# A general machine learning model of aluminosilicate melt viscosity and its application to the surface properties of dry lava planets

Charles Le Losq, Clément Ferraina, Paolo A. Sossi, Charles-Édouard Boukaré

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Charles Le Losq, Clément Ferraina, Paolo A. Sossi, Charles-Édouard Boukaré. A general machine learning model of aluminosilicate melt viscosity and its application to the surface properties of dry lava planets. Earth and Planetary Science Letters, 2025, 656, 10.1016/j.epsl.2025.119287. insu-05080911

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<figure><img src="image_1.png" /></figure>
# A general machine learning model of aluminosilicate melt viscosity and its application to the surface properties of dry lava planets

Charles Le Losq<sup>a,b, ID,*</sup>, Clément Ferraina<sup>a, ID</sup>, Paolo A. Sossi<sup>c, ID</sup>, Charles-Édouard Boukaré<sup>d, ID</sup>

<figure><img src="image_2.png" /></figure>

## ARTICLE INFO

Editor: O. Mousis

Dataset link: https://zenodo.org/records/14673467

**Keywords:**

Magma
Viscosity
Pressure
Machine learning
Magma ocean
Exoplanet
K2-141 b

## ABSTRACT

Ultra-short-period exoplanets like K2-141 b likely have magma oceans on their dayside, which play a critical role in redistributing heat within the planet. This could lead to a warm nightside surface, measurable by the James Webb Space Telescope, offering insights into the planet's structure. Accurate models of properties like viscosity, which can vary by orders of magnitude, are essential for such studies.

We present a new model for predicting molten magma viscosity, applicable in diverse scenarios, including magma oceans on lava planets. Using a database of 28,898 viscosity laboratory measurements on phospho-aluminosilicate melts, spanning superliquidus to undercooled temperatures and pressures up to 30 GPa, we trained a greybox artificial neural network, refined by a Gaussian process. This model achieves high predictive accuracy (RMSE ≈ 0.4 log₁₀ Pa·s) and can handle compositions from SiO₂ to multicomponent magmatic and industrial glasses, accounting for pressure effects up to 30 GPa for compositions such as peridotite.

Applying this model, we calculated the viscosity of K2-141 b's magma ocean under different compositions. Phase diagram calculations suggest that the dayside is fully molten, with extreme temperatures primarily controlling viscosity. Even in the absence of major volatiles (H, C, N), a tenuous rock-vapour atmosphere (0.1 bar) might exist around a 40° radius from the substellar point. At higher longitudes, atmospheric pressure drops, and by 90°, magma viscosity rapidly increases as solidification occurs. The nightside surface is likely solid, but previously estimated surface temperatures above 400 K imply a partly molten mantle, supporting geothermal flux through vertical convection.

## 1. Introduction

Magma oceans are fundamental processes that shape the internal structure and secondary atmospheres of rocky planets (Elkins-Tanton, 2012; Sossi et al., 2020; Gaillard et al., 2022). Although much of our understanding is theoretical, spectroscopic observations of ultra-short period (USP) exoplanets (e.g. 55 Cnc e and K2-141b; Chao et al., 2021) may provide insights into their nature and longevity. Indeed, these “lava planets” are thought to host vast magma oceans on their dayside because of intense stellar irradiation producing dayside temperatures well above 2000 K (Chao et al., 2021). Outgassing from magma oceans is expected to shape the atmospheric composition of lava planets, introducing elements such as H, C, N, Si, Fe, Na, and K (e.g. Charnoz et al., 2023). Spectroscopic observations of these atmospheres may provide

indirect evidence of the presence and composition of magma oceans, provided there is a robust understanding of the interactions between the molten mantle and the atmosphere. Integrating fluid dynamics simulations (e.g. Meier et al., 2023; Boukaré et al., 2023) with outgassing and atmospheric models (e.g. Charnoz et al., 2023; Falco et al., 2024) offers promising approaches to this challenge. However, even with new spectroscopic data (e.g. Hu et al., 2024) from MIRI and NIRSpec on the James Webb Space Telescope, interpreting such data in the context of geodynamic and geochemical processes remains difficult.

This is because information as to the key physical parameters that dictate the behavior of magma oceans is currently missing. Chief among these is magma viscosity (η, Pa·s), which controls melt mobility and elemental diffusion timescales, and thus the vigor of thermal convection and magma outgassing (Solomatov, 2000; Huang et al., 2024). Magma

<footnote><sup>a</sup> Université Paris Cité, Institut de physique du globe de Paris, CNRS, F-75005 Paris, France</footnote>

<footnote><sup>b</sup> Institut Universitaire de France, Paris, France</footnote>

<footnote><sup>c</sup> Institute of Geochemistry and Petrology, ETH Zürich, Zürich, Switzerland</footnote>

<footnote><sup>d</sup> York University, York, Canada</footnote>

<footnote>* Corresponding author.
E-mail address: lelosq@ipgp.fr (C. Le Losq).</footnote>
viscosity varies non-linearly over more than 15 orders of magnitude with temperature, melt chemical composition (including volatile concentration), crystal and bubble fractions, and pressure (see for a review Russell et al., 2022), posing a significant challenge for accurate calculations.

Models have been developed to calculate the viscosity of crystal- and bubble-free silicate melts (Russell et al., 2022), including empirical models (e.g. Shaw, 1972; Giordano et al., 2008), high accuracy models for specific compositions (e.g. Romine and Whittington, 2015; Russell et al., 2024), models based on theoretical frameworks (e.g. Le Losq and Neuville, 2017; Starodub et al., 2019), and machine learning (ML) (e.g. Le Losq et al., 2021; Langhammer et al., 2022; Cassar, 2023; Le Losq and Baldoni, 2023). Regardless of the method, most existing models focus on predicting the viscosity of crystal-free melts at room pressure over a restrained compositional domain (e.g. industrial or geologic compositions).

No existing model can predict magma viscosity across the wide range of melt compositions (X), temperatures (T), and pressures (P) found on USP planets, where compositions could range from Earth-like to refractory or even carbon-rich (Léger et al., 2011; Madhusudhan et al., 2012; Zieba et al., 2022), and where temperatures vary greatly between day- and night sides (e.g. Léger et al., 2011; Nguyen et al., 2020). Besides, P may exceed those found on Earth given that super-Earths can have masses up to 10 times greater than Earth's. Predicting the properties of molten rocks under such conditions requires a new generation of models that make the most of existing data on alumino-silicate (sensu lato) compositions, as these are the primary constituents of most magmas.

In this study, we present a new database of viscosity measurements as a function of T and P for melts with diverse compositions in the system SiO₂-FeO-Fe₂O₃-TiO₂-Al₂O₃-MnO-MgO-CaO-Na₂O-K₂O-P₂O₅-H₂O. We benchmark ML models to predict magma viscosity and propose a new model combining a Gaussian process with an artificial neural network to calculate melt viscosity and associated uncertainties. We then apply this model for the calculation of the rheology of the magma ocean at the surface of K2-141 b (Malavolta et al., 2018; Barragán et al., 2018), a "dry" USP planet (i.e. a planet with no atmosphere or a thin steamless atmosphere). We further assess the impact of internal temperature on the nightside surface, potentially providing insight on the interior state of dry lava planets from nightside temperature measurements.

## 2. Materials and methods

In this study, we aim to perform the following task: given T, P and X, an ML model G with adjustable parameters ψ will compute log₁₀η as:

$$ \log_{10} \eta = G(T, P, X, \psi) \qquad (1) $$

The parameters ψ are optimized during a training phase by solving a least-squares regression problem. Using laboratory measurements of η at known T, P, and X, ψ are adjusted via gradient descent to minimize the root mean square error (RMSE) between predicted and measured values of η. Training a ML model requires a dataset prepared in advance. Furthermore, different models may be suited to solving this problem. The following subsections present the dataset, along with the training and benchmarking strategies for several ML models.

### 2.1. Dataset

We began with the hand-curated i-Melt database (Le Losq and Baldoni, 2023), which includes η versus T measurements for 790 melt compositions in the system Na₂O-K₂O-CaO-MgO-Al₂O₃-SiO₂. This database was expanded through a comprehensive literature survey targeting multi-component compositions that include additional oxides such as FeO, Fe₂O₃, TiO₂, MnO, P₂O₅ and H₂O. As of 25/09/2024, the hand-curated database includes 15,440 viscosity measurements ranging from 10⁻³ to 10¹⁵ Pa·s for 2,155 melt compositions differing by at least 0.1

mol% at room pressure, and 1,227 viscosity measurements for 243 melt compositions at high pressures, up to 30 GPa. This compilation incorporates data from 245 publications.

Certain studies were excluded based on the identification of aberrant data points that deviated significantly from established trends. When available, the proportions of iron as ferrous and ferric were compiled directly from the source publications. When not provided, the Fe³⁺/Fe²⁺ ratios were calculated using the Borisov et al. (2018) model. For studies that do not provide estimates of oxygen fugacity, we assumed that measurements were conducted in air (log fO₂ = -0.68). The Borisov et al. (2018) model was selected for its simplicity and broad compositional applicability, although its use is not recommended for melts with SiO₂ content above 76 mol%. This limitation has a minimal impact, as iron-bearing felsic melts constitute only a small fraction of the database, and for many such compositions, experimental data on ferrous and ferric iron fractions were available.

The following features thus are available in the database: the molar proportions of SiO₂, TiO₂, Al₂O₃, FeO, Fe₂O₃, MnO, MgO, CaO, Na₂O, K₂O, P₂O₅, H₂O, pressure in GPa, temperature in Kelvin, and viscosity in log₁₀ Pa·s. This hand-curated database is available at Ferraina et al. (2025).

We complemented this hand-curated database with additional data from SciGlass using GlassPy (Cassar, 2020), amounting to 12,231 data points from 3,591 compositions (different by at least 0.1 mol%) that are absent from the hand-curated database. Fig. 1 provides a visual summary of the entire database, in which the number of compositions including the various elements are visible, together with the viscosity versus temperature and pressure coverage.

### 2.2. Data preparation

We first separated the dataset in three training (80%), validation (10%) and testing (10%) subsets (see Appendix A for details). The training subset is used for training ML algorithms and the validation subset for tuning their hyperparameters (see below). The testing subset contains data not used during training and hyperparameter tuning, such that it allows evaluating the predictive uncertainties of the models.

T, P and the fractions of SiO₂, TiO₂, Al₂O₃, FeO, Fe₂O₃, MnO, MgO, CaO, Na₂O, K₂O, P₂O₅, and H₂O are features that are provided as inputs to the ML models (eq. (1)). In an attempt to further decrease predictive errors in the ML models, we tested adding new features (such as the glass optical basicity) but no significant improvements were observed. Therefore, for the sake of simplicity, we used only the features listed above.

A final data preparation step entailed scaling the data. For the four classical models (see below), we used a standard scaler, and for other custom models we used a custom scaling where features are 1000/T (K), P/30 (GPa), and oxide fractions.

### 2.3. Machine learning models

Different ML models are appropriate for performing the task described by equation (1). In this work, we benchmark six models, including four classic supervised regression models that are easy to implement using SciKit Learn (Pedregosa et al., 2011) and XGBoost (Chen and Guestrin, 2016):

* LR is a linear model, used as a baseline;

* SVM is a support vector machine model;

* XGB is the XGBoost (eXtreme Gradient Boosting) model;

* ANN is a feed-forward artificial neural network model.

These models are described in machine learning textbooks (Goodfellow et al., 2016; Murphy, 2012), to which we refer the readers for exhaustive descriptions. Briefly, LR establishes a linear relationship between inputs (T, P and X) and outputs (log₁₀η). SVM performs a similar task, but it
<figure><img src="image_3.png" /><figcaption>Fig. 1. (a) Bar plots showing the number of compositions including a particular oxide component in the database. (b) Viscosity versus temperature diagram showing all data from the full database. Data from the SciGlass database are shown in grey. Colored symbols indicate high pressure data.</figcaption></figure>

first maps the data into a higher-dimensional space using a mathematical technique known as the kernel trick. This mapping allows non-linear relationships in the original data to become linear in the transformed space, enabling SVM to effectively model non-linear relationships between inputs and outputs. XGB is a machine learning algorithm based on gradient-boosted decision trees, designed for efficient and scalable implementation (Chen and Guestrin, 2016). It operates by sequentially building an ensemble of decision trees, where each tree attempts to minimize the residual errors of the previous trees. XGBoost incorporates several enhancements, including regularization techniques to reduce overfitting, a scalable tree construction algorithm for improved computational efficiency, and the ability to handle missing data. These features make XGBoost highly effective for regression tasks. Finally, the ANN model consists of interconnected nodes (a.k.a. neurons) that 'react' differently to their inputs, thanks to non-linear activation functions (e.g., tanh) used to modulate the response of each node to its inputs. This is a foundation structure used in deep learning models, including those for silicate melt viscosity predictions (Cassar, 2023; Langhammer et al., 2022; Le Losq et al., 2021; Le Losq and Baldoni, 2023).

Then, we test a greybox artificial neural network (hereafter called Greybox ANN) model that embeds the Vogel-Tammann-Fulcher equation (Russell et al., 2022):

$$ \log \eta = A + \frac{B}{T-C}, \qquad (2) $$

with T the temperature in K, A a common adjustable parameter (Russell et al., 2022), and B and C adjustable parameters that depend on melt composition X and pressure P. The artificial neural network predicts A, B and C given P and X, then equation (2) is used to calculate $log_{10} \eta$ as a function of T. The use of equation (2) constrains the $log_{10} \eta$ versus T dependence. This injects domain knowledge in the model and ensures reliable extrapolations at high temperatures, for instance in conditions pertinent for exoplanetary magma oceans. Other equations can

be used (e.g. see Le Losq et al., 2021; Le Losq and Baldoni, 2023). Here we chose the VFT equation as it is simple and its usefulness in viscosity modeling has been proven (Russell et al., 2022).

Finally, we tested a Gaussian process (GP) model. A GP is a collection of random variables, any finite number of which have consistent joint Gaussian distributions (Rasmussen and Williams, 2006). In other words, a GP places a probability density over functions. Here, we are interested in placing a probability density over the function G in equation (1). We do not know this function, but data can allow its probability density to be constrained. A GP is fully defined by its mean $m(x)$ and covariance (a.k.a. kernel) $k(x, x')$ functions:

$$ f(x) \sim GP(m(x), k(x, x')). \qquad (3) $$

$k(x, x')$ choice is critical as it embeds the mathematical formalism to describe how smooth and how variable is the expected function. For instance, a linear kernel implies that the target function is linear. For placing a probability distribution over non-linear functions, one will choose a non-linear kernel such as the squared exponential. Kernels can be multiplied and added, to create complex kernel functions that can reproduce very different shapes of functions. $k(x, x')$ usually has hyperparameters that will control the behavior of the function (smoothness, periodicity, etc.) and that can be adjusted by gradient-based optimization. For details regarding kernel design and selection, see Rasmussen and Williams (2006).

The mean function usually is chosen as a constant value equal to 0 in most applications. This is because the mean function is directly updated as we perform predictions on new points $x^*$. It thus usually does not have a strong influence on posterior predictions at those new values $x^*$. However, in the present case, we are dealing with a rather sparse data coverage of a large multidimensional problem. We further may want to extrapolate predictions at the high temperatures typical of exoplanetary magma oceans (2300-3000 K). In such a case, setting
$m(x) = 0$ is not the best choice because upon extrapolating, the GP will tend toward the initial value of $m(x)$, thus toward 0. To avoid this, prior information about the problem can be provided in the mean function, in the form of known analytical equations or of a model (Rasmussen and Williams, 2006; Ziatdinov et al., 2021; Zhang et al., 2022). Here, we use the previously described Greybox ANN as the GP mean function. The Greybox ANN predictions are prior estimates of $\log_{10} \eta$ given $T$, $P$ and $X$ that are updated by the GP at inference time using the kernel function $k(x, x')$ and the data, in order to provide posterior values of $\log_{10} \eta$ at the desired $P \cdot T \cdot X$ conditions.

The interest of GPs in this work is twofold: first, using appropriate kernels and mean functions, we ensure model smoothness and can leverage domain knowledge, and, second, the GP formalism allows calculating uncertainties on posterior predictions. Uncertainties can be obtained from artificial neural networks using methods such as Monte Carlo Dropout and Conformal predictions (e.g. see Le Losq and Baldoni, 2023), but those are not necessarily straightforward to implement. GP predictions are rooted in Bayesian statistics and the mathematical formalism is well established. GP thus can provide reliable prediction uncertainties, an important point for evaluating the quality of predictions.

## 2.4. Hyperparameter tuning and model training

To tune the hyperparameters of the models, we monitored the root-mean-squared errors (RMSEs) on the training and validation data subsets. The hyperparameters of the LR and SVM models were tuned using Scikit-Optimize, with a Bayesian optimization algorithm. For XGB hyperparameters, we used the Optuna framework. For the ANN model, manual tests and a grid search indicated that a simple yet effective architecture is a network composed of two layers of 200 ReLU activation units (Glorot et al., 2011). The Greybox ANN was implemented using PyTorch (Paszke et al., 2019), with the same architecture as the ANN model but GELU activation functions (Hendrycks and Gimpel, 2020), as in i-Melt (Le Losq and Baldoni, 2023). During training, early stopping and dropout (Srivastava et al., 2014) were adopted to prevent overfitting. The GP model is implemented using GPyTorch (Gardner et al., 2021). After testing different kernels, we selected a Matérn 5/2 kernel with 14 different lengthscales. The GP mean function is a pre-trained Greybox ANN model.

Models with optimal hyperparameters were trained on the combined training and validation data subsets, except for the Greybox ANN because early stopping uses the validation dataset during training. For the Greybox ANN and the GP, training may not always proceed well due to the stochasticity in artificial neural network training. This problem is solved by performing 100 different training runs and then selecting only the models for which the metrics were the lowest for the Greybox ANN. The library gpvisc (Le Losq et al., 2025) allows using trained Greybox ANN and GP models to perform predictions in new situations via Python coding or a more convenient GUI interface.

# 3. Results

## 3.1. Performance evaluation

Over the very broad range of compositions we investigate (Fig. 1), the different models predict the viscosity of melts with variable degrees of accuracy. Table 1 reports the RMSE, median absolute error (MAE), and coefficient of determination ($R^2$) for the different methods and data subsets (see also Supplementary Fig. 1 for histograms of the residuals of the different models). Errors on the test data subset are indicative of predictive errors because the models did not see melt compositions included in this data subset. We further report the metrics for the test data subset excluding data points from SciGlass (Test\* column) because those are known to be of variable quality (e.g. Wu et al., 2015). Our hand-curated database is of better quality, and error metrics on it may be more indicative of true model errors.

Metrics indicate that the LR model performs poorly, with RMSE higher than $1 \log_{10} Pa \cdot s$ (Table 1). This is expected because melt viscosity depends non-linearly on $T$, $P$ and $X$. Accordingly, non linear models such as SVM and XGB perform better, with test RMSE of 0.53 and 0.47 $1 \log_{10} Pa \cdot s$, respectively. XGB has an issue though: the training RMSE is significantly lower than the test RMSE. This indicates that XGB tends to overfit its training data.

The ANN performs well with minimal overfitting as indicated by close training-validation and testing RMSEs, respectively of 0.36 and $0.44 \log_{10} Pa \cdot s$. This indicates that artificial neural networks are a method of choice for the problem at hand; a result that is not unexpected when considering their success in previous studies (Le Losq et al., 2021; Langhammer et al., 2022; Le Losq and Baldoni, 2023; Cassar, 2023).

The Greybox ANN model also performs well with minimal overfitting, as indicated by good average metrics (Table 1) slightly higher than those of the blackbox ANN. This is probably due to a slight decrease in model flexibility resulting from the enforcement of the $T$ versus $\log_{10} \eta$ relationship through equation (2). However, this apparent disadvantage actually is a strength: the Greybox ANN model will always provide physically-realistic viscosity predictions as a function of temperature, even in the high temperature range of magma oceans. Indeed, in the Greybox ANN, as $T \to +\infty$, $\eta \approx -4.70$, a value in agreement with theory and other models (Russell et al., 2022, and references therein).

The GP model, based on the combination of the Greybox ANN with a Gaussian process, shows very good predictive performance (Table 1), with a test dataset RMSE of $0.44 \log_{10} Pa \cdot s$ ($0.32 \log_{10} Pa \cdot s$ when excluding SciGlass data). GP overfits very slightly its training dataset (RMSE = 0.35) compared to the Test dataset (RMSE = 0.44), but this remains acceptable. In addition, $\approx 95\%$ of the test data fall within the $95\%$ confidence intervals predicted by the GP model, validating the reliability of those intervals.

## 3.2. Model selection

All non-linear models show strong test metrics, with RMSE ranging between 0.44 and 0.53, MAE between 0.15 and 0.19, and $R^2$ between 0.986 and 0.990 (Table 1). The GP model appears to slightly outperform others based on these metrics, but the differences are small and require statistical validation to determine significance.

To do so, we employed two complementary statistical approaches. First, a permutation test evaluated whether differences in MAE between models could have arisen by chance. This non-parametric method, which is robust to non-normal distributions (Supplementary Fig. 1), works by randomly shuffling model residuals to generate a null distribution of metric differences. The p-value represents the proportion of permuted differences that are as extreme or more extreme than the observed difference. Secondly, we conducted a Wilcoxon test to evaluate whether the median of the differences between paired residuals from the different models differs significantly from zero. For those two tests, we used the test data subset to calculate model residuals.

The permutation test revealed that GP significantly outperforms LR, ANN, and Greybox ANN ($p < 0.05$) in terms of MAE, but not SVM and XGB ($p > 0.05$). According to the Wilcoxon test, GP's residuals are significantly different from those of LR, SVM, XGB and ANN ($p < 0.05$), but not from those of Greybox ANN. Combining the results of both tests validates the overall superior performance of the GP model.

While statistical tests suggest the GP model's superior performance, practical considerations also influence model selection. In particular, predicted viscosity should form smooth and continuous functions of intensive variables such as pressure and temperature. For instance, when asking predictions for anorthite melt viscosity as a function of temperature, LR, SVM and XGB show limitations (Fig. 2a). In contrast, ANN, Greybox ANN and GP make smooth, physically-consistent predictions. Similarly, those models yield consistent and smooth predictions of albite melt viscosity under pressure at various temperatures below 7 GPa,
<caption>**Table 1**</caption>

Root Mean Squared Error (RMSE), Median Absolute Error (MAE) and coefficient of determination R² of the different models. For final training of the algorithms after hyperparameter optimization, the training and validation data subsets were merged, except for the Greybox ANN where the validation subset is always used to perform early stopping. The RMSE on the merged training-validation datasets is shown here. The second Test* column indicates the metrics for the hand-curated part of the test data subset, excluding SciGlass values.

<table><thead><tr><th>Model</th><th>Metric</th><th>Training-Validation</th><th>Test</th><th>Test*</th></tr></thead><tbody><tr><td rowspan="3">Linear regression (LR)</td><td>RMSE</td><td>1.58</td><td>1.58</td><td>1.51</td></tr><tr><td>MAE</td><td>1.08</td><td>1.09</td><td>0.91</td></tr><tr><td>R<sup>2</sup></td><td>0.873</td><td>0.872</td><td>0.903</td></tr><tr><td rowspan="3">Support Vector Machine (SVM)</td><td>RMSE</td><td>0.46</td><td>0.53</td><td>0.36</td></tr><tr><td>MAE</td><td>0.09</td><td>0.15</td><td>0.13</td></tr><tr><td>R<sup>2</sup></td><td>0.989</td><td>0.986</td><td>0.994</td></tr><tr><td rowspan="3">XGBoost (XGB)</td><td>RMSE</td><td>0.20</td><td>0.47</td><td>0.37</td></tr><tr><td>MAE</td><td>0.07</td><td>0.15</td><td>0.14</td></tr><tr><td>R<sup>2</sup></td><td>0.998</td><td>0.989</td><td>0.994</td></tr><tr><td rowspan="3">Artificial Neural Network (ANN)</td><td>RMSE</td><td>0.36</td><td>0.44</td><td>0.35</td></tr><tr><td>MAE</td><td>0.14</td><td>0.18</td><td>0.16</td></tr><tr><td>R<sup>2</sup></td><td>0.994</td><td>0.990</td><td>0.995</td></tr><tr><td rowspan="3">Greybox Artificial Neural Network (Greybox ANN)</td><td>RMSE</td><td>0.44</td><td>0.48</td><td>0.33</td></tr><tr><td>MAE</td><td>0.18</td><td>0.19</td><td>0.18</td></tr><tr><td>R<sup>2</sup></td><td>0.994</td><td>0.988</td><td>0.995</td></tr><tr><td rowspan="3">Gaussian process (GP)</td><td>RMSE</td><td>0.35</td><td>0.44</td><td>0.32</td></tr><tr><td>MAE</td><td>0.12</td><td>0.15</td><td>0.14</td></tr><tr><td>R<sup>2</sup></td><td>0.994</td><td>0.990</td><td>0.996</td></tr></tbody></table>

<figure><img src="image_4.png" /><figcaption>**Fig. 2.** Viscosity against reciprocal temperature of an anorthite melt (a), and against pressure for albite melt (b). Symbols are measurements from the hand-curated database (Ferraina et al., 2025, see references reported herein) In (a), visual inspection of model predictions highlights that LR is not appropriate, and that XGB predictions may lack smoothness and be unreliable outside data-dense regions. SVM, ANN, Greybox ANN and GP show smooth, physically-consistent predictions, with Greybox ANN and GP being the closest to the data. SVM predictions significantly deviate from data at $\eta > 10^{10}$. In (b), only the predictions from the ANN, Greybox ANN and GP models, which present the best metrics (Table 1), are shown.</figcaption></figure>

within the data range (Fig. 2b). Beyond 7 GPa, where no data are available, divergence between models highlights the uncertainty of extrapolation.

Overall, the GP model is generally preferred due to its better metrics and built-in uncertainty quantification. However, the Greybox ANN model offers practical advantages in specific scenarios, such as computationally intensive fluid dynamics calculations where prediction speed is critical. Indeed, the Greybox ANN significantly outperforms the GP for 10-point inference tasks: ≈ 524 microseconds versus ≈ 294 milliseconds on a 13th Gen Intel® Core™ i9-13900K CPU, and ~12 milliseconds versus ~282 microseconds on a NVIDIA RTX A4500 GPU. Besides, in some rare cases, we observed that the Greybox ANN model offered more

physically-sound predictions, for instance when asking to predict $\eta$ at very high temperatures for water-rich melts (see example notebooks at Le Losq et al., 2025). Both models are available in the gpvisc library (Le Losq et al., 2025). The GP model will be used for subsequent analyses in this paper.

### 3.3. *High pressure predictions*

We now explore in more detail the high pressure predictions using the GP model. It reproduces well existing high and low melt viscosities data (RMSE = 0.34 log₁₀ Pa·s, Fig. 3a), including those of polymerized melts such as albite up to 7 GPa (Fig. 3b) as well as those of ultramafic
<figure><img src="image_5.png" /><figcaption>Fig. 3. (a) Calculated viscosity with the GP model versus measurements. The black dashed line represent the 1:1 correspondence, and the grey dashed lines the average model 95 % confidence interval (±0.80). (b), (c) and (d) Comparison between Gaussian process (GP) viscosity predictions and data available in Ferraina et al. (2025) for albite NaAlSi₃O₈, wollastonite CaSiO₃ and peridotite melts. The viscosity at high pressure and high temperature (η(T, P)) is represented relative to that at the same temperature and 1 bar (η(T, P = 1 bar)), calculated from an interpolative fit of experimental data with the Vogel-Tammann-Fulcher equation. In (b) to (d), 1-sigma uncertainties on model predictions are represented.</figcaption></figure>

compositions of significance for magma oceans such as CaSiO₃ and peridotite up to 30 GPa (Fig. 3b,c).

However, despite such promising results, we should bear in mind
that the high pressure dataset is small and sparse in terms of pressure
and compositional coverage. The good predictive performance of the
GP model for a composition such as peridotite does not imply that the
model can predict viscosity across a range of melt compositions up to
30 GPa.

3.4. Limitations

One main limitation of ML models is that they are interpolative by
nature. Therefore, when asked to predict values for inputs far outside the
range of the training data, predictions may be erroneous. At 1 bar, the
dataset spans a very broad compositional range, making it unlikely that
users would query compositions significantly outside the training data.
Furthermore, the use of equation (2) in the GP model imposes math-
ematical constraints on the temperature dependence of melt viscosity,
ensuring that predictions at temperatures outside the data range remain

physically consistent. However, users should still exercise caution when
querying compositions beyond the training range, as extrapolations may
be less reliable.

To illustrate what happens in such a case, we report in Fig. 4a the
viscosity of sodium silicate melts as a function of silica concentration.
Predictions from three different GP models are reported because differ-
ent training runs yield slightly different models (in particular, slightly
different Greybox ANNs), and we can leverage this to test the robustness
of model predictions. The three GP models have comparable test RMSE
values, equal to or lower than 0.45. In Fig. 4a, above 50 mol% silica,
where data are available, all models yield similar predictions within un-
certainties. Below 50 mol% SiO₂, their predictions largely diverge. This
indicates that the models are providing predictions for compositions out-
side their training range; they are working in an extrapolative regime
and their predictions become unreliable.

We perform the same test at high pressure for a peridotite melt at
T = 3000 K (Fig. 4b). Below 30 GPa, predictions from all models are
comparable but above 30 GPa, they are distinct. Again, this indicates
<figure><img src="image_6.png" /><figcaption>**Fig. 4.** (a) Viscosity of Na₂O-SiO₂ melts as a function of silica content. (b) Viscosity of peridotite melt at 3000 K as a function of pressure. Solid lines represent mean predictions from three GP models in the interpolative regime, while dashed lines indicate extrapolations. Colored areas show 1-sigma confidence intervals on predictions. The grey shaded region marks areas with no available data.</figcaption></figure>

that the GP models are providing unreliable predictions for pressures outside the training range for this composition.

Those tests indicate that predictions from models such as the Grey-box ANN and GP become unreliable in the extrapolative regime (i.e., for queries outside the range covered by the training dataset). Extrapolations in temperature are generally accurate because the log₁₀ η versus T behavior is constrained by equation (2). On the other hand, extrapolations in X and P may be unreliable because there is no functional form to describe how X and P affect viscosity (Fig. 4). In particular, at high pressure, given the dataset sparsity, such regimes could be easily reached for compositions deviating from those in the training dataset. This could be improved by adding a term accounting for the effect of P in equation (2). A linear term could be added (e.g. Russell et al., 2024), but this will not work in all cases because melt viscosity dependence on pressure changes as a function of composition (Bottinga and Richet, 1995). This area requires further work to be improved.

In case of doubt, the robustness of the predictions can be tested by querying predictions from a few different GP models (Fig. 4). The gpvisc library (Le Losq et al., 2025) provides three trained Greybox ANN and associated GP models to do so, with examples. The test should be performed in case of doubt, because unfortunately the error bars calculated by the GP model are not informative in the extrapolation domain (Fig. 4) as they do not become very large. This may be inherent to the important constraint the Greybox ANN mean function is placing upon the results of the GP model. We also identify this as an area of improvement for future implementations.

Data from molecular dynamics (e.g. Huang et al., 2024) could be used to constrain the ML models in the very high pressure range. We did not perform this step here because we focused on presenting models trained on a database that only contains experimental measurements. However, it seems to be a natural way to improve future versions of our model.

Finally, like most ML models, our models perform best in regions with sufficient training data. Slight variations in model performance are observed across compositional domains (Supplementary Fig. 2), potentially linked to data density. To investigate this, we calculated the Pearson correlation between local RMSE and data density (within a 5 mol% radius), finding a weak negative correlation (r=-0.054, p=0.003). RMSE is variable in data-scarce regions, and converges toward ≈ 0.26 log₁₀ Pa·s when increasing the local data density. This suggests that higher data density is generally beneficial but not strictly

critical for performance. Addressing data imbalance during model training could further improve predictions, though this remains a challenging and active area of research for regression-based ML models (e.g. see Yang et al., 2021).

# 4. Discussion

## 4.1. Machine learning modeling of liquid viscosity

Using a new database containing viscosity data on very various melt compositions (from unary SiO₂ and H₂O to geological and industrial melts) and at high pressure (up to 30 GPa for peridotite), we find that non-linear ML models generally perform well. The GP model performs particularly well. Its performance metrics (Test RMSE of 0.44 log₁₀ Pa·s, decreasing to 0.32 when excluding SciGlass data, see Table 1) compare well with those from published models. Taking for instance as a reference point the Giordano et al. (2008) model of the viscosity of geological melts, its RMSE on unseen data is of 0.7 log₁₀ Pa·s. The more recent artificial neural network model of Langhammer et al. (2022) announces a RMSE on unseen data of 0.45 log₁₀ Pa·s. Those two models are limited to geological compositions and do not include the effect of pressure. They thus cannot be directly compared to the estimates proffered by models in the present work. Another reference point may be the generalist (i.e. very broad compositional domain) GlassNet machine learning model (Cassar, 2023). It uses a multitask blackbox artificial neural network to predict various properties, including glass-forming melt viscosity. Applied to the present dataset at room pressure, the RMSE between measurements and GlassNet predictions is equal to 0.95 log₁₀ Pa·s.

Overall, all non-linear ML models, and particularly the GP, either match or outperform existing models while enabling predictions across an extremely wide range of temperature, pressure, and compositional scenarios. The GP model, in particular, achieves a level of performance that allows it to be used for outlier detection. For example, during the construction of the hand-curated database, extreme deviations from model predictions often revealed potential issues with outlier data points, such as typos in the database or experimental errors. Finally, for applications requiring even greater precision in specific P-T-X regions, models such as the Greybox ANN or GP can be fine-tuned. Fine-tuning, a common ML technique, involves re-training a global model on a targeted subset of data to improve its predictive performance within a specific domain.
<figure><img src="image_7.png" /><figcaption>Fig. 5. Longitudinal temperature profile at the surface of K2-141 b. See text for details.</figcaption></figure>

## 4.2. Material state and viscosity at surface of dry lava planets

Using the GP model of melt viscosity, we now will explore the possible properties of the magma ocean on the dayside of a dry hot super Earth, K2-141 b (Malavolta et al., 2018; Barragán et al., 2018). This planet is of particular interest as it is part of the USP planets studied during JWST Cycle 1 General Observers program (e.g. Dang et al., 2021).

We first calculate K2-141 b surface temperature as a function of longitude. The ≈ 48° finite angular size of its star results in a high planetary illumination (Carter et al., 2024) that must be taken into account for the calculation. To do so, we use the analytical and numerical solution described in Carter et al. (2024) and implemented in Carter (2024), with the following assumptions: a zero bond albedo (Essack et al., 2020), an average nightside surface temperature of 950 K (Zieba et al., 2022), and the absence of a thick atmosphere as supported by no hotspot offset and no atmospheric heat redistribution (Zieba et al., 2022). A thin rock vapor atmosphere may be present but does not participate in redistributing heat on the planet's surface (Malavolta et al., 2018; Nguyen et al., 2020; Zieba et al., 2022). We neglect its influence on the surface temperature calculation.

The longitudinal temperature profile shows a maximum temperature of ≈ 3080 K at the substellar point (Fig. 5), in close agreement with Malavolta et al. (2018). The dayside generally is characterized by temperatures above 1700 K. A penumbral region extends between (-)90° and ≈ (-)115°. In this region, temperature drops from ≈ 1700 K down to ≈ 950 K, the value assumed for the nightside. Given the liquidus temperature of most alumino-silicate rocks, we expect extended melting on the dayside, and magma ocean solidification in the penumbral region.

To examine the effect of composition on magma viscosity, we consider three different compositional scenarios for the magma ocean: Bulk Silicate Earth (BSE) (Palme and O'Neill, 2014), BSE + 20% FeO (Fe-rich BSE) and a Type-B CAI composition (Stolper and Paque, 1986). The first scenario is an Earth-like case, the second assumes a higher Fe/Si for K2-141 b, and the last one is considered given the high dayside temperatures that may promote a shift of the mantle composition toward a refractory endmember due to evaporation (Léger et al., 2011). The compositions are approximated in simplified systems containing only SiO₂, Al₂O₃, MgO, CaO, and FeO (with Fe³⁺/∑Fe = 0.037 after Sossi et al., 2020); we did not include the alkalis or volatiles (H and C). Using the Equilib module of FactSage 8.2. (Bale et al., 2016), we calculated the equilibrium fractions of crystals, gas and melts as a function of temperature for the three different bulk compositions, at atmospheric pressures P_atm of 10⁻⁴, 10⁻¹ and 1 bar.

We first explore the possible longitudinal profile of evaporation/con densation, but neglect phenomena that could disrupt melt-gas equi

<table><caption>**Table 2**
Longitudes (degrees) at which the temperature allows for 1% gas and 99% melt at equilibrium for three compositional scenarios and three different atmospheric pressures P_atm.</caption><thead><tr><th>P_atm (bar)</th><th>BSE</th><th>Fe-rich BSE</th><th>CAI</th></tr></thead><tbody><tr><td>10<sup>-4</sup></td><td>77</td><td>79</td><td>72</td></tr><tr><td>10<sup>-1</sup></td><td>36</td><td>37</td><td>0</td></tr><tr><td>1</td><td>0</td><td>0</td><td>0</td></tr></tbody></table>

<figure><img src="image_8.png" /><figcaption>Fig. 6. Magma ocean outgassing potential as a function of longitude at the surface of K2-141 b for the BSE (solid lines), Fe-rich BSE (dashed lines) and CAI (dotted lines) compositional scenarios at two different pressures.</figcaption></figure>

librium (as computed in FactSage), such as magma ocean convection and atmosphere evaporation and circulation. Thus, local equilibrium in terms of space and time is assumed, and given the longitudinal temperature profile, we calculate the longitude at which the system comprises more than 1% gas at different imposed total pressures (Table 2). At P_atm = 1 bar, temperatures are not high enough to generate significant outgassing for the alkali-free compositions modeled in this work. At 10⁻¹ bar, outgassing starts at longitudes lower than ~ 37° for the BSE and Fe-rich BSE cases, and is completely inhibited for the CAI case. At 10⁻⁴ bar, outgassing should occur at longitudes lower than 70°-80° in the three compositional scenarios.

An alternative way to interpret these results is to equate higher gas fractions at a given P-T-X in the equilibrium closed-system scenario with more vigorous outgassing at the surface of the dynamic magma ocean, all else being equal. Plotting the equilibrium gas fraction as a function of longitude shows that a 10⁻¹ bar atmosphere is expected near the substellar point for BSE and Fe-rich BSE compositions (Fig. 6). At longitudes greater than 37°, a lower pressure atmosphere is expected, with P_atm as low as 10⁻⁴ around 78° longitude.

This simple evaluation neglects the presence of volatile and/or alkali elements in the magma. These elements would act to increase the total pressure for a given temperature, or, equivalently, increase the vapour mass fraction of the system at a given total pressure and temperature. Recent JWST observations indicate a CO- or CO₂-rich atmosphere might occur on another USP, 55-Cancri-e (Hu et al., 2024), suggesting volatile-laden atmospheres might be commonplace on close-in lava worlds. Oxygen fugacity fO₂ and temperature also influence total atmosphere pressure, as shown by the equilibrium models of Seidler et al. (2024). They find that, near 3000 K, surface pressures are 10⁻² bar at the iron-wüstite (IW) buffer and increase up to ≈ 3.5 bar under reducing
<figure><img src="image_9.png" /><figcaption>Fig. 7. Percent of melt and crystals at surface as a function of longitude for the BSE (solid lines), Fe-rich BSE (dashed lines) and CAI (dotted lines) compositional scenarios. Colored lines are melt fractions and grey lines are crystal fractions.</figcaption></figure>

conditions (ΔIW = -6) for a bulk silicate Earth-like (BSE) composition, consistent with our results. Another limitation is the absence of consideration for atmospheric circulation, which would imply evaporation near the sub-stellar point and a gradual condensation as gas moves toward the penumbral region (Kite et al., 2016). Considering such complexities is outside the scope of this work. Despite this, the present analysis agrees with, and corroborate the results from the study of Zieba et al. (2022). Their calculations suggest a rock-vapor atmosphere of 10⁻¹ bar up to ≈ 40° longitude at K2-141 b surface, its pressure decreasing when going toward the penumbral region. Assuming a lateral atmospheric current between the substellar point and the penumbral region, outgassing is thus expected in the 40° region around the substellar point, and rock vapor condensation should occur in the lower pressure regions, between 40° and 90°.

We now consider the fractions of melt and crystals the magma ocean contains at surface (Fig. 7). In all three cases, the magma ocean is fully molten at the dayside surface (Fig. 7). Differences only are visible close to the shores, in the penumbral region. The refractory CAI composition has the sharpest transition from fully molten to solid, whereas this transition occurs over the widest temperature range for the BSE. However, for all compositions, we expect an abrupt solidification of the magma ocean in the penumbral region (≈ 5 – 15°).

Given the variations in melt composition and crystal/liquid fractions, we calculate magma viscosity using the GP model for the liquid phase. To account for the effect of crystals, we apply the model of Costa et al. (2009) for liquid-rich magmatic mixtures (φ ≈ 65%, where φ is the crystal fraction). This model incorporates parameters for crystal size and shape distributions as well as the maximum packing fraction. For these parameters, we use values determined for peridotite compositions provided in Costa et al. (2009). For crystal-rich magmatic mixtures (φ ≥ 65%), we use the model from Scott and Kohlstedt (2006), assuming a reference viscosity for the solid mantle of 10¹⁸ Pa·s at 1273 K. We caution the reader that the rheology of crystal-rich magmas is non-Newtonian. Here, for this first-order analysis, we disregard non-Newtonian effects due to the complexity of incorporating them. Addressing these effects requires a dedicated study, which is beyond the scope of the present work.

We observe little variations in magma ocean viscosity between the different scenarios (Fig. 8). Across most of the dayside, magma viscosity is lower than 10⁻¹ Pa·s. At the substellar point, magma viscosity reaches values of ≈ 10⁻².³ and ≈ 10⁻².⁶ Pa·s for the CAI and BSE/Fe-BSE cases, respectively. Melt composition thus has a small impact (≈ 0.3 log units) on the viscosity of the fully molten magma and is overwhelmed by temperature (cf. equation (2)). When going toward the nightside,

the earlier appearance of crystals in the BSE case (Fig. 7) implies that magma viscosity increases sooner when transitioning from the dayside to the nightside. However, the difference is only of a few degrees compared to the Fe-rich BSE and the CAI scenarios.

Overall, this analysis highlights two points:

* in the high- to very high temperature regimes characteristic of magma oceans on hot super-Earths, temperature exerts the zeroth-order control on magma viscosity, erasing the influence of other factors usually critical in Earth-like volcanic systems (e.g. melt composition);

* when temperature decreases, crystallization dictates the rheology of magma, due to combined effects of the presence of solid particles and of the changes it imparts on the residual melt composition. This effect will be important on the shores of magma oceans, but possibly also at depth as crystallization may be favored by the increasing pressure. We do not explore this here because it requires an internal model of the planetary structure that is beyond the scope of this paper.

### 4.3. K2-141 b: nightside temperature

On the nightside, we expect a solid surface (Fig. 7) with a very high viscosity (Fig. 8). K2-141 b has a very tenuous atmosphere, unlikely to redistribute heat toward the nightside (Zieba et al., 2022). Therefore, an internal geothermal heat flow will drive heat redistribution between the dayside and nightside (Meier et al., 2023). The nightside surface temperature T_s will be controlled by a balance between the geothermal heat flux Q geo and the radiative surface heat flux Q rad = σ_B T_s⁴, with σ_B the Stefan-Boltzmann constant. Q geo can be sustained either by horizontal transport of heat from the dayside to the nightside by a thin magma ocean under a lid, or by vertical transport from the interior to the surface by the rocky mantle (potentially partially molten). In the case of horizontal convection, we have (Hughes and Griffiths, 2008):

$$Q_{geo} = k \frac{\Delta T}{L} R_a^{1/5}, \quad (4)$$

with k the average thermal conductivity, ΔT = T_s - T_i with T_i the average internal temperature of the convecting layer and L its height, and R_a the Rayleigh number equal to

$$R_a = \frac{\rho g \alpha \Delta T L^3}{D \eta}, \quad (5)$$

where g is the gravity acceleration, and ρ, α, D and η the average values of density, thermal expansivity, thermal diffusivity and viscosity of the material composing the convecting layer. For vertical convection, two cases can be distinguished: hard and soft convection. In the case of a mantle containing a large fraction of melt (below the rheological threshold at around 60 vol% solids), R_a will be very high (≈10³⁰ for an Earth-sized planet) and hard convection should be considered, such that we have (Solomatov, 2000):

$$Q_{geo} = 0.22k \frac{\Delta T}{L} R_a^{2/7} R_p^{-1/7} \lambda^{-3/7}, \quad (6)$$

where P_r = D/ρη is the Prandtl number and λ is the mean flow aspect ratio. We assume λ = 1, following Solomatov (2000). Now, if the mantle is mostly composed of crystals, it will undergo the rheological transition and R_a will decrease significantly. Compared to the fully molten magma ocean, we expect a decrease in R_a by approximately 15 orders of magnitude as viscosity drastically increases (Fig. 8). Soft convection should then be considered and the heat flux can be expressed as (Solomatov, 2000):

$$Q_{geo} = 0.089k \frac{\Delta T}{L} R_a^{1/3}. \quad (7)$$

By equating Q geo with Q rad, we can calculate the nightside surface temperature by iteration. We performed this for two cases: horizontal and
<figure><img src="image_10.png" /><figcaption>**Fig. 8.** Magma viscosity at surface given longitude, calculated assuming a Earth-like composition (Bulk Silicate Earth, BSE), a Fe-rich composition (Fe-BSE) and a refractory composition (Type B CAI, see Stolper and Paque, 1986). See text for details. Shaded regions on the side of the lines are 95 % confidence intervals.</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>**Fig. 9.** Average nightside surface temperature as a function of average nightside internal temperature of the convecting layer for two cases: horizontal (a, eq. (4)) and hard (b, eq. (6)) convection. The red square delimits the temperature range estimated for K2-141 b (Zieba et al., 2022). The curves are the mean values of 200 models, calculated by sampling the parameter distributions described in the text. Shaded areas around those curves represent the 95% confidence intervals of model predictions.</figcaption></figure>

hard convection. We neglect the case of soft convection as it is valid for crystal-rich cases (above the rheological threshold). Equation (7) does not actually apply to solid mantle convection and another functional form should be adopted (Solomatov, 2015). Here, we neglect this and use the hard convection case to provide an upper limit on $T_s$ for crystal-rich cases.

Fig. 9 shows the result of the calculation for a convecting layer of 1000 km, and for the three cases BSE, Fe-rich BSE and Type B CAI. For the calculations, we used $\rho \sim \mathcal{N}(4000, 400^2)$, $\alpha \sim \mathcal{N}(6e-5, 6e-6^2)$,

$D \sim \mathcal{N}(5e-7, 2e-7^2)$, $C_p \sim \mathcal{N}(1500, 150^2)$ following experimental values and calculations from Eriksson et al. (2003); Gibert et al. (2003); Suzuki et al. (1998); Richet and Bottinga (1985); Le Losq and Baldoni (2023). We neglect variations of those properties with melt composition as their influence is minimal because they do not vary by more than one order of magnitude. On the contrary, the convecting layer viscosity can vary by up to 27 orders of magnitude as a function of temperature and magma crystal content (Fig. 8). In particular, the influence of crystallization on melt viscosity is paramount, and unfortunately one uncertainty
remains: the volumetric fraction of crystals at which the rheological threshold, $\phi_c$, occurs. $\phi_c$ is usually taken to be at a solid fraction of 0.60 (Kolzenburg et al., 2022). Below $\phi_c$, the convecting layer has the rheological behavior of a crystal-bearing liquid. Calculated magma viscosities are lower than $10^6$ Pa·s and $R_a$ will be above $10^{25}$. However, above $\phi_c = 0.60$, the rheological behavior changes. The convecting layer adopts the rheological behavior of a partly-molten solid, magma viscosity largely increases and $R_a$ too. The $\phi_c$ value is thus critical. It varies significantly with crystal size and shape (Kolzenburg et al., 2022). Here, to take into account of the uncertainty surrounding $\phi_c$ value in the different scenarios, we assume for $\phi_c$ a normal distribution with a mean of 0.6 and a standard deviation of 0.06 (10% error uncertainty).

Fig. 9a reveals that horizontal convection is not efficient in redistributing the thermal energy. If this mode were dominant, we would not expect temperatures above 600 K at the nightside surface of a dry exoplanet such as K2-141 b, even in the presence of a molten magma ocean beneath a lid. Vertical hard convection is more efficient at providing a sustained geothermal flux (Fig. 9b). In this case, if the average internal nightside temperature is above $\approx 1600$ K, the nightside surface can reach temperatures above 400 K, the apparent lower limit for K2-141 b (Zieba et al., 2022). The $\approx 1600$ K interior temperature threshold is easily understandable: at this temperature, the rheological transition in BSE-like bulk compositions occurs and the convection of the liquid-rich magma becomes efficient in redistributing heat. The average nightside temperature then starts to closely mirror the average internal nightside temperature.

We finally note that little differences are discernible between the three different magma compositions. The visible difference concerns the onset of the nightside surface temperature rise for a given internal nightside temperature when the latter is around 1600 K. This arises and is due to differences in how the crystallisation sequences between the BSE, Fe-BSE and CAI magma depend on temperature (Fig. 7). In any case, most uncertainties for the determination of the curves shown in Fig. 9 arise from the uncertainty on $\phi_c$. The maximal influence of melt composition can be observed when comparing the endmember cases of partially polymerized MgSiO$_3$ and fully polymerized SiO$_2$ melts (Fig. 9). While the curve for liquid MgSiO$_3$ closely follows those of BSE, Fe-rich BSE and CAI (except below $\approx 1600$ K as MgSiO$_3$ is considered fully molten), a fully polymerized, very viscous SiO$_2$ liquid mantle will not redistribute efficiently heat to the surface.

The present calculations have inherent limitations, such as the absence of any consideration regarding temperature, pressure and associated property gradients. Besides, while horizontal convection appears unlikely to efficiently transport heat, the reality may be more complex as this mode is not well understood in the case of the laterally varying surface heating that applies for USP lava planets (Meier et al., 2023). Despite such limitations, the present calculations yield an important conclusion. For sustaining a nightside temperature above 400 K without atmospheric redistribution, as suggested for K2-141 b (Zieba et al., 2022), a portion of the nightside mantle should be partly molten. From Fig. 9, we estimate that this partly molten mantle portion should have a melt fraction at or slightly above 0.4.

## 5. Conclusion

In this paper, we presented a new database and its use to train machine learning models to predict the viscosity of water-bearing phospho-alumino-silicate melts as a function of chemical composition, temperature and pressure. Greybox artificial neural networks produce accurate descriptions of experimental data. Their predictions can further be refined using a Gaussian process (GP) to achieve high predictive accuracy (RMSE $\approx 0.4 \log_{10}$ Pa·s) and further compute confidence intervals on these predictions. In its current implementation, the GP model allows computation of the viscosity of magmatic melts in a wide variety of scenarios, including (water-rich) magmatic systems in subduction regions, alkali-rich intraplate volcanism, magma oceans, and extra-terrestrial

volcanisms involving compositions far from the Earth trends. Glass manufacturing may also use this model to calculate melt viscosity in glass-forming furnaces, with potential applications in furnace or compositional design for ensuring energy efficient and environmentally effective production of glass. Finally, while pressure effects can be reproduced up to 30 GPa for compositions like peridotite, the sparsity of high pressure viscosity data hampers the ability of ML models to find general laws expressing how melt viscosity varies with pressure for any composition. This invites to further work to better constrain the viscosity of silicate melts at pressures above 1 GPa.

Using the new model as well as phase equilibrium and temperature calculations, we evaluate the properties of a magma ocean at the surface of the dry USP exoplanet K2-141 b, given three different compositional scenarios. Assuming a null bond albedo, calculation of surface temperature indicates that K2-141 b dayside is fully molten in all scenarios. On the dayside, the extreme temperatures generally dictate melt viscosity, and other parameters such as melt composition are secondary. Phase calculations indicate that a tenuous rocky atmosphere with a pressure of $10^{-1}$ bar may be present in a 40° around the substellar point, in agreement with Zieba et al. (2022). When going to higher longitudes, the atmospheric pressure is expected to drop to very low values. When reaching the penumbral region at a longitude of 90°, magma viscosity increases rapidly as magma ocean solification proceeds. The composition of the magma slightly affects the position of the shores, but this effect is limited (a few degrees). In the nightside region, surface is expected to be solid. However, estimated nightside temperatures above 400 K (Zieba et al., 2022) suggest that the mantle below the surface is partly molten and feeds the geothermal flux through vertical convection.

## CRediT authorship contribution statement

**Charles Le Losq:** Writing – review & editing, Writing – original draft, Software, Resources, Project administration, Methodology, Investigation, Funding acquisition, Formal analysis, Data curation, Conceptualization. **Clément Ferraina:** Writing – review & editing, Software, Methodology, Investigation, Formal analysis, Data curation. **Paolo A. Sossi:** Writing – review & editing, Methodology, Investigation, Formal analysis. **Charles-Édouard Boukaré:** Writing – review & editing, Methodology, Investigation, Formal analysis, Conceptualization.

## Materials & Correspondence

The hand-curated viscosity database is available at Ferraina et al. (2025). The computer code to reproduce the results of this study is available as a Python library at the web address https://github.com/charlesll/gpvisc and on Zenodo (Le Losq et al., 2025). Correspondence can be addressed to the corresponding author.

## Funding

This study was supported by the Labex UnivEarthS, ANR-10-LABX-0023 and ANR-18-IDEX-0001.

## Declaration of competing interest

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.

## Acknowledgement

CLL thanks S. Charnoz (IPGP) for various discussions on the present topic and his help in catching the $R_p$ scaling of the illumination in Carter (2024) code.
# Appendix A. Data splitting

For performing the training-validation-testing splits, we first separated the dataset in two 1 atm and high pressure datasets, because the compositional coverage and the number of data at 1 atm and high pressures are very different and skewed towards the former. For 1 atm pressure data, we adopted a stratified group splitting approach that ensures that each subset will contain different compositions in relatively balanced proportions (see Le Losq and Baldoni, 2023, for details). For high pressure data, we did not use this method because the high pressure dataset contains too few compositions, with only a few data points at different pressures per composition. Considering that the compositional domain will be already well covered and represented in the different 1 atm pressure data subsets, and that the aim of the high pressure data is to provide knowledge about the effect of pressure, we decided to split randomly the high pressure data in the training, validation and testing data subsets regardless of melt compositions. To perform the different random splits, we selected optimal random seeds such that the training, validation and test subsets present similar statistical distributions of their features. To do so, we tried 1,500 different random seeds, and selected the 10 ones that ensured the lowest Wasserstein distances of the feature distributions between the various data subsets. Then, among those seeds, we rejected those that resulted in including endmember compositions (e.g., SiO₂ or H₂O) in the validation or testing data subsets. We then selected an appropriate seed given this constrain and the Wasserstein distances test. This approach ensures the similarity of the data distributions between the different training, validation and test data subsets (see for visual evaluation Supplementary Figs. 3 and 4). Finally, after data splitting, the high and low pressure parts of the training-validation-testing data subsets were concatenated.

# Appendix B. Supplementary material

Supplementary material related to this article can be found online at https://doi.org/10.1016/j.epsl.2025.119287.

## Data availability

Everything is available on Zenodo: https://zenodo.org/records/14673467

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