# Accurate VLE Predictions via COSMO-RS Guided Deep Learning Models: Solubility and Selectivity in Physical Solvent Systems for Carbon Capture

Edoardo Parascandolo†, Vincent Gerbaud‡*, David Camilo Corrales§, Noslen HernándezΩ, Sophie Thiebaud-Roux†, Ivonne Rodriguez-Donis†

†Laboratoire de Chimie Agro-industrielle (LCA), Université de Toulouse, INRAE, Toulouse INP, 31030 Toulouse, France

‡Laboratoire de Génie Chimique (LGC), Université de Toulouse, CNRS, Toulouse INP, 31432 Toulouse, France

§TBI, Université de Toulouse, CNRS, INRAE, INSA, 31077 Toulouse, France

ΩINTHERES, Université de Toulouse, INRAE, ENVT, 31076 Toulouse, France

*Corresponding author, E-mail address: vincent.gerbaud@toulouse-inp.fr

## Abstract

Carbon capture through physical solvents reduces energy consumption and lowers environmental impact compared to conventional chemical absorption methods. Typical properties for solvent screening are solubility and selectivity. However, they require accurate prediction of vapor-liquid equilibrium (VLE) that remains a critical challenge due to the lack of enough available experimental data. This could be supplemented by in silico data prediction provided that current predictions models are improved as this paper intends. When modeling physical solvents, a challenge arises due to the dominant role of non-bonding interactions and molecular geometry. For this purpose, a machine learning pipeline is developed using VLE results obtained from the quantum chemical-based thermodynamic model COnductor like Screening MOdel for Real Solvents (COSMO-RS) and experimental data. A Direct-Message Passing Neural Network (D-MPNN) architecture is employed, leveraging molecular representations, additional features and transfer learning to refine predictions. Two models, solubility and selectivity, are pre-trained over 30000 COSMO-RS simulated data points and fine-tuned with experimental VLE datasets for CO₂ and common gas impurities (H₂S, CH₄, N₂, H₂) respectively. The models' accuracy is significantly improved over COSMO alone by correcting bias in total pressure predictions. Experimental trends are successfully reproduced in the test data, confirming the physical consistency of the models. Sensitivity analysis confirms that molecular features have the highest impact on estimations, while the scaling effect of additional features is essential for accuracy. These results demonstrate the potential of the proposed methodology to systematically screen and optimize an extensive range of physical solvents on the basis of their chemical structure for carbon capture applications, reducing reliance on costly and time-consuming experimental measurements.

**Keywords:** Carbon Capture, Deep Learning, Physical Solvent, Principal Component Analysis, COSMO-RS
# 1 INTRODUCTION

At present, the global average surface temperature is already around 1.2 °C above pre-industrial levels¹, prompting heatwaves and other extreme weather events expected as climate change. Among greenhouse gases, CO₂ emissions alone are the most impacting to global warming² and they have not peaked yet¹. Within all the approaches to mitigate, carbon capture and storage (CCS) is predicted to be an auspicious and viable technique to meet the climate target without impacting too much the economic growth of any country when integrated with industrial and power plants³, and its worldwide use could mitigate CO₂ emission of approximately 70% by 2050⁴.

Nevertheless, some major drawbacks are still present in the carbon capture phase. Indeed, the most mature technology, that is chemical absorption using blended amines⁵, is limited in its application due to several significant disadvantages: above all, they require high energy input for solvent regeneration because of the strong chemical bonds formed with CO₂, leading to increased operational costs and reduced overall efficiency. Furthermore, the employment of amines is generally avoided given their toxicity, degradation, corrosivity and generally high environmental impact for amines disposal⁶.

Physical solvents are a viable alternative. Differently from chemical solvents, the dissolution in physical solvents is driven by physical affinities between the solvent and the solute in the absence of any reaction, thus having a positive impact on energy consumption and side-effects, such as corrosion. Moreover, physical solvents are known to have greater absorption capacity and hence to allow lower solvent recirculation rates⁷. On the other hand, their disadvantages concern the need for high acid gas partial pressures, the necessity to have a low concentration of inert gases, and the difficulty in meeting H₂S limit threshold specification⁸. Advantages and disadvantages of benchmark physical solvents are presented in Table 1.

The assessment of a new solvent, physical or chemical, requires the evaluation of the vapor-liquid thermodynamic equilibrium (VLE) for its binary mixture with CO₂ at different conditions of temperature and partial pressures. Indeed, solubility and selectivity are key parameters to monitor, as optimizing these properties could potentially extend the research on physical solvents to sources with low CO₂ concentrations and high content of H₂S and inert gases. The conventional method for VLE calculation is either the use of a thermodynamic Gibbs excess (G<sup>ex</sup>) model (Non-Random Two-Liquid, UNIversal QUAsiChemical, UNIversal quasichemical Functional group Activity Coefficients) or an equation of state (EoS) approach to account for the frequent non-ideal behavior in the liquid phase⁹. As an example, for physical solvents in particular, the Perturbed Chain-Statistical Associating Fluid Theory EoS has proven to be a reliable tool for representing the phase equilibrium in pre-combustion carbon capture¹⁰. However, the consistency of VLE model based on the aforementioned G<sup>ex</sup> or EoS approach holds on the regression of estimation parameters from experimental data.
Therefore, the use of G<sup>ex</sup> or EoS for the estimation of any new CO<sub>2</sub>-solvent mixture, whose experimental data are not available in literature, is both time-consuming and resource-intensive.

<table><caption>**Table 1.** Advantage and disadvantage of selected physical solvents (adapted from literature<sup>7</sup>).</caption><thead><tr><th>Solvent</th><th>Advantage</th><th>Disadvantage</th></tr></thead><tbody><tr><td>Selexol<sup>TM</sup></td><td><ul><li>Low vapor pressure (&lt;0.001 mmHG @ 25°C)</li><li>Applicable in wide temperature range (–18 to 175 °C)</li><li>Proper for selective H<sub>2</sub>S removal</li><li>Process can be configured to provide the Claus process with both a rich H<sub>2</sub>S feed and a bulk CO<sub>2</sub> removal.</li></ul></td><td><ul><li>High viscosity (&gt; 7 cP @ 25 °C)</li><li>High energy requirements for solvent recirculation</li></ul></td></tr><tr><td>Methanol</td><td><ul><li>Available and low price</li><li>Flexible with different possible flow schemes</li></ul></td><td><ul><li>High vapor pressure (92 mmHg @ 25 °C)</li><li>Complex process configuration</li><li>Higher selectivity for H<sub>2</sub>S over CO<sub>2</sub></li></ul></td></tr><tr><td>Propylene Carbonate</td><td><ul><li>High solubility of CO<sub>2</sub> and other gases.</li><li>Low heat of solution for CO<sub>2</sub></li><li>Low vapor pressure (0.045 mmHg @ 25 °C)</li><li>Low solubility of light hydrocarbons</li><li>Nonreactive toward all natural gas components</li><li>Noncorrosive</li></ul></td><td><ul><li>Solvent is not suitable if H<sub>2</sub>S is present in more than trace concentrations</li><li>The reaction between solvent and water and CO<sub>2</sub> is slow and irreversible at 90 °C</li></ul></td></tr><tr><td>N-Methyl-Pyrrolidone</td><td><ul><li>Able to treat gas flows that contain less than 0.1% CO<sub>2</sub></li><li>Applicable to other processes such as acetylene recovery from pyrolysis gases</li></ul></td><td><ul><li>Significant vapor pressure (0.345 mmHg @ 25 °C)</li><li>Higher selectivity for H<sub>2</sub>S over CO<sub>2</sub></li><li>Water washing for solvent recovery</li></ul></td></tr></tbody></table>

As an alternative to parameter regression, the fully predictive quantum chemical-based thermodynamic model COnductor like Screening MOdel for Real Solvents (COSMO-RS) is able to perform VLE calculations directly from molecular structures without requiring any experimental data through careful parametrization<sup>11,12</sup>. Several studies<sup>13,14</sup> confirm the growing interest for COSMO-RS in predicting CO<sub>2</sub>-solvent equilibria, including recent efforts focusing on alternative solvents for greenhouse gas capture<sup>15–17</sup>. Nevertheless, COSMO-RS predictions have been classifiable as “qualitatively good” than “quantitatively good”<sup>18</sup>. This diminishes the intrinsic interest of COSMO-RS for VLE predictions involving systems with limited data.

Recently the attention has moved on machine learning models (ML), which have already proven to be useful for VLE estimations of different CO<sub>2</sub>-solvent systems<sup>19,20</sup>. Although prediction is progressively improved in accuracy as more information becomes available<sup>21</sup>, it requires a limited set of experimental data to build-up a predictive model. Nevertheless, spite its potential, the sole use of machine learning presents significant
limitations. Notably, an insufficiently diverse training dataset can lead to overfitted or underfitted models, resulting in a poor generalization beyond the cases seen during training phase. Additionally, some models may lack interpretability, leading to predictions that fall outside their physical boundaries. As previously reported<sup>22</sup>, it is the case of physical solvents where one requires precise descriptors that could identify and differentiate multiple molecular structures and CO<sub>2</sub>-interaction mechanisms, while accounting for non-idealities.

In this context, combining COSMO-RS to ML offers a powerful approach to enhance the predictive accuracy and computational efficiency of thermodynamic property estimations. The utilization of COSMO-RS descriptors as input during the training of ML models is a viable technique that is a viable technique for reducing the error produced from quantum chemistry calculations alone, achieving more accurate and generalizable predictions particularly in cases where traditional methods fall short due to limited experimental data availability. Different studies<sup>18-20</sup> show how adding ML to COSMO-RS can reduce the deviation of the calculated CO<sub>2</sub> solubility in DESs. A frequent approach to combine ML models using COSMO-RS is to use descriptors as sigma profiles, which can provide information about donor and acceptor ability of molecular surfaces<sup>18,23</sup>. While effective for systems characterized by strong electrostatic interactions and hydrogen bonding, such as DES, sigma profiles are less suited to describe physical solvents, where non-bonding interactions and molecular geometry play a more critical role.

Graph Convolutional Neural Networks (GCNNs) offer a powerful alternative. Unlike sigma profiles or direct physicochemical features, GCNNs use graph-based molecular descriptors that incorporate bond connectivity and atomic environments, enabling them to capture more complex molecular properties. This results in a more reliable property prediction, particularly in cases where non-electrostatic interactions play a significant role<sup>24</sup>.

In the present study, COSMO-RS calculations are refined with Chemprop<sup>24,25</sup>, a directed-message passing algorithm to address both issues of enhancing solubility and selectivity estimations for physical solvents in carbon capture. Indeed, while most previous studies focus exclusively on CO<sub>2</sub> solubility, selectivity is rarely considered, despite being a crucial parameter in phase separations involving multicomponent gas mixtures. After pre-training two different multi-component regression models on more than 30,000 COSMO-RS predictions, the former specifically for CO<sub>2</sub> while the latter for H<sub>2</sub>S, CH<sub>4</sub>, N<sub>2</sub> and H<sub>2</sub>, and fine tuning with respectively two experimental datasets, made of 1091 VLE datapoints (CO<sub>2</sub> - 43 solvents) and 660 VLE datapoints (contaminants – 31 solvents), the error in predicting the equilibrium pressure was substantially reduced over a wide range of experimental conditions. Such a correction was general enough to improve the accuracy of VLE predictions from quantum chemistry for all the binary systems taken into account. The effectiveness of the learned molecular representations and additional features was further assessed through a 3D principal component analysis (PCA), combining visual and numerical clustering, as well as a sensitivity analysis to identify the most influential descriptors. These results and the limited need of experimental data could
represent a significant advancement in the search for promising CCS physical solvents candidates by reducing computational times and costs while ensuring a systematic exploration of selectivity and performance trade-offs.

## 2 MATERIALS AND METHODS

COSMO-RS works by first performing quantum chemical calculations on individual molecules in a virtual conductor-like environment. Each solute and solvent molecule is surrounded by a dielectric medium, and the surface charge densities are computed. COSMO-RS then uses statistical thermodynamics to evaluate pairwise interactions between these surface segments to estimate solvation effects and predict phase equilibria from chemical potentials. It is worth noting that the model inherently assumes complete miscibility across the entire composition range, without considering the occurrence of solubility limits in the phase diagram. Nevertheless, as long as the interactions are properly estimated, the predicted equilibrium pressure in regions of non-solubility is expected to be high and its trend to remain coherently increasing with the mole fraction of solute in the liquid phase. Consequently, this limitation does not significatively impact the ability to differentiate effective solvents from unsuitable ones. Additionally, this limitation is partially addressed in our final model through a fine-tuning phase, ensuring improved alignment with experimental or more refined data.

All COSMO-RS calculations were performed with COSMOtherm²⁶ (v. 24), using the BP-TZVPD-FINE parametrization. For solvents already included in the database, the conformer supplied by COSMOtherm was used without further optimization; for all other solvents, the .cosmo input files were produced by geometry optimization and basic conformer search at the DFT B-P/def2-TZVPD level in Turbomole²⁷ (v. 7.4), selecting the lowest-energy structure. Solubility and selectivity are therefore assessed in this paper as the equilibrium pressure of each gas in the same solvent at matched temperature and composition, while composite indicators such as ratios for selectivity can readily be derived a posteriori.

Second, a preliminary analysis of the VLE data was performed, finding a trend in the equilibrium curves and a consistent distribution of the data with the exception of systems with H₂O or H₂O₂. These deviations can be attributed to the intrinsic limitations of the COSMO-RS model when applied to highly polar solvents with dominating H-bond interactions²⁸. In particular the continuum solvation approach, while effective for most systems, is not accurate enough in predicting the interactions whenever these are driven by strong hydrogen bonding²⁹. This discrepancy leads to less reliable predictions for equilibrium behavior in such systems, resulting in inconsistent VLE curves. Although these trends were not found in all the impurities-water/oxygenated water mixtures, all pairs exhibited pressure values lying outside or at the upper boundary of those observed with the other solvents. Therefore, all the simulations done with H₂O and H₂O₂ were not taken into account when building the databases. After these pre-processing considerations, a ten-step procedure for building the two final models was developed as summarized in Figure 1.
<figure><img src="image_1.png" /><figcaption>**Figure 1.** Machine Learning pipeline representing the 10-steps development of the solubility and selectivity model.</figcaption></figure>

## 2.1 Steps of data preparation

**Step 1a.** First, 1241 liquid molecules at room temperature without reactive groups towards CO₂ were selected. The equilibrium pressure of each binary mixture CO₂-molecule was estimated via the COSMOtherm VLE module for mole fraction of solute dissolved in the liquid, going from 0.1 to 0.9 (with step 0.1) at three temperatures (0°C, 25°C and 50°C).

**Step 1b.** The procedure was implemented as described and then repeated, limiting the mole fraction for the impurities from 0.03 to 0.15 (with step 0.03), for each gas species, in order to get the raw databases for pre-training the two models, referred to herein respectively as solubility model and selectivity model.

**Step 1c.** Finally, experimental data for fine-tuning the models were collected using the NIST ThermoData Engine SRD#103b v7.1³⁰ accessed via AspenPlus v14³¹. The former database was assembled with 1091 equilibrium points belonging to 43 CO₂-physical solvents binary mixtures, while the latter with 660 equilibrium points fitted from 64 different impurity-physical solvents pairs.

To visualize the diversity of the datasets, Morgan fingerprints were evaluated and represented in 2D through t-distributed Stochastic Neighbor Embedding (t-SNE), as illustrated in Figure 2. Morgan fingerprints are a technique used to represent molecular structures numerically, capturing their features in a compact form. This allowed to assess the distribution of molecular structures and ensure that the dataset covered a wide range of chemical compositions.
<figure><img src="image_2.png" /><figcaption>Figure 2. Dimensional reduction with t-distributed Stochastic Neighbor Embedding (t-SNE) for the Morgan fingerprints of all the solvents in the training.</figcaption></figure>

Table 2 and Table 3 show that the ranges of mole fraction, temperature, and pressure in these datasets are centered around the values used in the pre-training databases, with the exception of the H₂S-solvent systems. For this class of mixtures, no constraints were considered on the experimental mole fraction of solute dissolved in the liquid. This adjustment is justified by the higher solubility of H₂S compared to CH₄, N₂, and H₂, thus a lower pressure range. Therefore, limiting equilibrium points in a region of low H₂S solubility can introduce significant scaling errors in the model, that is a higher relative error for these systems. Nevertheless, an imbalanced distribution of data has the potential to hinder the learning process by emphasizing patterns that are statistically overrepresented but physically less relevant. This can result in a loss in the overall predictive performance. In conclusion, the H₂S data were collected into a separated dataset, together with a minor percentage (20%) of the other solute-solvent mixture data, to then be employed for a further less invasive fine-tuning phase. This additional step allows for targeted optimization in the estimations of H₂S-solvent systems while preserving generalizability across other pairs, without introducing significant biases in the parameters.

<table><caption>Table 2. Statistical metrics for the solubility dataset.</caption><thead><tr><th rowspan="2">Variable</th><th colspan="6">Pre-Training</th></tr><tr><th>Mean</th><th>Std</th><th>Min</th><th>Max</th><th>Skewness</th><th>Kurtosis</th></tr></thead><tbody><tr><td>CO<sub>2</sub> Liquid Fraction (-)</td><td>0.5</td><td>0.258</td><td>0.1</td><td>0.9</td><td>0</td><td>-1.3</td></tr><tr><td>Temperature (°C)</td><td>25</td><td>20.4</td><td>0</td><td>50</td><td>0</td><td>-1.5</td></tr><tr><td>Pressure (bar)</td><td>32.3</td><td>24.9</td><td>1.20</td><td>120.5</td><td>1.04</td><td>0.35</td></tr></tbody><thead><tr><th rowspan="2">Variable</th><th colspan="6">Fine-Tuning</th></tr><tr><th>Mean</th><th>Std</th><th>Min</th><th>Max</th><th>Skewness</th><th>Kurtosis</th></tr></thead><tbody><tr><td>CO<sub>2</sub> Liquid Fraction (-)</td><td>0.466</td><td>0.288</td><td>0.00299</td><td>0.991</td><td>0.106</td><td>-1.26</td></tr><tr><td>Temperature (°C)</td><td>36.4</td><td>13.9</td><td>0</td><td>75</td><td>0.255</td><td>0.433</td></tr><tr><td>Pressure (bar)</td><td>41.1</td><td>26.1</td><td>0.26</td><td>125.7</td><td>0.386</td><td>-0.523</td></tr></tbody></table>
<table><caption>Table 3. Statistical metrics for the selectivity dataset.</caption><thead><tr><th rowspan="2">Variable</th><th colspan="6">Pre-Training</th></tr><tr><th>Mean</th><th>Std</th><th>Min</th><th>Max</th><th>Skewness</th><th>Kurtosis</th></tr></thead><tbody><tr><td>CO<sub>2</sub> Liquid Fraction (-)</td><td>0.09</td><td>0.04</td><td>0.03</td><td>0.15</td><td>0</td><td>-1.3</td></tr><tr><td>Temperature (°C)</td><td>25</td><td>20.4</td><td>0</td><td>50</td><td>0</td><td>-1.5</td></tr><tr><td>Pressure (bar)</td><td>58.9</td><td>67</td><td>0.01</td><td>928.1</td><td>2.53</td><td>13.12</td></tr></tbody><thead><tr><th rowspan="2">Variable</th><th colspan="6">Fine-Tuning</th></tr><tr><th>Mean</th><th>Std</th><th>Min</th><th>Max</th><th>Skewness</th><th>Kurtosis</th></tr></thead><tbody><tr><td>Gas Liquid Fraction (-)</td><td>0.119</td><td>0.204</td><td>0.0002</td><td>0.993</td><td>2.45</td><td>5.53</td></tr><tr><td>Temperature (°C)</td><td>34.5</td><td>11.03</td><td>15</td><td>55</td><td>0.17</td><td>-1.14</td></tr><tr><td>Pressure (bar)</td><td>38.9</td><td>42.1</td><td>0.47</td><td>353.5</td><td>2.69</td><td>11.56</td></tr></tbody></table>

Skewness and kurtosis are two key parameters that provide insight into the distribution of data. Skewness measures the asymmetry of the dataset distribution. A skewness value close to zero indicates a symmetric distribution, while positive skewness suggests a longer right tail, and negative skewness indicates a longer left tail. Kurtosis, conversely, is a measure of the peak's sharpness of a distribution in comparison to that of a normal distribution. The comparable distributions suggest that the COSMO-RS pre-training simulations are physically representative. Observed differences in statistical metrics arise from the higher system variability during the pre-training phase. Despite these variations, the overall distribution shapes remain consistent, validating the representativeness of COSMO-RS data. The positive deviation in the skewness and kurtosis of the gas liquid fraction distribution is justified by the inclusion of higher solubility data for H₂S, as mentioned above. The high kurtosis observed in the gas pressure data reflects the rapid increase in equilibrium pressure for systems involving non-condensable gases. This behavior is consistent with the physical nature of such systems, where small changes in composition can result in large variations in pressure due to the limited solubility of the gases. As previously introduced, some pre-training data exhibit unrealistic pressure values due to COSMO-RS assumption of complete miscibility. Additionally, a summary of statistical outliers per chemical family (alkanes, cycloalkanes, aromatics, ...) is presented in Tables S1 and S2. Outlier incidence is observed across different groups, with relatively higher counts in esters, aromatics and halogenated molecules, reflecting the degrees of chemical complexity and the inherent limits of COSMO-RS depending on the functional groups. While this outlier assessment is inherently qualitative - relying on a broad 1.5 × Interquartile Range (IQR) threshold and retaining all data for model training - it highlights systematic deviations across solvent classes and provides an initial strategy for targeted thermodynamic refinement in future studies.

**Step 1d, 1e.** Finally, nested cross validation with random repetitions (3 splits, 3 repeats) for pre-training and a standard 9-fold cross validation for each fine-tuning phase with arbitrarily split into testing and training (20 %/80 %) were used.
## 2.2 Message Passing Neural Network

Graph Convolutional Neural Networks (GCNNs) are deep learning models designed to process graph-structured data. The graph representation of molecules provides a more robust approach compared to traditional methods like group contribution, the convolution mechanism enables automated and optimized feature extraction, eliminating the need for manual selection of fingerprints or physicochemical descriptors³². Chemprop²⁴,²⁵ is a software package leveraging a Direct-Message Passing Neural Network (D-MPNN), a class of GCNNs, to learn the atomic embeddings from the encoded local features. It is furthermore provided with an aggregation function to construct the molecular embedding and a Feed Forward Network (FFN) to predict the target from the molecular embeddings.

As illustrated in Figure 3, atoms and bonds retrieved from the input SMILES string are respectively associated to vertices and edges of the graph through RDKit³³. Following, a one-hot encoding of the main properties of atoms and bonds generates initial features vector. The features for each vertex and an edge that describe a bond are respectively concatenated in a directed edge features vector (each edge features vector has two possible directed edge features vectors, one per direction) that is passed to the D-MPNN. A non-linear activation function generates the hidden edge feature vectors, that are iteratively updated based on the local environment to produce the final atomic embeddings with information from incoming directed edge hidden states. Finally, an aggregation function derives the molecular embedding used in the FFN to make the prediction. The detailed theory behind the model can be found in literature²⁵ and is summarized in Figure 3.

<figure><img src="image_3.png" /><figcaption>**Figure 3.** Overview of the architecture of Chemprop implementing Direct-Message Passing Neural Network technique. The message passing update of the hidden vector for directed edge 2 → 1 is expanded for demonstration. Reproduced with permission from Ref. 25, Copyright 2023 Journal of Chemical Information and Modeling.</figcaption></figure>

Since molecules with different roles (i.e., solutes and solvents) are required as input, a default multi-molecule configuration was chosen. Two separated D-MPNN, one for the solute and one for the solvent, are trained with shared hyperparameters to then concatenate the resulting embeddings prior to the FFN.
## 2.3 Additional Features

In the case of VLE, the sole information provided by the molecular embeddings is not sufficient for the prediction of properties. Mixture properties, along with equilibrium conditions, influence the distribution of molecules between the vapor and liquid phases. In Chemprop, additional features can be implemented on the three levels (atom, bond, molecule). In particular, additional molecular features are concatenated to the final molecular embedding via

$$ \mathbf{h}'_m = \mathrm{cat}(\mathbf{h}_m, \mathbf{f}_m) \qquad (1) $$

where `cat()` denotes simple concatenation (vectors are indicated in bold and scalars in light fonts), `f_m` contains additional molecular features and `h'_m` is the final molecular embedding (by default, it is simply the aggregation over atomic embeddings `h_m` introduced in Section 2.2). Nevertheless, it is reported that concatenating a numerical array is an operation more suited for intrinsic properties²⁵ rather than bulk mixture properties. In this work the molecular embeddings have been scaled with the weighted values of molar fraction and equilibrium temperature through multiplication

$$ \mathbf{h}'_m = (w_x x_{m,i})(w_T T_{m,i}^{eq}) \mathbf{h}_m \qquad (2) $$

where $x_{m,i}$ is the molar fraction of the molecule $m$ for the equilibrium point $i$ at temperature $T_{m,i}^{eq}$ after normalization over 373.15 (in K). The weights $w_x$ and $w_T$ were tailored using the relevance factors on the experimental pressure of molar fraction and temperature respectively, derived from

$$ w_i(X_i, Y) = \frac{\sum_{j=1}^{n}(X_{i,j} - \bar{X}_i)(Y_j - \bar{Y})}{\sum_{j=1}^{n}(X_{i,j} - \bar{X}_i)^2 \sum_{j=1}^{n}(Y_{i,j} - \bar{Y})^2} \qquad (3) $$

with $X_i$ input parameter, $Y$ experimental value of every $j^{th}$ point for the total number of data $n$. More in detail, H₂S equilibrium points with molar fraction above the upper limit of the pre-training dataset where not considered in the correlation matrix. Indeed, the transformation of the embedding directly affects only the D-MPNN phase, that is focused on learning molecular representations and features. Therefore, using out-of-scale weights for molar fractions in the learning phase is associated with the risk of introducing a consistent bias in the feature extraction. The final models take as inputs the D-MPNN-derived molecular embeddings for solute and solvent, generated from their SMILES-encoded molecular structures, along with the equilibrium temperature and solute mole fraction, and return the predicted equilibrium pressure of the gas-solvent system.

## 2.4 Steps of Hyperparameter Optimization, Pre-training and Transfer Learning

**Step 2a.** The optimization of the hyperparameters started with 10 randomly sampled trails followed by 20 targeted iterations using two Tree-structured Parzen Estimator algorithms³⁴,³⁵. The search was conducted over the variables shown in Table 4.
<table><caption>Table 4. Optimized hyperparameters in the Feed-Forward Network training.</caption><thead><tr><td>Parameter</td><td>Description</td></tr></thead><tbody><tr><td>batch_size</td><td>the minibatch size</td></tr><tr><td>depth</td><td>the number of message-passing iterations</td></tr><tr><td>dropout</td><td>the dropout probability after each layer in the D-MPNN encoder and FFN</td></tr><tr><td>ffn_hidden_size</td><td>the size of each hidden layer in the FFN</td></tr><tr><td>ffn_num_layers</td><td>the number of layers in the FFN</td></tr><tr><td>hidden_size</td><td>the message size in the D-MPNN encoder</td></tr><tr><td>max_lr</td><td>the maximum learning rate used in the learning rate scheduler</td></tr><tr><td>init_lr</td><td>the initial learning rate expressed as the ratio of init_lr to max_lr</td></tr><tr><td>final_lr</td><td>the final learning rate expressed as the ratio of final_lr to max_lr</td></tr><tr><td>warmup_epochs</td><td>the number of epochs over which to ramp up the learning rate up from init_lr to max_lr expressed as a fraction of the total training epochs</td></tr></tbody></table>

**Step 2b.** During pre-training with COSMO-RS results, the number of epochs was limited to 30 while early stopping was evaluated with a patience of 5 epochs (maximum consecutive epochs without improvement in validation performance before interrupting the training). The Rectified Linear Unit (ReLU) was selected as activation function while scaled sums were used to aggregate atomic into molecular feature vectors. The Mean Squared Error (MSE) was used as the loss function and z-scores were employed during the training process (the target values are standardized with new mean equal to 0 and standard deviation equal to 1).

Transfer learning is an approach consisting in using the acquired information from an initial model to improve the training of a related model. This strategy is often used to transfer knowledge from a model trained on a large, low-quality dataset (referred as pre-training model) to a model trained on a smaller, high-quality dataset (referred as fine-tuning model) to improve its performance. In this work, the pre-training models were adapted to the experimental data with two tailored approaches:

**Step 2c.** All the CO2 dataset was involved in a one-step fine-tuning procedure where both the message-passing blocks of the solute and the solvent were frozen, updating the weights of the FFN only, ensuring minimal adjustments were needed while preserving the pretrained molecular representations. A visual rendering of the approach employed is provided in Figure 4 for illustration.
<figure><img src="image_4.png" /><figcaption>Figure 4. Transfer-learning selected option when using the CO₂ dataset. Adapted with permission from Ref. 25, Copyright 2023 Journal of Chemical Information and Modeling.</figcaption></figure>

**Steps 2d, 2e.** H₂, N₂, CH₄ and H₂S were separated into different datasets to be used respectively in a two-step fine-tuning procedure. First, the model was fine-tuned on experimental data of the less-soluble gases by freezing the message-passing network of the solute only. Although increasing the risk of overfitting through catastrophic inference, this approach was preferred over the previous to prioritize experimental data over simulations, due to the intrinsic limitations of COSMO-RS introduced in Section 2. Subsequently, a second, more restrictive fine-tuning phase was performed using mainly H₂S data, which exhibited higher solubility compared to the other gases. Since these data correspond to higher mole fractions and exhibit lower pressures compared to the rest of the dataset, to mitigate the risk of overfitting, all the message-passing blocks and first FFN layer were frozen during the training, allowing adjustments in the weights of the latest FFN layers only. This ensures the model preserves its generalizability while adapting to the specific behavior of the high-solubility data.

## 2.5 Model Performance Evaluation

To evaluate the performance of the models, a set of suitable statistical metrics was employed. For regression tasks where the target variables span significantly different scales, absolute metrics as the Mean Absolute Error (MAE) and the Root Mean Squared Error (RMSE) are preferred over percentage-based metrics.

MAE measures the average deviation of the predicted value $\tilde{Y}_i$ from the sampled value $Y_i$, providing a clear interpretation of the model performance in absolute terms.

$$MAE(Y_i, \tilde{Y}_i) = \frac{1}{n_{samples}} \sum_{i=1}^{n_{samples}} |Y_i - \tilde{Y}_i| \quad (4)$$

RMSE, on the other hand, penalizes larger deviations through squaring, making it more sensitive to outliers and extreme errors.

$$RMSE(Y_i, \tilde{Y}_i) = \sqrt{\frac{1}{n_{samples}} \sum_{i=1}^{n_{samples}} (Y_i - \tilde{Y}_i)^2} \quad (5)$$

In both MAE and RMSE, lower values indicate better model performance. The combined use of these metrics is preferred as it offers a balanced perspective on both error magnitude and outlier sensitivity³⁶.

The coefficient of determination (R²) is employed to assess the target variance explained by the models. R² provides a normalized measure of the model fitting ranging from 0 to 1, where values closer to 1 indicate a better fit.
$$R^2(Y_i, \tilde{Y}_i) = 1 - \frac{\sum_{i=1}^{n_{samples}} (Y_i - \tilde{Y}_i)^2}{\sum_{i=1}^{n_{samples}} (Y_i - \bar{Y})^2} \qquad (6)$$

Finally, the Mean Bias Error (MBE) is then introduced to evaluate any systematic tendency in the models. A positive MBE is associated to overprediction, while a negative MBE suggests underprediction.

$$MBE(Y_i, \tilde{Y}_i) = \frac{1}{n_{samples}} \sum_{i=1}^{n_{samples}} (\tilde{Y}_i - Y_i) \qquad (7)$$

The mean of each metric across all the individual models was evaluated to summarize the overall performance of the respective final model (standard deviations are reported for clarity). In addition to the quantitative evaluation of statistical metrics, graphical analysis was employed to further assess the accuracy of the models.

## 3 RESULTS AND DISCUSSION

In section 3, the statistical metrics were originally evaluated for the sole predictions of COSMO-RS on the experimental datasets. The values were then compared with those obtained from each respective model, based on the data divided into training and test sets. In all the cases examined, the D-MPNN exhibited higher accuracy and reliability across all indicators, as illustrated in Table 5 and Table 6.

<table><caption>Table 5. COSMO-RS model and D-MPNN model metrics evaluated on the solubility dataset.</caption><thead><tr><th rowspan="2">Results</th><th colspan="4">Statistical Criteria</th></tr><tr><th>MAE (bar)</th><th>RMSE (bar)</th><th>R2</th><th>MBE (bar)</th></tr></thead><tbody><tr><td>COSMO-RS</td><td>7.07</td><td>9.78</td><td>0.889</td><td>-2.21</td></tr><tr><td>Training</td><td>3.48 ± 0.22</td><td>5.64 ± 0.32</td><td>0.953 ± 0.006</td><td>0.035 ± 0.628</td></tr><tr><td>Test</td><td>3.84 ± 0.27</td><td>6.11 ± 0.52</td><td>0.944 ± 0.014</td><td>0.139 ± 0.894</td></tr></tbody></table>

<table><caption>Table 6. COSMO-RS model and D-MPNN model metrics evaluated on the selectivity dataset.</caption><thead><tr><th rowspan="2">Results</th><th colspan="4">Statistical Criteria</th></tr><tr><th>MAE (bar)</th><th>RMSE (bar)</th><th>R2</th><th>MBE (bar)</th></tr></thead><tbody><tr><td>COSMO-RS</td><td>17.4</td><td>27.9</td><td>0.64</td><td>-11.7</td></tr><tr><td>Training</td><td>7.36 ± 1.28</td><td>10.46 ± 1.41</td><td>0.938 ± 0.041</td><td>-0.306 ± 1.316</td></tr><tr><td>Test</td><td>8.3 ± 0.27</td><td>11.68 ± 0.52</td><td>0.903 ± 0.009</td><td>1.075 ± 0.681</td></tr></tbody></table>

The simultaneous reduction of both MAE and RMSE indicates that the D-MPNN model is capable of providing more accurate predictions, with lower average deviations and a reduced occurrence of large errors. This reflects both improved alignment with experimental values and enhanced reliability across the full pressure range considered. Although the selectivity model is inherently less precise compared to the solubility model, it shows greater relative improvement over COSMO-RS. These tendencies are further confirmed by the higher R² values obtained with the D-MPNN, suggesting that the model is successfully reproducing the experimental trends
across different systems. Regarding the MBE, COSMO-RS predictions are characterized by a pronounced
negative value, indicating a systematic underestimation in its predictions. This behavior is in line with its
thermodynamic model, where the assumption of ideal mixture of ideal gases for the vapor phase becomes
inconsistent at medium-high pressures. In contrast, all the D-MPNN achieve a near-zero MBE, demonstrating
its ability to correct this bias and provide more balanced predictions across all the pressure range.

3.1 Graphical Evaluation of the Models

To achieve a comprehensive understanding of the performance, the cross-plots of predictions of the D-MPNN
networks vs actual data were initially evaluated. Given the data preparation procedure illustrated in Section 2.1,
the results are shown on three representative models for each of the 9 in the final models in Figure 5.

<figure><img src="image_5.png" /><figcaption>Figure 5. Cross plots of the three representative models for solubility (a, b, c) and selectivity (d, e, f).</figcaption></figure>

First, data in both training and test partitions of the datasets exhibit a balanced distribution across the full range
of experimental conditions. Second, the cross-plot evaluation highlighted the absence of significant outliers and
systematic deviations. Additionally, individual models showed similar performances on both training and test
data, reinforcing consistency in the overall estimations.
The frequencies of predicted data for all the sub-models as a function of the residual error are reported in Figure 6. The two models present centered distributions without significant tails, further confirming the lack of systematic deviations and outliers. Consistently with its inherent lower precision, the selectivity model exhibits a broader spread distribution compared to the solubility model. More than 70% of the solubility predictions present a residual error that is below the associated MAE value in both training and test datasets, while the percentage is reduced to 60% in the case of selectivity predictions (see Table 5 and Table 6).

<figure><img src="image_6.png" /><figcaption>**Figure 6. Residual error distribution for the solubility model (a) and the selectivity model (b).**</figcaption></figure>

To assess the physical consistency, the trends of the estimations as a function of the model input parameters were investigated for various datapoints only in the test set. The results revealed that the predicted trends align with the experimental ones: for a given mixture, the equilibrium pressure increases with an increase in the equilibrium temperature or an increase in the liquid mole fraction of the solute. As an example, predicted trends at different mole fractions and temperature between CO₂ and two benchmark physical solvents (N-Methyl-Pyrrolidone, Methanol) are illustrated in Figure 7. Lines connect predicted points at the same values of CO₂ molar fraction than the measures.

<figure><img src="image_7.png" /><figcaption>**Figure 7. Predicted equilibrium pressure at 25 °C and at 40 °C for the two mixtures CO₂-NMP and CO₂-methanol (test dataset).**</figcaption></figure>

Statistical metrics (i.e., MAE, RMSE, MBE) for all test points grouped by chemical family were then evaluated to assess the models' robustness across different functional groups (Tables S3 and S4). In the solubility dataset (Table S3), most families achieve low absolute errors (< 3 bar) with low bias (< 1 bar), whereas amides and sulphurated solvents exhibit larger errors (MAE between 5 and 11 bars, MBE up to 9 bars). In the selectivity
dataset (Table S4), alkanes and amides show moderate error (MAE between 5 and 6 bars), while aldehydes and phenols display the highest deviations (MAE of 17.0 bars and 16.1 bars, respectively). Across both tasks, families with strong, specific interactions (e.g., amides, phenols, sulphurated) are associated with larger residuals, suggesting that future work could target improved conformer sampling or descriptor refinement for these classes. Overall, the models maintain low prediction errors for most solvent families, confirming its general applicability across diverse chemical classes without requiring family-specific adjustment.

## 3.2 Sensitivity Analysis

To evaluate the impact of each input parameter on the predictions, the relevance factor introduced in Section 2.3 can be calculated with Equation 3 substituting the experimental values $y_j$ with the predicted values $\tilde{y}_j$. Nevertheless, the relevance factor relies on the assumption that the relationships are primarily linear, making this approach not well-suited for neural networks, where activation functions introduce non-linearity in the model (see Section 2.2). Moreover, relevance factors face challenges in retrieving the relationship between the abstract molecular features $\mathbf{h}'_{m,i}$ and the final prediction $\tilde{Y}_i$.

Therefore, to initially confirm an effective learning from the molecular structures, the learned representation of the solvents after the D-MPNN phase was reduced with Principal Component Analysis (PCA)³⁷ on three axes to then analyze points with similar reduced properties through agglomerative clustering³⁸ and visualization. In PCA, explained variance is a measure for the proportion of total variability in the original data captured by each axis: high cumulative variance ensures minimal information loss during dimensionality reduction, preserving the relationships between the points in the latent space. In this study, the three axes exhibit high cumulative explained variance in both models (76% for the solubility model, 90% for the selectivity model) indicating that the projection retains the majority of structural information encoded by the D-MPNN. The agglomerative clustering identified homogeneous groups of molecules characterized with shared geometries, polarity profiles, or functional groups specially in the solubility model, where the weights to produce the fingerprints remained unvaried in the fine-tuning phase. These resulting patterns already indicate that the D-MPNN successfully encoded chemically meaningful features into its latent representations. The graphical visualizations of the reduced embeddings are provided in Figure 8.
<figure><img src="image_8.png" /><figcaption>**Figure 8.** 3D PCA and Agglomerative Clustering (colors) and visual chemical family clustering (circles) of the embeddings in solubility (a) and selectivity (b) D-MPNN.</figcaption></figure>

Finally, to quantify the effect of all the independent parameters in the model (i.e., molecular structure, liquid molar fraction of the solute, equilibrium temperature) on the equilibrium pressure, a specific sensitivity analysis was performed. In particular, all data belonging to solvent molecules with at least one hydroxyl group were used unvaried in order to obtain reference estimations with the models. Following, all the features were individually perturbed: $x_{m,i}$ and $T_{m,i}^{eq}$ were randomly varied of ±25%, still respecting the physical boundary values (0, 1) for both the molar fractions, while the hydroxyl function was removed to include the impact of structural features.

The MAE between the reference predictions and the predictions obtained perturbing the features was selected as numerical criteria to estimate their contribution in the model.

The findings indicated that all features play a meaningful role in the predictions, with a non-negligible magnitude of produced deviations (Figure 9). In both models, the same trend was observed, with structural features emerging as the most impactful, while temperature-related features had a comparatively lower influence. Additionally, the molar fractions exhibited a higher weight in the selectivity model, likely due to the rapid pressure increase caused by the presence of non-condensable gases as solutes.

<figure><img src="image_9.png" /><figcaption>**Figure 9.** Barplot of the MAE introduced due to feature perturbations in the two models.</figcaption></figure>
4 CONCLUSIONS

Integrating first estimations from the quantum chemical-based model of COSMO-RS and experimental data
into a D-MPNN algorithm has proven to be an effective strategy for improving vapor-liquid equilibrium (VLE)
predictions of physical solvents for carbon capture applications. By leveraging featurization and transfer learning
with directly suited strategies, the final architecture successfully discerned the quality of input data without
introducing overfitting or unwanted biases, ensuring that corrections were applied selectively while preserving
the learned information.

Compared to COSMO-RS alone, the developed models exhibited a consistent improvement in the statistical
metrics across all examined systems, including CO₂-solvent mixtures and impurities-solvent mixtures. The
analysis on test results showed alignment with the thermodynamic trends for solute molar fractions and
temperatures, while the bias error introduced in COSMO-RS due to ideality assumptions in the vapor phase
was corrected.

Visual and numerical analysis of the data in train and test sets confirmed the reliability of the models and the
absence of significant outliers. These observations were further established analyzing the distributions of their
residuals. Finally, clustering the molecular embeddings and performing a sensitivity analysis highlighted the
importance of the selected architecture and the parameterization of the model. Molecular features were found
to have the highest contribution in the estimations, emphasizing the role of the D-MPNN in processing
structural information. Nevertheless, the scaling effect of additional features proved to be essential for accurate
predictions.

Although the model is developed with a focus on non-reactive solvents, it can be readily applied to neoteric
solvents such as deep eutectic solvents and ionic liquids, which typically exhibit lower variability in molecular
descriptors and interaction patterns, making them more suitable from a learning perspective. Moreover, the
methodology accounts not only for solubility but also for selectivity, a critical performance aspect often
overlooked in similar studies, enabling a more realistic and informative assessment of solvent behavior.

A key strength of the methodology lies in the extensive pre-training phase on more than 30,000 simulated data
points derived from COSMO-RS. This large-scale simulation dataset enhances the generalizability of VLE
predictions beyond the limited experimental results in the fine-tuning. Therefore, while models directly trained
on smaller and curated datasets may achieve improved metrics in narrow application windows, the proposed
approach allows the systematic exploration of solvent candidates with minimal experimental input, accelerating
the identification of viable physical solvents while maintaining high accuracy across a wide range of conditions.
Future developments will focus on enriching the model pipeline with additional molecular descriptors and
adapting it for chemical solvents, further improving the prediction accuracy and extending its applicability to a
broader range of carbon capture technologies.
# 5 ACKNOWLEDGMENTS

The authors acknowledge the financial support received by State funding administered by the National Research Agency under the France 2030 initiative, (reference No. ANR-22-PESP-0008), and through the "Investissements d'Avenir" program, (reference No. ANR-18-EURE-0021).

# CONFLICTS OF INTEREST

The authors declare no competing financial interest.

# 6 NOMENCLATURE

<table><tr><td>CCS</td><td>carbon capture and storage</td></tr><tr><td>CH<sub>4</sub></td><td>methane</td></tr><tr><td>CO<sub>2</sub></td><td>carbon dioxide</td></tr><tr><td>COSMO-RS</td><td>Conductor-like screening model for real solvents</td></tr><tr><td>DES</td><td>deep eutectic solvents</td></tr><tr><td>D-MPNN</td><td>directed-message passing neural network</td></tr><tr><td>EoS</td><td>equation of state</td></tr><tr><td>FFN</td><td>feed-forward network</td></tr><tr><td>GCNN</td><td>graph convolutional neural network</td></tr><tr><td>G<sup>ex</sup></td><td>Gibbs excess</td></tr><tr><td>H<sub>2</sub></td><td>hydrogen</td></tr><tr><td>H<sub>2</sub>S</td><td>hydrogen sulfide</td></tr><tr><td>ILs</td><td>ionic liquids</td></tr><tr><td>IQR</td><td>Interquartile Range</td></tr><tr><td>MAE</td><td>mean absolute error</td></tr><tr><td>MBE</td><td>mean bias error</td></tr><tr><td>ML</td><td>machine learning</td></tr><tr><td>MSE</td><td>mean squared error</td></tr><tr><td>N<sub>2</sub></td><td>nitrogen</td></tr><tr><td>NRTL</td><td>Non-Random Two-Liquid</td></tr><tr><td>PCA</td><td>principal component analysis</td></tr><tr><td>R<sup>2</sup></td><td>coefficient of determination</td></tr><tr><td>RMSE</td><td>root mean squared error</td></tr><tr><td>t-SNE</td><td>t-distributed stochastic neighbor embedding</td></tr><tr><td>UNIFAC</td><td>UNiversal quasichemical Functional-group Activity Coefficients</td></tr><tr><td>UNIQUAC</td><td>UNiversal QUAsiChemical</td></tr><tr><td>VLE</td><td>vapor-liquid equilibrium</td></tr></table>
# 7 AUTHOR CONTRIBUTIONS (Credit)

**Edoardo Parascandolo**: Funding Acquisition, Conceptualization, Methodology, Software, Validation, Formal analysis, Investigation, Data curation, Visualization, Writing – original draft

**Vincent Gerbaud**: Supervision, Writing – review & editing

**David Camilo Corrales**: Methodology, Validation, Writing – review & editing

**Noslen Hernández**: Methodology, Writing – review & editing

**Sophie Thiebaud-Roux**: Supervision, Resources, Project administration, Writing – review & editing

**Ivonne Rodriguez-Donis**: Supervision, Resources, Project administration, Writing – review & editing

# 8 AUTHOR INFORMATION (ORCID<sup>ID</sup>)

**David Camilo Corrales**: 0000-0003-4717-3040

**Vincent Gerbaud**: 0000-0003-2738-7922

**Noslen Hernández**: 0000-0003-2438-1912

**Ivonne Rodriguez-Donis**: 0000-0002-4232-6792

**Edoardo Parascandolo**: 0009-0003-3536-2132

**Sophie Thiebaud-Roux**: 0000-0002-9409-8306

# 9 DATA AND SOFTWARE AVAILABILITY

All data and scripts used in this work are available in a public GitHub repository at https://github.com/edparasca/CO2. The repository contains the COSMO-RS simulated data, the experimental VLE datasets, the scripts for training, transfer learning and prediction, and a modified version of chemprop/models/multi.py used to scale molecular fingerprints with equilibrium conditions.

The Supporting Information includes additional data and analyses to complement the main text. Specifically, it provides: (i) the analysis of solvent families represented in the COSMO-RS dataset, pre-training and fine-tuning sets, including the occurrences, statistical outliers, and the ranges of temperature, mole fraction, and equilibrium pressure explored; (ii) model performance metrics (MAE, RMSE, MBE) grouped by solvent family for both solubility and selectivity predictions.

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## 11 GRAPHICAL ABSTRACT

<figure><img src="image_10.png" /></figure>