and a 0 GPa isobar. Only

Theoretical computations provide mineral elastic properties across the lower mantle's pressure and temperature conditions (e.g.

The finite-strain equation of

Experiments that are performed on polycrystalline materials (e.g.

To quantify the uncertainties in shear wave speeds from the MDN-predicted pdf's of the shear modulus and density, we chose a pragmatic approach. We simply extracted the mean and standard deviation from the pdf's. However, one can also take the most probable Gaussian kernel, i.e. the kernel with the largest weight (in Fig.

With the MDN approach, we quantify uncertainties in the shear modulus (in Figs.

Furthermore, the MDN-based approach can be extended to model material properties that have more complicated dependencies on pressure and temperature, such as the transition from a high- to low-spin state of iron in (Fe,Mg)O ferropericlase. A reduction in the bulk modulus of ferropericlase on transition from the high-spin to low-spin state has been reported by various studies (e.g.

The shear modulus of MgO is constrained by experiments at low-

In general, at low pressures, shear moduli based on an explicit finite-strain equation whose fitting parameters (i.e.

Comparisons with MDN-predicted pdf's show that an explicit finite-strain equation represents one possible solution within the range of uncertainties, which is sometimes, although not always, the most likely value of the pdf's.

Data-driven approaches identify inconsistent data. Brillouin-scattering experiments on polycrystalline MgO are currently the only available measurement type that span the lower mantle's temperature and pressure conditions. However, these measurements follow a different trend from the remaining low-pressure experimental data from Brillouin scattering on single crystals and ultrasonic measurements on polycrystals. Even at ambient temperature, these different experimental datasets are inconsistent.

From a purely data-driven point of view, our pdf's show a large uncertainty in the shear properties of MgO, especially for pressures larger than about 30 GPa.

There are

The MDN approach provides realistic estimates of the uncertainties in the pressure–temperature range where measurements have been taken, which should be considered a lower bound if one extrapolates shear elastic properties outside of this region using, for example, a finite-strain formalism. Although the formalism will appear better-constrained, it could potentially be biased as some of our examples have shown in this study.

Currently, MgO is the mineral with the most data in the lower mantle. Therefore, one can expect larger uncertainties for other minerals.

A mixture density network (figure modified after

The code used in this paper is freely available by contacting the corresponding author.

The data used in this paper were collected from already published literature which is referenced in the caption of Fig.

AR contributed in terms of data curation, methodology, software, validation, visualization, formal analysis, investigation, visualization and writing – original draft. LC contributed in terms of conceptualization, supervision, funding acquisition, and writing – review and editing. JT contributed in terms of supervision and writing – review and editing. HM and JMJ contributed in terms of writing – review and editing.

The contact author has declared that none of the authors has any competing interests.

Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

We would like to thank the two anonymous reviewers for constructive comments which improved the manuscript. Ashim Rijal and Laura Cobden received funding from the Dutch Research Council (NWO) under grant number 016.Vidi.171.022. Hauke Marquardt acknowledges the support provided through the European Union's Horizon 2020 research and innovation programme (ERC grant 864877). Jennifer M. Jackson is thankful for support of this research by the National Science Foundation's Collaborative Studies of the Earth's Deep Interior (EAR-2009735).

This research has been supported by the Nederlandse Organisatie voor Wetenschappelijk Onderzoek (grant no. 016.Vidi.171.022), Horizon 2020 (DEEP-MAPS (grant no. 864877)) and the National Science Foundation (grant no. EAR-2009735).

This paper was edited by Etienne Balan and Carmen Sanchez-Valle and reviewed by two anonymous referees.