Thermodynamic modeling of hydrous-melt–olivine equilibrium using exhaustive variable selection

Journal Pre-proof Thermodynamic modeling of hydrous-melt–olivine equilibrium using exhaustive variable selection

Kenta Ueki, Tatsu Kuwatani, Atsushi Okamoto, Shotaro Akaho, Hikaru Iwamori PII:

S0031-9201(19)30169-4

DOI:

https://doi.org/10.1016/j.pepi.2020.106430

Reference:

PEPI 106430

To appear in:

Physics of the Earth and Planetary Interiors

Received date:

19 June 2019

Revised date:

29 November 2019

Accepted date:

17 January 2020

Please cite this article as: K. Ueki, T. Kuwatani, A. Okamoto, et al., Thermodynamic modeling of hydrous-melt–olivine equilibrium using exhaustive variable selection, Physics of the Earth and Planetary Interiors(2018), https://doi.org/10.1016/ j.pepi.2020.106430

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Thermodynamic modeling of hydrous-melt–olivine equilibrium using exhaustive variable selection Kenta Ueki1,* ORCID: 0000-0002-5936-3856 Tatsu Kuwatani1,2 ORCID: 0000-0001-7161-9458 Atsushi Okamoto3 ORCID: 0000-0001-5757-6279 Shotaro Akaho4 ORCID: 0000-0002-4623-2718 Hikaru Iwamori1,5,6 ORCID: 0000-0001-5913-6410

Research Institute for Marine Geodynamics, Japan Agency for Marine-Earth Science and

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1

Technology, 2-15 Natsushima-cho, Yokosuka 237-0061, Japan 2

PRESTO, Japan Science and Technology Agency (JST), 4-1-8 Honcho, Kawaguchi,

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Saitama 332-0012, Japan

Graduate School of Environmental Studies, Tohoku University, Sendai 980-8579, Japan

4

Human Informatics Research Institute, National Institute of Advanced Industrial Science and

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3

5

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Technology, 1-1-1 Umezono, Tsukuba, Ibaraki 305-8568, Japan Earthquake Research Institute, The University of Tokyo, 1-1-1 Yayoi, Bunkyo-ku, Tokyo

6

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113-0032, Japan

Department of Earth and Planetary Sciences, Tokyo Institute of Technology, 2-12-1

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Ookayama, Meguro-ku, Tokyo 152-8550, Japan

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*Corresponding author: Kenta Ueki ([email protected]) Tel.: +81-46-867-9628

Keywords: Thermodynamics ‧ Machine learning ‧ Hydrous melt ‧ Mantle melting ‧ thermodynamic equilibrium model ‧ Basalt Highlights A machine learning method for petrological thermodynamic modeling is presented The method selects key variables from experimental data on hydrous-melt systems A robust thermodynamic model of the hydrous-melt–olivine equilibrium is constructed Model calculations reproduce measured effect of H2 O on olivine liquidus temperature Improved understanding of the underlying mechanisms of hydrous-melt systems

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Abstract Water in silicate melt influences the phase relations of a hydrous- melt system. Given the importance of water in silicate melts, a quantitative thermodynamic understanding of the nonideality of hydrous melt is necessary to properly model natural magmatic processes. This paper presents a novel method for quantitative thermodynamic modeling of hydrous- melt– olivine equilibrium. Specifically, a machine learning method, exhaustive variable selection (ES), is used to model the non- ideality of hydrous melts. Using the ES method, we quantitatively validate the predictive capacities of all possible combinations of variables and then adopt the combination with the highest predictive capacity as the optimal model

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equation. The ES method allows us to obtain the underlying thermodynamic relationship of the hydrous- melt–olivine system, such as the relative importance of different variables to the thermodynamic equilibrium, as well as to construct a robust and generalized model. We show

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that the combination of a linear term and a squared term of the total water concentration of

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melt is significant for describing the hydrous- melt–olivine equilibrium. This result is interpreted in terms of the microstructural changes related to the dissociation of water in

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silicate melt. Calculations using the optimal model reproduce the experimentally determined effects of water on the olivine liquidus and the distribution coefficient for Mg between

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olivine and hydrous melt. Our study demonstrates that the ES method yields a thermodynamic equilibrium model that captures the essential thermodynamic relationship

1 Introduction

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explaining the high-dimensional and complex experimental data.

Understanding the phase relations of hydrous melt is fundamental to ascertaining the processes of mantle melting under hydrous conditions and the differentiation of hydrous magmas in arc settings (e.g., Tatsumi, 1989; Sisson and Grove, 1993; Kawamoto, 1996; Grove et al., 2003). Melting experiments have revealed that water drastically changes the phase relations of hydrous- melt systems (e.g., Kushiro et al., 1968; Kushiro, 1969; Green, 1973; Mysen and Boettcher, 1975a, b; Médard and Grove, 2008; Green et al., 2010, 2014). Such experiments are performed by varying the pressure, temperature, melt water concentration, and bulk composition of the system (Fig. 1). Thermodynamic modeling allows such discrete experimental data to be integrated and a model calculation to be conducted under various conditions (e.g., Ghiorso and Sack, 1995; Ghiorso et al., 2002; Gualda et al., 2012; Ueki and Iwamori, 2013, 2014; Jennings and Holland, 2015).

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Natural magmatic systems are open systems in which the bulk chemical composition and total energy of the system evolve with melting, solidification, and melt migration (e.g., Langmuir et al., 1977). The relationships between pressure (P), temperature (T), bulk composition, phase stability, and phase properties (composition, fraction, and volume), and the mass and energy balances between phases need to be considered to properly investigate a dynamic and open-system magma environment. Thermodynamic calculations can yield consistent relationships between mass and energy balances. Various equations from an ideal mixing model (e.g., Ueki and Iwamori, 2013) and from more complex models that consider the interactions between various end- member components (e.g., Ghiorso and Sack, 1995)

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have been proposed to model the thermodynamic relationship between the composition of silicate melt and its Gibbs free energy. Ueki and Iwamori (2013, 2014) demonstrated that a relatively simple thermodynamic equation for silicate melt could reproduce the melting phase

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relation of anhydrous alkaline-absent spinel lherzolite. However, hydrous melts have not

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been considered in some recent thermodynamic models of mantle melting (e.g., Ueki and Iwamori, 2013, 2014; Jennings and Holland, 2015) because of the difficulty of performing

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such modeling.

As the detailed microstructural and chemical properties of hydrous melt under high

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pressure and temperature are complicated, the experimental and theoretical understandings of the structure- free energy relation (i.e., the thermodynamic configurations and equations for

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modeling) are correspondingly poorly constrained. Consequently, despite its importance, the hydrous melt has

not yet been adequately

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phase relation of multi-component

thermodynamically modeled. Experimental studies on the microstructure (e.g., Mysen, 2007, 2014; Yamada et al., 2011) and first-principles molecular dynamics simulations (Mookherjee et al., 2008; Karki and Stixrude, 2010; Karki et al., 2010) have shown that water in silicate melt acts to depolymerize the melt via the reaction

(e.g.,

Wasserburg, 1957; Stolper, 1982). Ueki and Iwamori (2016) argued that a non- ideal interaction that considered the dissociation of water in silicate melt would be required to model the phase relation of a hydrous- melt system on the basis that such a system shows strongly non- linear relationships between melt compositions and phase relations (e.g., Hirschmann et al., 1998; Médard and Grove, 2008). To model the thermodynamic behavior of hydrous melt in continuous pressure–temperature–composition space, the appropriate model equation and parameters for composition-related non-ideality must be developed. Machine learning (ML) techniques (i.e., the science of using computers for the automatic detection of patterns in data; e.g., Bishop, 2006) can be used to understand the relationships

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between variables and construct predictive models. ML techniques are being rapidly introduced to petrological and geochemical studies. For example, ML has been used for discrimination and feature extraction using labeled geochemical data (e.g., Kuwatani et al., 2014; Petrelli and Perugini, 2016; Ueki et al., 2018) and for pattern recognition in highdimensional geochemical data (e.g., Iwamori and Albaréde, 2008; Ueki and Iwamori, 2017; Yoshida et al., 2018). The present paper reports the novel results of an ML-based model-selection approach to the thermodynamic modeling of hydrous melt. “Model selection” refers to the ML task of selecting a statistical model that best reflects the various possible underlying mechanisms that

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might have produced the patterns in the data. Using the ML method of exhaustive variable selection (ES), we are able to construct a predictive model from high-dimensional and complex data, and understand the physical mechanisms and processes underlying the data

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(e.g., Nakamura et al., 2017).

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The objective of the present study is to model the thermodynamic equilibrium of hydrous basaltic melt and olivine under conditions of the uppermost upper mantle. A large number of

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possible equations are expected for modeling multicomponent silicate melts. For example, to select an equation that properly models the thermodynamic relationships among the multiple

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components in silicate melt, we need to evaluate which components, including their interactions, are thermodynamically more important. In this study, we use an ML-based

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model-selection approach, in which we evaluate a set of candidate model equations for the

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equilibrium of hydrous basaltic melt and olivine and select the optimal model equation. The compiled experimental data set (experimental temperature, pressure, and melt and olivine compositions) is used as input data to determine the optimal equations for the thermodynamic model and its parameters. We use the leave-one-out cross-validation technique to select the optimal equation from the various possible equations. The leave-one-out cross- validation method leaves out one data case from the data set (i.e., validation data) and performs multiple linear regression on the remaining data to construct the model. Using the validation data, we evaluate how well the model can predict unknown data; i.e., how well the model predicts the equilibrium between olivine and melt for the temperature, pressure, and melt and olivine compositions of the validation data. This procedure is repeated for all data points in the data set, and the mean of the squared residuals between the model prediction (difference in molar Gibbs free energy between melt and olivine) and the validation data (= equilibrium state) is considered as the predictive capacity of the model equation for the unknown data (e.g., Bishop, 2006). The total number of candidate model equations is 1024 in the present study.

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We evaluate all possible model equations on the basis of the predictive capacities of the equations and select the optimal model equation. Through the modeling of this fundamental system for magma generation, we demonstrate that the ML approach used in this study is a robust method that allows a generalized model of complex magma system to be constructed. The ML method also allows an understanding of the underlying thermodynamic relationship through, for example, quantifying the relative importance of different explanatory variables. The ML method and the constructed model should act as reference points for the future development of multi-component and multiphase melting thermodynamic models (Ueki and Iwamori, 2013, 2014) as well as for the

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modeling of various geological and geochemical processes, mechanisms, and conditions, including metamorphic reactions (e.g., Okamoto and Toriumi, 2004) and ultrahigh-pressure

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phase relations such as metal–melt equilibria (e.g., Walter and Cottell, 2013).

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2 Basic thermodynamics

We model the thermodynamic equilibrium between hydrous basaltic melt and olivine at a

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pressure range corresponding to spinel lherzolite stability (~1.0–2.5 GPa). Specifically, we model the thermodynamic relationship between the silicate components, water concentration,

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pressure, and temperature of melt and the composition of olivine (Mg/(Mg + Fe)). Olivine is the primary constituent mineral of the upper mantle and is also the primary crystalline phase

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derived from basaltic melt. Therefore, the thermodynamic modeling of this system yields

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essential constraints on hydrous-mantle melting (e.g., Hirschmann et al., 2009) and on the differentiation of hydrous basaltic magma (e.g., Sisson and Grove, 1993). To model the hydrous basaltic melt–olivine equilibrium, we consider the following reaction: Mg2 SiO4 melt = Mg2 SiO 4 ol where superscript

melt

(1)

denotes the silicate melt phase, and superscript

ol

denotes the olivine

phase. The molar Gibbs free energy of the melt phase is modeled on the basis of the thermodynamic equation formulated by Ueki and Iwamori (2013, 2014), in which the difference in molar Gibbs free energy ( ) between the melt ( solid end-member component ( The

) is expressed as

of the reaction in Eq. 1 is written as (2)

(

) and the corresponding ).

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is the difference in molar Gibbs free energy between pure melt and the

corresponding pure solid end- member component (see Appendix A for details). Consequently,

is dependent on temperature and pressure, and is independent of

melt and olivine compositions.

denotes the reaction quotient (e.g., Putirka, 2017). As a

reaction proceeds, the reaction quotient changes to reduce the total Gibbs free energy of the system. When an equilibrium state is achieved (

),

is equal to the equilibrium

constant, and Eq. 2 can be written as (

(3)

is the activity.

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where

)

By assuming the dissociation of the forsterite end- member in melt as Mg2 SiO 4 melt = 2MgOmelt + SiO 2 melt ,

in Eq. 3 can be rewritten as (4)

is the molar fraction of end- member component i in a phase, and

composition-related non-ideality.

is a

e-

where

)

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(

corresponds to an ideal solution.

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In the present study, we focus on . Selection of a non- ideality model of silicate melt is critical to constructing a model that adequately reproduces the experimentally determined

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melting phase equilibria (e.g., Beattie, 1993; Putirka, 2017). It has been suggested that nonideal interactions between H2 O and other melt end- member components are required to

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reproduce the non- linear relationship between the total water concentration of a melt (i.e., the total amount of dissolved water in the melt) and the phase equilibria (e.g., Médard and Grove,

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2008; Ueki and Iwamori, 2016). The immiscibility between silicate melt and aqueous fluid (e.g., Hunt and Manning, 2012) also indicates the non- ideal mixing behavior of water in silicate melt. The

term (Eq. 4), which describes the relationship between the total water

concentration of melt and the Gibbs free energy, needs to be considered in hydrous- melt thermodynamics. To develop such a numerical model for melt–olivine equilibrium, we use an ML approach to formulate a robust equation that describes the relationship between melt composition and Gibbs free energy.

3 Variable selection and model optimization using an ML approach 3.1 Basic configuration of the thermodynamic model We select and optimize the variables in , the composition-related non-ideality (Eq. 4), that adequately capture the underlying thermodynamic relationship of the hydrous- melt– olivine thermodynamic system. For this purpose, we first present a basic equation for variable

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selection. We enumerate all possible terms in the basic equation and then select the essential terms using an ML approach. We consider an equation based on a symmetric regular solution as the basic equation. The composition-related non-ideality of multi-component silicate melt can be thermodynamically modeled using a symmetric regular solution type equation (e.g., Ghiorso et al., 1983). The composition-related non- ideality of an n end- member component symmetric regular solution is written as follows (e.g., Ghiorso et al., 1983): (5) (J/mol) represents the interaction parameters of end-member component i.

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where

The maximum number of unknown thermodynamic parameters (

) in Eq. 5 is

because symmetric interactions between two different end- member

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components are considered in this equation. The model-selection approach determines the variables. Consequently, the total number of

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optimal combination of variables from the possible equations in this case is

. Such a large number of unknown parameters would

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make the computational cost too high to adopt the ML method employed in this study (i.e., the problem of combinatorial explosion). In addition, a large dataset would be required to

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obtain generalized and robust results when optimizing a large number of parameters (e.g., Bishop, 2006). Therefore, we employ a simplified equation based on Eq. 5 as the basic

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equation for the non-ideality model of this study.

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Carmichael (2002) analyzed

of experimental hydrous melts, and found the

linear relationship between the molar fractions of major-element oxide end- member components (

and

) and

range of basalt to andesite. In particular,

of hydrous melts in the compositional was found to strongly affect

. Their

results suggest that the linear term (first term of the right-hand side of Eq. 5) could satisfactorily model the effects of major-element oxide end- member components on the nonideality of mafic silicate melts. Following Carmichael (2002), we consider the following equation as the basic equation of the non-ideality model developed in the present study: (6) Eq. 6 is a simplified equation obtained by omitting the second term of the right-hand side of Eq. 5. As a result, the maximum number of unknown thermodynamic parameters (

) is

10 in this study. The total number of possible model equations is reduced to 210 with the

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simplified equation. As shown in section 4, this simplified equation properly models the thermodynamic relationship between hydrous melt and olivine. Although only SiO 2 and MgO are involved in the chemical reaction considered here (Eq. 1), alkalis (Na2 O and K2 O) also have substantial effects on the thermodynamic properties of mafic silicate melts (e.g., Kushiro, 1975; Carmichael, 2002; Hirschmann et al., 1998; Putirka et al., 2007), meaning that other end-member components (in addition to SiO 2 and MgO) may also need to be considered in modeling the hydrous-melt–olivine equilibrium. Therefore, the importance of the various oxide end- member components in the thermodynamic equation should be quantified when modeling the thermodynamic behavior of silicate melt.

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Consequently, we consider SiO 2 –Al2 O3 –FeO–MgO–CaO–Na2O–K2 O as the major-element oxide end-member components of melt.

To describe dissociated water in silicate melt, we consider two terms in addition to total

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water concentration in the melt (e.g., Stolper, 1982; Nowak and Behrens, 1995). Hunt and

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Manning (2012) performed thermodynamic modeling of the SiO 2 –H2 O system by considering the molecular H2 O, bridging oxygens, and OH group attached to a silicate

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polymer in silicate melt and reproduced experimentally derived SiO 2 solubility in the aqueous fluid, hydrous melting of quartz, and the critical endpoint between silicate melt and

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aqueous fluid. Their results based on the SiO 2 –H2 O system suggest that thermodynamic modeling considering the speciation of water may improve the model’s ability to reproduce

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the thermodynamic behavior of hydrous silicate melt.

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The dissociation reaction of water in silicate melt is written as follows (Stolper, 1982): (7)

where H2 Omolecular, O0 , and OH denote the molecular H2 O, bridging oxygens, and OH group attached to a silicate polymer in silicate melt, respectively. The mass balance of the reaction in Eq. 7 is written as ⁄ where

(8)

denotes the concentrations of the species in a silicate melt, and H2 Ototal denotes the

total water concentration of a silicate melt. With a fixed equilibrium constant of the reaction in Eq. 7,

is proportional to

, and

is proportional to

(Stolper, 1982). Bridging oxygens are consumed in the reaction of Eq. 7, meaning that the amount of bridging oxygen decreases in proportion to Consequently, in addition to the linear term (

(Stolper, 1982).

), we consider two terms for H2 O

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,

9

) to model the dissociation of water.

Consequently, we have

(9) where the linear term (

) represents the total amount of dissolved water in the melt,

the squared term (

) is a proxy thermodynamic term for

with the consumed bridging oxygen, and the square-root term (

) is a proxy

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thermodynamic term for OH (melt). Finally, substituting Eq. 9 into Eqs. 3, 4, and 6, we have (10)

The left-hand side of Eq. 10 can be calculated using experimental results (pressure,

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temperature, and melt and olivine compositions) and the non- ideality model of olivine (Sack

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and Ghiorso, 1989). Therefore, we can use linear regression to determine W from experimental results (Fig. 1). The maximum number of explanatory variables in the present

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case is 10. On the basis of Eq. 10, various combinations of variables are possible, such as

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where only the linear term is considered for the

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effect of water,

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where the water-related terms are not considered,

and more sparse equations such as ,

and

. The total number of

such candidate model equations is 1024 (=210 ). We evaluate the various combinations of variables to model the equilibrium of hydrous basaltic melt and olivine using an ML-based model-selection approach. All possible combinations of variables are evaluated, and the optimal combination of end- member components for representing the hydrous-melt–olivine equilibrium is selected.

3.2 Exhaustive variable selection On the basis of the basic configuration (Eq. 9), we implement an ML-based model-selection approach to understand the underlying thermodynamic relationship that explains the data and construct a robust and generalized model of the thermodynamic equilibrium state between

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hydrous melt and olivine, in which the most informative set of variables to describe the relationship between melt composition and the molar Gibbs free energy is incorporated. The problem is to find the appropriate combination of variables from the major-element oxide end- member components

,

, and

,

,

,

, and the water-related terms

,

, , and

in Eq. 9. This type of problem can generally be formulated within multiple linear regression, a type of multivariate analysis that is used to construct a multinomial model that relates many

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explanatory variables to the objective variable (e.g., Chatfield and Collins, 1980; Le Maitre, 1982). In the optimization of thermodynamic equilibrium experiments, the multinomial equation can be written as follows:

indicates the objective variable calculated using the experimental results as ;

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where

pr

(11)

indicates the number of explanatory terms;

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indicates the explanatory variables, which are the observed composition of melt ( and the squared and square-root terms of water composition, , respectively; and ,

and

indicates each unknown thermodynamic parameter to be

,

,

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estimated (i.e.,

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,

,

,

, and

).

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In cases where the required explanatory variables have already been specified and where the amount and quality of the observed data are suitable, the unknown parameters can be appropriately estimated, meaning that multiple linear regression can be used to construct a statistical model that has a high predictive capacity (e.g., Chatfield and Collins, 1980). However, in most problems of thermodynamic parameter optimization the parameters are presumably unknown and the variation in the experimental conditions of the observed data (pressure, temperature, and composition) are generally insufficient for properly determining all the thermodynamic parameters that exhibit high predictive capacity in the entire pressure, temperature, and composition range of interest. In the present study, the model equation is unknown, and the P, T, and melt composition of the input experimental data are discrete and heterogeneous. Applying a simple multiple linear regression analysis to such a problem could cause overfitting (e.g., Bishop, 2006); i.e., the model is incorrectly fitted to (noisy) observed data using too many irrelevant and unnecessary variables, which results in an extremely low

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predictive capacity of the obtained model when applied to a new dataset. In multiple linear regression analysis, a model that uses irrelevant (redundant) variables can reproduce the input data set used for regression but may not predict unknown data. To construct a model that has high predictive capacity, appropriate combinations of explanatory variables should be determined by removing unnecessary variables. In doing so, an alternative evaluation criterion is required based not only on the reproducibility of observed data (e.g., sum of squared residuals) but also on predictive capacity when applied to new datasets (e.g., Bishop, 2006; Kuwatani et al., 2014). The ES method is the most robust approach for variable selection (Nagata et al., 2015;

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Igarashi et al., 2016). This method automatically extracts the essential variables for describing high-dimensional data and constructs the model with the highest predictive capacity. The ES method has been used in many previous studies in a range of scientific

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fields, including psychology (Ichikawa et al., 2014; Nagata et al., 2015), astronomy (Igarashi

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et al., 2018), and Earth sciences (Kuwatani et al., 2014; Nakamura et al., 2017). The ES method evaluates the predictive capacity of the constructed model for all combinations of

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explanatory variables. In testing all possible combinations of variables, there are two possible states for each variable: a used state and a discarded state. If this is expanded to all variables,

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the total number of possible states is 2n , where n is the number of explanatory variables. In this study, we adopted a cross-validation (CV) technique to evaluate the predictive capacity

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of the model for unknown data (i.e., the generalization capability of the model). The CV

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technique can estimate the generalization capability for new datasets using only the available observed dataset. The technique divides the available dataset into two parts: the regression data, which are used to construct the model, and the validation data, which are used to evaluate the generalization capability of the model. By using the ES method with the CV technique, the generalization capability of thermodynamic equations for magmatic equilibria can be objectively evaluated, and the model with the best generalization capability can be identified. We used the ES method for variable selection based on Eq. 10. The maximum number of unknown thermodynamic parameters (

) is 10, meaning that the predictive capacity of the

model for 1024 (= 210 ) combinations of variables was evaluated using the CV technique. In this study, we used the leave-one-out CV technique, whereby M − 1 data (where M = the total number of data in the dataset = 296 in the present study) of the melting experiments were considered as the regression data {

} and

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12 . Multivariate linear

regression was applied to the regression data to construct the training models

.

Generalization capability (predictive capacity) was evaluated using the constructed model and the validation data. To obtain the generalization capability, we used the cross- validation error (CVE), defined by the root mean square error which is obtained by repeating the above calculation for all M samples: √

.

(12)

Using this calculation, all possible ways (= M ways) of dividing the original data into the regression data and the validation data were able to be tested, and the root mean squared

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residual between model prediction and the validation data (i.e., CVE) was regarded as the measure of the validity of the model equation (i.e., its generalization capability). Using the

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leave-one-out CV technique, we uniquely determined the CVE of each equation. To implement the calculations, we used the R language (R Core Team, 2016), an open-source

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programming language for statistical computing. The R source code and the input data are

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provided in Appendix C (Supplementary Materials 1 and 2, respectively).

3.3 Dataset for optimization

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The dataset for model-selection and parameter optimization comprised previously reported experimental results that were obtained according to the following three criteria:

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1. The data describe ultramafic and mafic bulk compositions.

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2. The system consists of at least the following six major-element oxide components: SiO 2 , Al2 O 3 , Fe2 O 3 or FeO, MgO, CaO, and Na2 O, without CO2 . 3. The system includes olivine and melt ± orthopyroxene ± clinopyroxene ± spinel, without garnet or plagioclase.

Accordingly, 296 individual experimental runs, 72 of which are hydrous experiments, were used for model-selection and optimization during this study. Some of the experimental results were obtained from the Library of Experimental Phase Relations open database of melting experiments (Hirschmann et al., 2008). Anhydrous experiments were obtained from experimental datasets compiled by Ueki and Iwamori (2014). P–T ranges and the total water concentrations of melt are displayed in Fig. 1, and data sources are given in the caption to that figure. The dataset used for the present study is given in Appendix C (Supplementary Material 2). The P–T range of the dataset is 0.8–2.5 GPa and 1050–1600 °C, and the total water concentration of melt ranges from 0 (anhydrous) to 8.96 wt% (Fig. 1). As we focused

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on those experiments that were based on mantle systems, almost all experimental melts (294 experimental results) exhibit basaltic to basaltic-andesite compositions. Two experimental melts exhibit andesitic and dacitic compositions. For the water concentration of melt, we used the values reported in each paper; i.e., we did not perform an independent estimation, such as a calculation based on a solubility model (e.g., Newman and Lowenstern, 2002). Fig. 1 shows that the experimental conditions are discrete and heterogeneous. For example, the number of experimental runs decreases with increasing pressure. Both partial- melting and crystallization experiments are included, as are K 2 O-present and K2 O-absent experiments.

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4 Results and discussion

4.1 Model assessment and the relative importance of explanatory variables By investigating the CVE-based ranking of models and comparing the top models, we can

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identify informative variables in describing the data, and relationships between variables. Fig.

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2 presents the relationship between the CVE (the measure of model generalization capability) and the number of variables used in the equations. The variables used, parameters, and CVEs

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of the best 100 models are shown in Fig. 3 (see Appendix C, Supplementary Material 3, for the full list of models). The variables used, optimized parameters, and CVEs of the top 10

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model equations are given in Table 1. Residuals obtained by CV analysis (determined as the differences between the values predicted using the constructed model and each validation

) in Fig. B1 of Appendix B. The root mean squared residual of Fig. B1 was regarded as

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(

in Eq. 12) are plotted against total water concentration in the melt

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data; i.e.,

the measure of the validity of the model equation. The non-ideality model with the lowest CVE (i.e., the model with the best generalization capability or optimal model) was obtained using nine variables (SiO 2 , Al2 O3 , FeO, MgO, CaO, Na2 O, K2 O, H2 O, and H2 O2 ) and is written as

. (13) We note that using all 10 variables yields a model with the second-lowest CVE, indicating that using all 10 variables results in overfitting (Fig. 3). Although intuitively it might seem that the most complex model (i.e., the 10-variable model) should yield the highest generalization capability, the use of redundant explanatory variables that explain negligible

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variation in the objective variable acts to reduce the generalization capability of models of the hydrous-melt–olivine equilibrium. The ideal-solution model (Eq. 6 = 0) yields the highest CVE; i.e., the worst generalization capability (Fig. 2). SiO 2 , Al2 O3 , FeO, MgO, CaO, and Na2 O are present in all of the top 10 models, and K2 O, in addition to the six major elements, is present in all of the top 5 models (Fig. 3), implying that these seven major elements are particularly informative in describing the phase equilibrium between hydrous basaltic melt and olivine. Also of note, all of the top 100 models used the SiO 2 and MgO terms. FeO, CaO, H2 O, H2 O2 , Al2 O3 , Na2O, H2 O0.5 , and K2O were selected in 79, 74, 65, 64, 63, 63, 53, and 50 of the top 100 models, respectively.

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These results mean that of the major-element end- member components, the SiO 2 , MgO, and FeO concentrations of melt are particularly informative in modeling of the silicate- melt– olivine equilibrium. Although this result might appear to be self-evident, given that we

pr

consider Mg–Fe olivine here, the result should be emphasized because this information was

e-

objectively and automatically defined using the ES method. The CVE values of the top 10 models range from 2.32 to 2.47 kJ, comparable with the experimental uncertainty on the

Pr

enthalpy of fusion for forsterite of ±2.6 kJ (Sugawara, 2005). is the most commonly used water-related term (among in the top 100 models. Conversely,

al

, and

, is

rn

the least commonly adopted of these terms in the top 100 models and is not adopted in the best model. All of the top five models include all seven major elements, and different

Jo u

combinations of water-related terms are present in the top five models. Regarding the parameters of the water-related terms, among the

tends to exhibit the highest absolute values

of the three water-related terms. In contrast,

absolute values of all

exhibits the lowest

(Table 1), which means that of the water-related terms, the square-

root term is the least informative regarding phase equilibrium. Fig. 2 indicates that models using SiO 2 , MgO, FeO, and the water-related terms tend to exhibit better generalization capabilities (i.e., lower CVEs) than those of models not using these terms. Models considering only water-related terms and/or alkaline (Na2 O and K2 O) terms tend to exhibit high CVEs. In addition, SiO 2 -absent models exhibit high CVEs and have poor predictability. The high CVE values of the models using water-related terms exclusively (i.e., models in which

,

, and/or

were used but major oxide end-

components were not) can be interpreted in terms of the input experimental data. Models

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15

considering only the water-related terms would be expected to perform poorly in predicting the results of anhydrous experiments, and the model calculation for the anhydrous condition is the same as the ideal model in this case. In addition, the analytical error on the water concentration might be large compared with those on the major oxide components. Therefore, models using water-related terms exclusively have the lowest predictive capacities of all variable combinations tested here. Generally, concentrations of alkalis (Na2 O and K2 O) in silicate melt have a strong correlation. As the model-selection approach involves highly correlated values, only one variable with less noise tends to be selected from a highly correlated group of variables as a

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significant variable. However, Na2 O and K2 O are present in all of the top five models, meaning that both of these elements have a considerable influence on the thermodynamic equilibrium between silicate melt and olivine (e.g., Kushiro, 1975; Hirschmann et al., 1998).

pr

In contrast, models using exclusively alkaline (Na2 O and K2 O) terms tend to exhibit high

e-

CVEs, and K 2O is the least commonly adopted variable in the top 100 models. Similar to the case of water, both K2 O-present and K2 O-absent experiments were included in the

Pr

experimental dataset. In terms of the thermodynamic equation (Eq. 4), directly contribute to the molar Gibbs free energy of melt through the

term, whereas

al

the concentrations of other oxide end- member components, as well as

the

and

, indirectly . As such,

rn

contribute to the molar Gibbs free energy of melt by the dilution of

and

terms represent the excess contribution of elements to free energy above that term. As we are concerned with ultramafic and mafic systems in

Jo u

contributed by the

this study, only trace amounts of alkaline elements were involved in the experiments, meaning that contributions of their non-ideal excess energy (the

terms) to the total Gibbs

free energy of the system are small in comparison with the contributions made through dilution. In addition, experimental and analytical errors on the trace amounts of concentrations of alkalis might be large compared with those of other major-element concentrations. Consequently, the experimental data on alkalis are relatively noisy, and K2 O and Na2 O are the least and second-least commonly selected major elements in the top 100 models, respectively, and models using exclusively alkaline (Na2 O and K 2O) terms tend to exhibit high CVEs. Our results obtained using the ES method suggest that experimental settings and related errors should be considered when constructing a robust thermodynamic equilibrium model based on the experimental results.

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16

To test reproducibility, anhydrous melt–olivine equilibrium temperatures calculated for a given pressure and melt and olivine compositions using the optimal model are compared with the results of the melting experiments of peridotite at 1–3 GPa of Hirose and Kushiro (1993) in Supplementary Material 4. The results calculated using the optimal model are also compared with those calculated using the olivine–anhydrous melt thermometry of Beattie (1996) in Supplementary Material 4. The experimental results of Hirose and Kushiro (1993) have not been used for parameter calibration and are suitable to test the reproducibility of our optimal model under the anhydrous condition. The olivine–melt thermometry of Beattie (1996) accurately reproduces olivine– anhydrous melt equilibrium temperatures in the

oo f

pressure range of 0.0001 to 15.5 GPa (Putirka, 2007) and is suitable for testing consistency between models for the anhydrous condition. Overall, the equilibrium temperatures derived using the optimal model are consistent with the experimental temperatures and with those

pr

calculated using the thermometry of Beattie (1996). However, the calculation results for 3

e-

GPa exhibit systematically lower temperatures than those of the experimental results and those derived using the approach of Beattie (1996), indicating that the model cannot be

Pr

extrapolated with respect to pressure; the pressure range of our dataset is 0.8–2.5 GPa. To demonstrate the effects of water on liquidus depression, the olivine liquidus depression

al

calculated using the lowest-CVE model (Eq. 13) as a function of the total water concentration of a melt is presented in Fig. 4. The experimental results plotted in this figure (Tatsumi, 1982;

rn

Pichavant et al., 2002; Almeev et al., 2007; Médard and Grove, 2008) are from experiments

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that were not used for regression and model selection in the present study. As pressure has little influence on the effect of water on liquidus depression (Médard and Grove, 2008), experiments conducted at the pressure range corresponding to the extrapolation of our data set (<0.8 GPa) are also plotted in Fig. 4 for comparison. Liquidus depression lines calculated using the empirical models of Falloon and Danyushevsky (2000), Sugawara (2000), Ariskin and Barmina (2004; “COMAGMAT”), Almeev et al. (2007), and Médard and Grove (2008), and the thermodynamic model pMELTS (Ghiorso et al., 2002), are also plotted for comparison. The liquidus depression was calculated as the difference between experimental wet liquidus and dry liquidus temperatures. The result obtained using a linear model (i.e., calculated using ,

,

,

, and the linear term (

,

,

,

) of the water-related terms) is

also shown in Fig. 4. Calculations were performed using the same melt compositions as those of the plotted experimental results. Representative values of each experiment were used for

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17

pressure and olivine composition in the calculations. For pMELTS and linear- model calculations, the melt composition reported by Pichavant et al. (2002) was used. Although our thermodynamic formulation was a function of melt composition, olivine composition, pressure, and temperature, the liquidus calculations were simplified using fixed olivine composition and pressure to demonstrate the effects of water in our models and to provide a comparison with independent results from previously reported experimental results and models for liquidus depression. Fig. 4 shows that our thermodynamic calculations for the hydrous-melt–olivine equilibrium using the optimal model reproduce the independent experimental results well at all water concentrations; the slope of the liquidus depression

oo f

curve decreases with increasing H2 O content. In contrast, the calculation using only the linear term to represent the effect of water failed to reproduce the non- linear relationship between the total water concentration and olivine liquidus temperature, although it is the fifth-best

pr

model overall (Table 1; Fig. 3). Liquidus temperature decreases monotonically in the result

Pr

4.2 Thermodynamic implications

e-

obtained by the linear model.

Ghiorso et al. (1983) argued that H2 O does not behave like a regular solution component

al

and proposed an additional term to describe the relationship between the total water concentration and the Gibbs free energy of silicate melt. The equation proposed by Ghiorso et

rn

al. (1983) was used for the MELTS (Ghiorso and Sack, 1995; Gualda et al., 2012) and

Jo u

pMELTS (Ghiorso et al., 2002) software programs for the thermodynamic modeling of phase equilibria in magmatic systems. However, MELTS and pMELTS failed to reproduce the relationship between the total water concentration of melt and the liquidus temperature of olivine (Almeev et al., 2007; Médard and Grove, 2008). In addition, some critical factors, such as the dissociation of water in silicate melt (Stolper, 1982) and its interaction with other oxide end- member components in the melt (e.g., Hunt and Manning, 2012), were not considered in the MELTS and pMELTS models. Putirka et al. (2007) argued that the total water concentration of melt affects the

(logarithm of distribution coefficient for Mg

between olivine and melt) for olivine–melt, and therefore those authors parameterized for olivine–melt as a function of temperature, pressure, water, and the Na2 O and K2 O concentrations of melt. Putirka et al. (2007) parameterized the effect of water on

as a

function of the total water concentration of melt. Carmichael (2002, 2004) parameterized the composition-related non- ideal excess energy of hydrous melt as a linear function of

,

Journal Pre-proof , and total alkalines (

18

). Hirschmann et al. (1999, 2009) and Aubaud et al.

(2004) modeled the effect of dissolved water in melt on the peridotite solidus by considering dilution by dissolved water through the

term. In the present study, we adopted a

model-selection method to formulate a model of the non- ideality of hydrous melt. The result produced by the model-selection approach shows that

(linear term) and

(squared term) were retained in the selection process; consequently, we infer that those two terms are important for describing the relationship between the total water concentration and the molar Gibbs free energy of the silicate melt, in addition to the

oo f

term and seven major elements including alkalis (SiO 2 , Al2 O3 , FeO, MgO, CaO, Na2 O, and K2 O).

The optimal model equation selected using the ES approach (Eq. 13) is considered to

pr

describe the underlying melt structure that explains the data; i.e., the relationship between free energy and the microscopic chemical features of hydrous melt. As shown above,

and the amount of bridging

Pr

(squared term) represents

e-

(linear term) represents the effect of the total water concentration, and

oxygen consumed by the dissolution of water. To demonstrate the contribution of the

al

polynomial function to hydrous- melt–olivine equilibrium and to examine the relationship between thermodynamic equilibrium and the microscopic chemical features of hydrous melt,

for olivine–melt was calculated using Eq. 13 as

Jo u

Fig. 5.

values are presented as a function of the total water concentration of melt in

rn

calculated

. The

temperature was varied from 1100 to 1500 °C, the pressure was fixed at 1 GPa, and melt and olivine compositions were fixed as the mean values of the entire experimental dataset. values of the experimental dataset are also plotted in Fig. 5. The experimental

values

exhibit wide variations at lower temperatures (<1200 °C), possibly related to the difficulties in performing low-temperature near-solidus hydrous experiments (e.g., Green et al., 2012). Although some of the experimental between experimental

values exhibit scatter, a parabolic relationship

and the total water concentration of melt is observed in Fig. 5 and

is particularly recognizable at temperatures of 1200–1300 °C. Fig. 5 indicates that our optimal model, which uses both the

and

terms for the effect of

water, captures the parabolic structure of the variation in concentration. Both the experimental and calculated

with changing total water

values decrease with an increasing

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19

total water concentration of melt from 0 to 5 wt%, and then increase with a further increase in the total water concentration to 9 wt%. In contrast, the linear model, in which only of the water-related terms was used, does not reproduce the relationship between and the total water concentration of melt;

decreases monotonically with increasing

total water concentration for the result obtained by the linear model. for anhydrous- melt–olivine depends on the melt composition and, consequently, on the microstructure of melt, and it exhibits a parabolic relationship as a function of NBO/T (nonbridging oxygen per tetrahedrally coordinated cations) (Kushiro and Mysen, 2002; Mibe et al., 2006; Filiberto and Dasgupta, 2011). Kushiro and Mysen (2002) attributed the

oo f

parabolic relationship to the change in the coordination states of Mg in melt associated with depolymerization (Fiske and Stebbins, 1994). Mg2+ in polymerized melts (low NBO/T) is

pr

predominantly four coordinated, and it changes to six coordination with an increase in NBO/T. As a result, Mg in polymerized melts tends to enter olivine in which Mg is s ix

e-

coordinated, and with an increase in six-coordinated Mg in the melt, more Mg enters the melt (Kushiro and Mysen, 2002; Mysen and Dubinsky, 2004; Mysen and Shang, 2005). With

Pr

respect to hydrous melt, Putirka et al. (2007) showed that

is sensitive to the water

concentration of melt. Pu et al. (2017) attributed this dependence of

to the coordination

al

environments of Mg. Other studies have suggested that the reaction between Mg and OH in

rn

melt could affect the partitioning behavior of Mg and mafic minerals (Crabtree and Lange, 2011; Frey and Lange, 2011; Waters and Lange, 2013). NBO/T changes systematically with

Jo u

the total water concentration of melt (e.g., Mysen, 2014). Therefore, the proportions of the different coordination states of Mg could be a function of the total water concentration of melt, and, consequently, melt. The minimum

could also be a function of the total water concentration of

value of mantle peridotite partial melt is located at NBO/T > 2.5

(Kushiro and Mysen, 2002; Mibe et al., 2006), which is more depolymerized than NBO/T values for anhydrous-based compositions of the input data of this study (0.15 to 1.88, calculated according to Jaeger and Drake, 2000). As dissolved water increases NBO/T, the actual NBO/T values of hydrous melts in the input data could be higher than the anhydrousbased calculated values. The minimum

value at ~5 wt% H2 O (Fig. 5) might represent

the NBO/T value corresponding to the maximum proportion of six-coordinated Mg in hydrous melt. As explained in Section 3.1, the amount of consumed oxygen associated with dissociation of water in melt is proportional to

(Stolper, 1982). Accordingly,

Journal Pre-proof we infer that

20

represents the structural change of the silicate melt associated

with the dissociation of water. The relationship between

, the total water concentration, and the microscopic structure

of melt could be further investigated with a detailed experimental analysis of the microscopic structures of Mg-bearing hydrous melts. Furthermore, first-principles calculations (e.g., De Koker and Stixrude, 2009; Taniuchi and Tsuchiya, 2018) may provide quantitative understandings of the detailed relationship between, and the mechanism connecting, thermodynamic properties and microscopic structures, potentially including the water

oo f

speciation, coordination environment, and NBO/T of hydrous silicate melt.

4.3 Machine learning for petrological thermodynamic modeling

pr

We constructed a thermodynamic model based on the selection of important variables by using data for melting experiments. This type of data-driven approach is becoming

e-

increasingly popular in various fields of natural science (Hey et al., 2009; Igarashi et al., 2016; Ueki et al., 2018). Specifically, we used the ES method to select the important

Pr

variables and the CV technique to quantitatively validate the predictive capacity of various equations. On the basis of the predictive capacities of the model equations, we were able to

al

obtain a generalized and robust equation for the hydrous-melt–olivine equilibrium, as well as

rn

to capture the underlying thermodynamic relationship, such as the importance of variables, the most informative combination of variables (Figs. 2 and 3), and the reproducibility of

Jo u

observed data (Appendix B). The proposed method for model-selection using heterogeneous input data could be applied to other petrological and geochemical problems, such as the modeling of metamorphic reactions and of high-pressure experimental data corresponding to lower-mantle conditions.

We simplified the thermodynamic equation and the model settings, and we will expand our modeling approach in the future for application to multi-component and more complex natural magmatic systems. Although we considered a maximum of 10 possible terms, the number could be increased, and the use of non- linear terms could also be further explored. For example, if we consider the full equation of the symmetric regular solution (Eq. 5), 245 combinations of variables must be evaluated because the maximum number of unknown thermodynamic parameters (W) is 45. The maximum number of unknown thermodynamic parameters increases as more complex equations (e.g., asymmetric regular solutions, or temperature- or pressure-dependent non-ideality) are introduced. Furthermore, other aspects,

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21

such as the influence of magma oxygen fugacity (e.g., Moore et al., 1995; Toplis and Carroll, 1995), and of trace components in olivine (e.g., Ca; Libourel, 1999; Aubaud et al., 2004; Gavrilenko et al., 2016), could be important for application to natural magmatic systems. In such cases, the potentially large number of combinations of variables and the size of the input dataset will need to be considered when employing the ES method (i.e., the problem of combinatorial explosion). For such a case, an alternative would be to use a different statistical treatment, such as ℓ 1-norm constraints (e.g., Tibshirani, 1996) or Akaike’s Information Criterion (e.g., Akaike, 1974), to obtain optimal model equations.

oo f

As we used experiments based exclusively on mantle conditions, the investigated compositional range was limited to mafic primary magma compositions. Evolved melt compositions corresponding to natural arc magmas should be incorporated in future modeling

pr

by expanding the compositional range of the input dataset. Also, our present model considers only the forsterite (Mg2 SiO4 ) component in olivine, meaning that we have fewer equations to

e-

constrain genetic or equilibrium conditions based on the composition of natural basaltic

Pr

samples using Eq. 13. By expanding our present model to include the fayalite (Fe2 SiO4 ) endmember component in olivine, we could use the Fe/Mg exchange reaction to constrain the temperature and melt water concentration simultaneously from the magma and olivine

al

compositions. Our results indicate that the ML-based model-selection approach is a robust

rn

and useful approach for constructing a thermodynamic petrological model. The model construction can be expanded to include multiple phases, and could be incorporated into a

Jo u

total energy minimization and bulk conservation algorithm for thermodynamic calculation of multi-component systems (Ueki and Iwamori, 2013, 2014) to model multi-component and multi-phase natural magmatic systems.

5 Summary We presented a novel method and outcome for thermodynamic modeling of the hydrous basaltic melt–olivine equilibrium. Specifically, we applied a machine-learning-based modelselection approach, the exhaustive variable selection method, to formulate a robust and generalized thermodynamic model that describes the underlying thermodynamic relationship of the hydrous basaltic melt–olivine equilibrium. A linear term and a squared term of the total water concentration of melt (

and

) were found to be significant

explanatory variables in the optimal model of the hydrous- melt–olivine equilibrium. This relationship may be attributed in physical terms to the microstructural changes that occur as a

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22

result of the dissociation of water in silicate melt. Calculations using the constructed optimal model reproduced the experimentally determined effects of water on the liquidus temperature of olivine and the distribution coefficient for Mg between olivine and melt. Our results indicate that ML is a useful approach for constructing a petrological numerical model and for analyzing experimental results characterized by high-dimensional and complex data, from which we are able to extract fundamental information, in the present case regarding hydrousmelt–olivine thermodynamics.

oo f

Acknowledgments

We thank Keith Putirka for a constructive and detailed review, and Kei Hirose for careful editorial handling of the manuscript. Maurizio Petrelli and an anonymous reviewer are

pr

thanked for useful and detailed comments on an earlier version of the manuscript. We thank

e-

Taku Tsuchiya, Ryuichi Nomura, and Atsushi Yasumoto for insightful discussions. We gratefully acknowledge grants from The Joint Usage/Research Center programs of the

Pr

Earthquake Research Institute, University of Tokyo, Japan, 2015-B-04 (Geochemical Data Analysis Using Machine Learning) and 2018-B-01 (Data-driven Geoscience: Application to

al

Dynamics in Mobile Belts). K.U. was supported by Japan Society for the Promotion of Science KAKENHI Grant Numbers JP15H05833, JP17H02063, and JP19K04026, and T.K.

rn

was supported by Japan Society for the Promotion of Science KAKENHI Grant numbers

Jo u

JP15K20864 and JP25120005 and by Japan Science and Technology Agency PRESTO Grant Number JPMJPR1676. S.A. and T.K. were supported by Japan Science and Technology Agency CREST Grant No. JPMJCR1761. Some of the experimental data used for the modelselection and parameter optimization were obtained from the Library of Experimental Phase Relations open database of melting experiments (Hirschmann et al., 2008). The R source code and dataset used in this study are provided in Appendix C.

Appendix A: Molar Gibbs free energy of pure solid and melt end-member components Following the formulation by Ueki and Iwamori (2013) and Putirka (2017), the Gibbs free energy of the melt–olivine (ol) reaction is written as * (A1)

+

Journal Pre-proof where

, and

23

are the temperature, enthalpy, and entropy of the fusion of solid

end- member component j at 1 bar, respectively.

denotes the reaction quotient (e.g., Putirka,

2017). When an equilibrium state is achieved (

),

the activity coefficient of end-member component i.

is equal to

, where

is

denotes the difference in molar

Gibbs free energy between the melt and the corresponding solid end- member components. is the difference in molar specific heat between the solid end- member component j and the melt end- member component i, and is written as

.

is the

difference in molar volume between solid end- member component j and melt end- member .

oo f

component i, written as

Although the P–V relation of silicate melt exhibits a non- linear relationship (e.g., Ueki and Iwamori, 2016),

values during melting within the mantle can be assumed to be constant

pr

at the pressures of interest in this study, regardless of temperature (Ueki and Iwamori, 2014).

e-

In addition, the total volume and compressibility of silicate melt can be treated as linear combinations of these parameters of oxide end- member components (Lange and Carmichael, was simply

Pr

1987, 1990) and the H2 O end- member component of the melt. Therefore,

derived by a linear regression between the partial molar volumes of silicate melt and olivine using the equation of state for hydrous melt (Ueki and Iwamori, 2016) and that for olivine used in this study is a function of pressure [P (bar)], as follows:

rn

al

(Berman, 1988).

As noted in our previous study (Ueki and Iwamori, 2014), only a weak pressure dependence ) is obtained.

Jo u

(

Previous experimental studies have shown that excess specific heat related to the mixing of end- member components, including water, in silicate melt is negligible (e.g., Bouhifd et al., 2006; Di Genova et al., 2014). Ueki and Iwamori (2013, 2014) reported that the value of during melting can be assumed to be constant at the temperatures and pressures of interest in the present study. Values of experimental results, with

,

,

, and

= 2174 K (Sugawara, 2005),

and Bottinga, 1985; Gillet et al., 1991; Médard and Grove, 2008), (Sugawara, 2005), and Ghiorso (1989) was used for

were taken from previous = 100 (J/mol-K) (Richet = 138.90 (kJ/mol)

= 63.89 (J/K-mol) (Sugawara, 2005). The model of Sack and .

To summarize the temperature- and pressure-dependent and melt-composition- and olivine-composition- independent terms, we introduced a term

as

Journal Pre-proof *

+

. (A2)

, Eq. A1 can be rearranged into Eq. 2 of the main text.

Jo u

rn

al

Pr

e-

pr

oo f

By introducing the term

24

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25

al

Pr

e-

pr

oo f

Appendix B: Residuals vs total water concentration (mole fraction) of the top 10 models

validation data; i.e.,

rn

Fig. B1. Residuals (difference between values predicted using the constructed model and the in Eq. 12) obtained during CV analysis plotted against

Jo u

the total water concentration in melt (

). Residuals were calculated using Eq. 10

(difference between the right- and left- hand sides of Eq. 10). Each circle represents an independent experimental result.

Conflict of Interest: The authors declare that they have no conflicts of interest.

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Rank

WSiO2

WAl2 O3

WFeO

WMgO

WCaO

WNa2 O

WK2 O

WH2 O

WH2 O2

2319

1

165702

115981

198584

142204

164325

318651

240832

123461

189732

2326

2

165676

116006

198491

142368

164235

318708

241339

132531

168468

2330

3

164845

118056

198309

143594

164276

319304

243483

195683

2357

4

166833

113856

199559

139773

164586

317045

234356

2376

5

162164

126073

200690

143310

167548

319937

235383

2422

6

172617

117492

199796

134838

152627

299425

123550

181418

2432

7

172620

117486

199815

134807

152651

299422

121646

185885

2436

8

171770

119765

199627

136087

152583

299892

191284

2454

9

173500

115469

200762

132621

153283

298405

2470

10

169080

127119

201786

136056

155965

301073

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CVE (J)

Table 1

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ePr al rn Jo u

−2158 −15638

497525

27805

157992

453 −14409

488172 156599

Parameters (J/mol) and cross-validation error (CVE) values of the top 10 (lowest CVE) model equations.

WH2 O0.5

27951

37

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pr

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Fig. 1

Variations in temperature and pressure (a) and in temperature and the total water

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concentration of melt (b) for the dataset of experimental results used in this study. The anhydrous-melting experimental results are those reported by Ueki and Iwamori (2014) (i.e.,

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Takahashi, 1980; Kinzler and Grove, 1992; Falloon et al., 1997, 1999, 2001; Kinzler, 1997;

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Kogiso et al., 1998; Robinson et al., 1998; Wagner and Grove, 1998; Falloon and Danyushevsky, 2000; Pickering-Witter and Johnston, 2000; Schwab and Johnston, 2001; Longhi, 2002; McDade et al., 2003; Wasylenki et al., 2003; Laporte et al., 2004; Parman and Grove, 2004), and the hydrous-melting experimental results are those reported by Hirose and Kawamoto (1995), Hirose (1997), Gaetani and Grove (1998), Falloon and Danyushevsky (2000), Laporte et al. (2004), Parman and Grove (2004), and Mitchell and Grove (2015).

38

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Fig. 2

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Bivariate plot of cross- validation error (CVE) values versus the number of variables used in the model equations. Equation types are indicated by circles and colors: the filled red circle

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represents the ideal model; filled light- gray circles represent models using water-related terms exclusively; filled dark-gray circles represent models using water-related terms and

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K2O exclusively; filled dark-blue circles represent models using water- and alkaline (Na2 O and K2 O)-related terms exclusively; filled yellow circles represent models using the SiO 2 , FeO, and MgO terms; open black and open blue circles represent models using and not using the SiO 2 term, respectively; and open green circles represent models not using water-related terms.

39

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Fig. 3

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CVE values, variables, and parameters of the top 100 (lowest CVE) equations. Each row

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represents a single equation. The CVE of each equation is shown at the far right. Parameter values in each equation are represented by color according to the scale shown. A blank (white

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space) in the matrix indicates that a particular variable was not used in that equation.

40

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Fig. 4

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Calculated and experimentally determined olivine liquidus depression as a function of the total water concentration of melt. Liquidus depression was calculated as the difference

rn

between the experimental wet liquidus temperature and the dry liquidus temperature. The black region in the diagram represents the result derived using the optimal model (Eq. 13).

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Open and closed symbols represent data for just above and just below olivine liquidus conditions, respectively, obtained from the experiments of Tatsumi (1982), Pichavant et al. (2002), Almeev et al. (2007), and Médard and Grove (2008). The experimental results plotted in this figure were obtained from experiments that did not satisfy our data compilation criteria or from studies that lacked the analytical data required for model-selection and parameter optimization; consequently, these data were not used for these tasks in the present study. Calculations were performed using the same melt compositions as those of the plotted experimental results. Representative values of each experiment were used for pressure and olivine composition in the calculations. Therefore, the temperature range of the calculated liquidus depression in Fig. 4 represents the variations in the melt and olivine compositions and pressure conditions of the experiments. Liquidus depression lines were calculated using the empirical models of Falloon and Danyushevsky (2000), Sugawara (2000), Ariskin and Barmina (2004; “COMAGMAT”), Almeev et al. (2007), Médard and Grove (2008), and the

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thermodynamic model pMELTS (Ghiorso et al., 2002). The result obtained using the linear model [a model containing only the linear term of the water-related terms (in addition to the element-oxide terms); see text for details] is also shown. For pMELTS and linear- model calculations, the melt composition reported by Pichavant et al. (2002) were used. The results obtained using previous models are indicated by the following abbreviations: Av, Almeev et al. (2007); Ct, COMAGMAT; FD, Fallon and Danyushevsky (2000); MG, Médard and

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rn

al

Pr

e-

pr

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Grove (2008); pM, pMELTS; Sg, Sugawara (2000).

42

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Fig. 5

Calculated and experimentally determined values of the distribution coefficient

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for melt–olivine plotted against the total water concentration of melt. The solid lines show the results obtained using the optimal model, and the dashed lines show the

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results obtained using the linear model (i.e., the model using only the linear term of the

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water-related terms in addition to the element-oxide terms; see text for details). For melt and olivine compositions, the averaged values of the experimental dataset were used. The pressure was fixed at 1 GPa. Also shown are

values calculated from the experimental

results that were used for model-selection and parameter optimization.

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Declaration of interests

☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:

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Author Statement Kenta Ueki: Conceptualization, Methodology, Software, Formal analysis, Investigation, Data Curation, Writing - Original Draft Tatsu Kuwatani: Conceptualization, Methodology, Writing Original Draft Atsushi Okamoto: Supervision Shotaro Akaho: Software, Supervision Hikaru

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e-

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Iwamori: Writing - Review & Editing, Supervision

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Highlights 

A machine learning method for petrological thermodynamic modeling is presented



The method selects key variables from experimental data on hydrous-melt systems



A robust thermodynamic model of the hydrous-melt–olivine equilibrium is constructed Model calculations reproduce measured effect of H2 O on olivine liquidus temperature



Improved understanding of the underlying mechanisms of hydrous-melt systems

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