Shear properties of MgO inferred using neural networks

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.</p

Shear properties of MgO inferred using neural networks

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

Applications of machine learning to mineral physics data and the inference of the thermochemical structure of the Earth's mantle.

The physical and chemical properties of the Earth’s mantle govern the cause of natural disasters, such as earthquakes and volcanoes. Since we do not have direct access to mantle materials, their properties are often inferred from laboratory measurements and surface observations (e.g. seismic data from earthquake recordings). This thesis addresses some key problems we face while utilising these data to constrain the thermal and chemical properties of the mantle. Firstly, we propose a data-driven approach based on machine learning to explain the laboratory measurements and quantify their uncertainties in the absence of an adequate physical model. Our results show that although conventional approaches based on fitting the measurements to an assumed model may appear better constrained, they could potentially provide biased results. Secondly, we use the data-driven approach to explore which thermochemical parameters can be constrained (and to what extent) with limited seismic observables- wave speeds and density. Our results show that these observables constrain temperature and major chemical parameters (silicon, magnesium, and iron), and they indicate the presence of thermochemical heterogeneities at the lowermost mantle. The dense and slow piles at the bottom of the lower mantle seen in seismic data can be explained by an enrichment in silica and iron content- characteristic feature of enstatite chondrites. The inferred heterogeneities have profound implications for the dynamics of the mantle and outer core. The methodology developed in this thesis is extremely efficient. It can easily incorporate additional observables and thus, has wide applications in the seismology and mineral physics community

Applications of machine learning to mineral physics data and the inference of the thermochemical structure of the Earth's mantle.

The physical and chemical properties of the Earth’s mantle govern the cause of natural disasters, such as earthquakes and volcanoes. Since we do not have direct access to mantle materials, their properties are often inferred from laboratory measurements and surface observations (e.g. seismic data from earthquake recordings). This thesis addresses some key problems we face while utilising these data to constrain the thermal and chemical properties of the mantle. Firstly, we propose a data-driven approach based on machine learning to explain the laboratory measurements and quantify their uncertainties in the absence of an adequate physical model. Our results show that although conventional approaches based on fitting the measurements to an assumed model may appear better constrained, they could potentially provide biased results. Secondly, we use the data-driven approach to explore which thermochemical parameters can be constrained (and to what extent) with limited seismic observables- wave speeds and density. Our results show that these observables constrain temperature and major chemical parameters (silicon, magnesium, and iron), and they indicate the presence of thermochemical heterogeneities at the lowermost mantle. The dense and slow piles at the bottom of the lower mantle seen in seismic data can be explained by an enrichment in silica and iron content- characteristic feature of enstatite chondrites. The inferred heterogeneities have profound implications for the dynamics of the mantle and outer core. The methodology developed in this thesis is extremely efficient. It can easily incorporate additional observables and thus, has wide applications in the seismology and mineral physics community

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

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 (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

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

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 (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

Shear properties of MgO inferred using neural networks

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

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

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 (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