# Defect formation energy of impurities in 2D materials: How does data engineering shape machine learning model selection?

A. El Alouani, M. Al Khalfioui, A. Michon, S. Vézian, P. Boucaud, Minh-Tuan Dau

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A. El Alouani, M. Al Khalfioui, A. Michon, S. Vézian, P. Boucaud, et al.. Defect formation energy of impurities in 2D materials: How does data engineering shape machine learning model selection?. Materials Today Physics, 2026, 61, pp.102006. 10.1016/j.mtphys.2025.102006. hal-05504904

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# Defect formation energy of impurities in 2D materials: How does data engineering shape machine learning model selection?

A. El Alouani, M. Al Khalfioui, A. Michon, S. Vézian, P. Boucaud, M.T. Dau*

Université Côte d'Azur, CNRS, CRHEA, 06560 Valbonne, France

## ARTICLE INFO

**Keywords:**

Machine learning workflow

Descriptor engineering

Model selection

Statistical models

Artificial neural network

2D materials

Doping

Interstitial

Adsorption

## ABSTRACT

Regressive Machine learning (ML) has emerged as an alternative method for theoretically evaluating materials properties. Most of ML-based study of materials properties including dataset handling, feature spaces, transformers and estimators have been reported without questioning the ML foundation of these components and their interactions with the outcomes because of the availability of homogeneous datasets, standardized training conditions and technical implementation challenges. In this paper, a database of defect impurities in 2D materials was used to assess the impact of ML workflow's components on the training-inference process. By investigating descriptor engineering (vectorized matrix properties) and model algorithms (statistical tree-based and artificial neural network - ANN models) on two sub-datasets derived from the database (interstitial-int and adsorbate-ads impurities), we report a comprehensive study of the ML-based prediction of the energy formation of the impurities in 2D materials. Quantitatively, for the statistical models, the training errors lower than 1.4 eV and 1.1 eV were found thanks to the descriptor engineering for the int and the ads datasets, respectively. Regarding the ANN models, these values are 2.1 eV and 1.3 eV. The prediction errors on the unseen data (test sets) were found lower than the ones obtained without descriptor engineering for all models. However, the overfitting effect remains visible but less pronounced for the ads dataset than for the int dataset. This finding reveals the impact of the dataset characteristics on the performance of the ML ecosystem involving data engineering and model algorithms. Beyond the search of best performances in regressive ML prediction of 2D materials properties, our work demonstrates a full-scale study of the ML process starting from the data engineering to model evaluation and selection, allowing to benchmark the criteria for further ML assessment in terms of training, models and prediction. Our results could be reference for further works in ML-led prediction physics of materials science.

## 1. Introduction

Since A. Geim and K. Novoselov discovered the great potential of graphene, one monolayer thick graphite in 2004 [1], numerous other two-dimensional (2D) materials have emerged, attracting significant scientific interest. For instance, transition metal dichalcogenides (TMDs) exhibit a direct bandgap in monolayer form, unlike their bulk counterparts, which have an indirect bandgap [2]. This direct band gap makes TMDs particularly appealing for applications such as sensitive photodetectors electronics, renewable energy technologies and quantum computing [3]. Similarly to bulk semiconductor materials, doping plays a crucial role in controlling the type and concentration of charge carriers in 2D materials, enabling customized properties for specific applications [4]. In the case of pure graphene, substitutional doping with elements such as N [5], B [6], S [7] and Si [8] disrupts the hybridization of sp² carbon atoms, leading to the opening of its

Dirac cone and enabling the modulation of graphene-based electronic devices [9,10]. Achieving both n-type and p-type behaviour in TMDs doping is also of great interest because it enables the achievement of in-plane or vertical TMD heterojunctions [11–14]. In particular, doping TMDs with transition metals has been shown to enable long-range ferromagnetism at room temperature, which is crucial for spintronic and valleytronic applications [15–18]. Moreover, doping in TMDs has also been explored for catalytic applications. For instance, Nb-doped MoS₂ has been shown to exhibit significantly enhanced hydrogen evolution reaction activity thanks to the improved charge transfer kinetics and the increase in density of active site [19].

A key factor in assessing the viability of a dopant within a 2D lattice is the formation energy, which predicts its stability. Classical computations like Density Functional Theory (DFT) are commonly used to estimate the formation energy. Despite their reliability, DFT methods

<footnote>* Corresponding author.
E-mail address: mtd@crhea.cnrs.fr (M.T. Dau).</footnote>
are computationally intensive and often too slow for large-scale tasks, such as screening multiple dopants or performing high-throughput studies [20,21]. In recent years, machine learning (ML) has emerged as an alternative tool in predicting band gaps and other specific physical properties of materials, both independently and in conjunction with DFT [22–24]. ML models offer high accuracy while significantly reducing computational costs. Indeed, it has been demonstrated that a deep learning model could predict Heyd–Scuseria–Ernzerhof (HSE06) bandgaps for 1503 binary compounds at speeds 10⁴ times faster than DFT calculations [25]. Beyond band gap predictions, ML has also been employed to identify optimal 2D candidates capable of hosting defects for quantum and neuromorphic information processing [26]. Therefore, by leveraging existing datasets of material properties, ML models may efficiently and accurately predict the formation energy of new dopants, significantly reducing computational costs.

In the realm of ML, feature engineering including descriptor construction could provide additional informative inputs whereas model selection remains a pivotal process to capture the relationship between the input features and the target properties. Although the feature engineering of the ML pipeline has been investigated within the framework of descriptor transformation to predict material properties [24,27–29], there is still a lack of investigation on the interplay between the feature engineering, ML model algorithms, datasets used for the ML pipeline and the target properties. To address this, we have performed a ML process to infer the formation energy of impurities in 2D materials. The ML process was investigated in equivalent conditions for a fair comparison, allowing to scrutinize the impact of the above fundamental components of the ML ecosystem on the prediction performance.

In this paper, we make use of Impurities in 2D Materials Database (IMP2D) [30], a robust data register based on DFT calculations that was split into two distinct datasets: interstitial (int) and adsorbate (ads) defects. The descriptor engineering was inspired by the strategy developed in our earlier work on predicting band gaps and work functions of 2D materials [31]. In that study, we demonstrated the effectiveness of data preprocessing and descriptor engineering, which were derived from vectorized property matrices and empirical property functions, in improving the performance of tree-based models. In the present work, we further introduce an additional descriptor based on Euclidean distance (see Methods) to build a full set of descriptors. Regarding the model algorithms, we employ two model classes: statistics-based (decision tree-based) and artificial neural network (ANN) models to predict the formation energy of both int and ads defects.

The assessment of the model evaluation and selection within the context of descriptor engineering is challenging due to the nested features of the outcomes in terms of training performance and prediction accuracy. To enable a rigorous analysis and comparison, the paper is structured as follows. First, we present the training-inference results from the descriptor engineering in two separate sections with respect to the statistical models (Section 3.1) and the ANN models (Section 3.2). By grouping similar ML algorithms, the analysis is straightforward and consistent between the two model classes with similar conditions in terms of datasets and features. This also permits to gauge the impact of descriptor engineering on the ML cycles with respect to the int and ads datasets of the IMP2D database. It has been shown that the descriptor engineering has positive impact on the statistical model-based training with root mean square error (RMSE) lower than 1.5 eV and 1.1 eV for the int and ads datasets, respectively. A similar positive trend was also observed for ANN-based training with RMSE lower than 2.1 eV and 1.3 eV with respect to the int and ads datasets, respectively. Regarding the inference on the unseen data (test set), the prediction on the int dataset shows better results compared to the ads dataset for both ML model classes (see Table 1). However, the overfitting remains pronounced when embedding the descriptor engineering. The overfitting was reduced or exacerbated depending on the datasets and the model complexity. Second, in the last (Section 3.3), the model evaluation and selection are addressed by underlying the roles of model

algorithms and datasets with respect to the findings of the previous sections. The advanced analysis permits to point to nested interactions between the data engineering, model algorithms and dataset characteristics. Therefore, it is imperative that these ML components and their interdependencies should be mutually and carefully considered for benchmarking and comprehensive ML-assisted studies of materials properties.

## 2. Methods

The ML pipeline can be broken down into four main building blocks or components. Each component dynamically contributes to the training, accuracy, analysis, and interpretability of the models (Fig. 1). Since this study aims to provide a comprehensive analysis of the ML pipeline for the regression-based materials properties, we primarily focus on two critical components: descriptor/feature engineering and model selection. The methodology applied throughout this work is detailed in Fig. 1 for both the int and ads defect datasets.

### 2.1. Data pre-processing

We initiated our work by pre-processing IMP2D database, which contains structural, thermodynamic and electronic data for over 17 500 single-impurity defects across 53 experimentally synthesized 2D monolayers [32]. These defects are categorized into int and ads sites that constitute the two defect datasets used in this study. Within each dataset, we selected only the elements exhibiting relaxation fully convergence. We then filtered out entries containing missing values to maintain data integrity. Finally, we further refined the datasets by retaining only the elements with defect formation energies within the predefined ranges [−10 eV; 30 eV] for int dataset and [−10 eV; 50 eV] for ads dataset, ensuring the selection of physically relevant defects and the removal of several outliers with high-energy values. The formation energy distributions of the datasets present a Gaussian shape around the mean values (see Supplementary Material).

### 2.2. Feature engineering

One of the key approaches enabling to tune ML model performance is feature engineering, which involves selecting the most informative descriptors that significantly impact the target property. In this study, we utilize a total of eight descriptors (Fig. 1 and Supplementary Material). Four of these descriptors (unit cell volume, unit cell mass, depth parameter and expansion factor) are directly extracted from the database (referred as features set 1). Among them, the depth parameter and the expansion factor are essential descriptors that are strongly related to the position, type of the defects and the distortion of the host [30]. These two descriptors encode the geometric response of the lattice to the presence of a defect. The depth parameter distinguishes adsorbed from interstitial configurations, while the expansion factor captures the in-plane dilation caused by impurity-induced strain. The remaining descriptors are introduced using specific computational approaches. The first derived descriptor, electronegativity, is calculated using an empirical function [33]. It has been established that difference in electronegativity governing charge redistribution, bond polarity and the extent of local structural relaxation around impurities, influences the defect stability [34,35]. For a molecular composition $A_aB_bC_c$, the electronegativity $\chi$ of the molecule is determined as: $\chi = (\chi_A^a\chi_B^b\chi_C^c) \frac{1}{a+b+c}$. The second descriptor, derived from a mean-centred Euclidean transformation, quantifies the Euclidean distance of each data point from the mean of the two important descriptors (depth parameter and expansion factor) of the dataset. It is calculated as: $d = \sqrt{(x - \bar{x})^2 + (y - \bar{y})^2}$, where x, y, $\bar{x}$, $\bar{y}$ are two features from IMP2D (depth parameter, expansion factor) and their centroid values. Finally, the two last descriptors (covalent radius and dipole polarizability)
<figure><img src="image_2.png" /><figcaption>**Fig. 1.** Schematic of the ML pipeline. The workflow of this work includes data pre-processing, feature engineering (using direct and generated descriptors) and model training-evaluation using statistical models (RF, GB, XGB) and ANN (MLP, PINN-like) for predicting the formation energy of int and ads defects in 2D materials.</figcaption></figure>

<caption>**Table 1**</caption>

Summary of RMSE metrics of all models and datasets investigated in this work. The mean and the standard deviation of the dataset's target are also shown.

<table><thead><tr><th rowspan="2"></th><th rowspan="2"></th><th colspan="5">Model</th></tr><tr><th>RMSE (eV)</th><th>RF</th><th>GB</th><th>XGB</th><th>MLP</th><th>PINN-like</th></tr></thead><tbody><tr><td rowspan="2">int<br>μ = 4.22 eV<br>σ = 4.59 eV</td><td>train</td><td></td><td>1.22</td><td>1.37</td><td>0.36</td><td>2.05</td><td>1.9</td></tr><tr><td>test</td><td></td><td>2.2</td><td>2.54</td><td>2.21</td><td>2.93</td><td>2.77</td></tr><tr><td rowspan="2">ads<br>μ = 1.94 eV<br>σ = 2.69 eV</td><td>train</td><td></td><td>1.03</td><td>0.69</td><td>0.25</td><td>1.15</td><td>1.2</td></tr><tr><td>test</td><td></td><td>1.72</td><td>1.51</td><td>1.61</td><td>1.57</td><td>1.59</td></tr></tbody></table>

are vectorized descriptors. They encode intrinsic size and electronic response properties of the dopant atoms. The covalent radius reflects atomic size mismatch, a primary source of local strain and bond-length modification around substitutional impurities [35–37]. The dipole polarizability describes the ability of an atom or molecule to deform its electron cloud, affecting local screening and relaxation [38–40]. These inherent properties would have a strong physico-chemical correlation with the defect formation energy. To consider the doping effect, we included the dopant atoms into the molecule formulas with a stoichiometry of 1. The vectorization method derived from the property matrices for these descriptors was well detailed in our previous work [31]. Collectively, the eight descriptors (the four direct descriptors and the four generated descriptors) form the feature set 2 (Fig. 1 and Supplementary Material).

## 2.3. Model selection

As a representative statistical approach, we used tree-based models such as Random Forest (RF), Gradient Boosting (GB) and Extreme Gradient Boost (XGB). RF operates by constructing multiple independent decision trees and averaging their predictions, leading to improved stability and reduced variance. GB and XGB, on the other hand, build trees sequentially, where each new tree corrects the errors of the previous ones, resulting in highly accurate models that excel in structured tabular datasets. As ANN models, we used Multilayer Perceptron (MLP) and Physics-Informed Neural Networks-like (PINN-like) model inspired by PINN architecture [41]. MLP consists of multiple layers of fully connected neurons that process information through activation functions and backpropagation technique, allowing them to detect intricate patterns in different manner than the tree-based models. PINN, a fine-tunable model, incorporates physical laws essentially derived from partial differential equations and boundary conditions into the
learning process, ensuring that predictions adhere to established phys-
ical principles. This integration is particularly valuable in materials
science, where the domain knowledge can enhance model reliability
and interpretability. Furthermore, recent studies have demonstrated the
efficacy of ANN-based approaches in predicting defect formation en-
ergies [42,43], showcasing the potential of ANN-based methodologies
in this domain. In this work, we introduce our PINN-like model (see
Supplementary Material) alongside the MLP to ensure a fair compar-
ison of model architectures compared to the statistical models and to
demonstrate the potential of this modified physics-induced ANN in the
regressive ML of materials properties.

## 2.4. Setup and resources

The ML pipeline was build using sklearn package for statistical mod-
els and Pytorch Lightning 1.8 framework for ANN-based models [44].
All dependencies used are compatible with Python 3.8 environment.
The 4-cores and 8-cores CPUs with 16 GB RAM were used for running
the descriptor generator and model training-testing, respectively.

# 3. Results and discussion

## 3.1. Descriptor engineering for statistical model-based training and evaluation

This section describes the results from the descriptor engineering applied to the tree-decision-based models, which are trained following statistical rules. The data engineering is performed on the int and ads datasets. Grid search combined with the cross validation technique was employed to tune the hyperparameters of the models (see Supplementary Material). The results presented below are obtained from the models with the best scores.

3.1.1. *Model training on the int defect dataset*

The training performance of the ML models on the int dataset using feature set 1, which is composed of four descriptors extracted from the IMP2D database, is illustrated in Fig. 2a. The models achieve R² values of 0.89, 0.85 and 0.88 on the training set for RF, GB and XGB, respectively, indicating a reasonable correlation between predicted and true defect formation energies. However, a noticeable dispersion of the data points, particularly at higher energy values, suggests a training limitation regarding the feature set 1. We note that the value range is relatively large (from -10 eV to 30 eV) as compared to the band gap or work function values [31]. Upon integrating four additional descriptors into the feature set 1 (feature set 2), a significant improvement in model performance is observed, as shown in Fig. 2b. The R² values increase to 0.93, 0.91 and 0.99 for RF, GB and XGB, respectively, with a narrow dispersion of predicted values along the ideal linear fit. This indicates excellent fits between the outcomes and the true values of the defect formation energy upon the training process. However, the strong fits, especially with R² = 0.99, should be carefully evaluated because they may stem from overfitting which will be discussed in the model evaluation on the test set (see Section 3.1.3)

3.1.2. *Model training on the ads defect dataset*

A similar trend is observed for the ads defects dataset. As shown
in Fig. 3a, the models trained with the feature set 1 achieve decent
R² values of 0.8, 0.91 and 0.99 for RF, GB and XGB, respectively. In
the limit range of energy (from -10 to 50 eV), the RF and GB models
seem to provide a good fit for the data having a distribution less than
10 eV (high density of data point), whereas the data above 10 eV
exhibit significant dispersion. On the contrary, an excellent fit on the
entire dataset obtained by XGB model (R² = 0.99) with few database-
based inputs is likely due to a strong overfitting, which is confirmed by
evaluating the test set (see Section 3.1.3). When the additional derived
descriptors are included in the feature set 2, a notable improvement

in predictive performance is observed for the RF and GB models, with
increased R² values and a tighter point cloud around the linear fit. Fur-
thermore, the presence of additional descriptors does not downgrade
the training of the XGB model since R² is still close to 1 and their
positive impact will be assessed below by considering the prediction
of the unseen dataset (test set).

3.1.3. *Evaluation of the test sets (int and ads)*

As mentioned above, the model training involving generated de-
scriptors was improved in terms of R² metrics. This section focuses on
the performance of the trained models with respect to the unseen data.
To this end, the RMSE error, which was computed from the inference
of the test subsets for both datasets (int and ads), is shown in Fig.
4. One may identify two characteristics of the resulting metrics. First,
the relative errors obtained from the prediction of the test sets are
clearly reduced regardless of the employed models by at least 10% (RF
model on ads dataset) and up to 30% (GB model on ads dataset) when
involving the generated descriptors as compared to the results without
the generated descriptors. Second, the discrepancies in RMSE errors
between the training and test sets (RMSE<sub>test</sub> - RMSE<sub>train</sub>) are found to
be in the range of [0.98 eV; 1.85 eV] and [0.69 eV; 1.36 eV] for the
models obtained with generated descriptors on the int and ads dataset,
respectively (Fig. 4b, d). These ranges are lower than the ones of the
corresponding models without generated descriptors (Fig. 4a, c), except
for the performance of the XGB model on the int dataset (0.89 eV). In
general, the discrepancy between the training and the test performance
is a good indicator to assess the impact of the overfitting: greater the
discrepancy is, more the overfitting emerges. Therefore, it is shown that
the generated descriptors reduce the overfitting for most of the models
with respect to the datasets. Considering the exceptional case of XGB
model, the data engineering reaches somehow the limit when using an
high-variance-generating algorithm such as XGB, which is known to fit
excessively the training data. The overfitting becomes overwhelming
with respect to the int dataset. Owing to a low amount of training data
in the int dataset, the descriptor engineering might generate a great
number of irrelevant features, giving rise to the magnification of the
overfitting. On the other hand, the overfitting was reduced for the ads
dataset compared to the trained models without generated descriptors.
This is likely thanks to the generated descriptors that permit to remove
some noise or inconsistencies of the input features, enabling the model
to capture slightly better the dataset patterns.

To provide insights into the influence of the generated descriptors
on the model performance, the feature importance score regarding
the prediction of the defect formation energy is presented in Fig.
5. It is clearly shown that the descriptors generated by vectorizing
property matrices greatly contribute to the training-inference of the
defect formation energy for both datasets and all models, especially
covalent radius and dipole polarizability. However, the contribution
of generated descriptors (Euclidean distance, electronegativity) is not
exceptional, especially for the int dataset. The explainable feature
importance and their local contributions to specific prediction of the
descriptors computed based on Shapley value are also in line with
the model-based feature importance (Supplementary Material). These
findings indicate the versatility and efficiency of our data engineer-
ing approach regarding the training-inference performance of diverse
datasets. This finding is also consistent with respect to ML algorithms
when using ANN-based models as discussed below.

3.2. *Descriptor engineering for ANN-based model training and evaluation*

This section addresses the application of the generated descriptors
on the training-inference process of the ANN-based models. We have
constructed fully connected neural networks (a MLP) and a regularized
model by adding a physics-induced regularizer (PINN-like model) to
assess the impact of the features on the model performance. Several
model architectures were constructed up to 6 hidden layers containing
<figure><img src="image_3.png" /><figcaption>(b) Feature Set 2</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>**Fig. 2.** Predicted values versus true values of int defect formation energy upon the model training using statistical models with feature set 1 (a) and feature set 2 (b). The lines are linear regression of the data points with confidence interval of 95% as a guide to the eye.</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>(b) Feature Set 2</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>**Fig. 3.** Predicted values versus true values of ads defect formation energy upon the model training using statistical models with feature set 1 (a) and feature set 2 (b). The lines are linear regression of the data points with confidence interval of 95% as a guide to the eye.</figcaption></figure>

various number of neurons. Hyperparameters tuning was also performed to select decent architectures for the study. Significant results (presented hereafter) were obtained by an architecture containing 3 hidden layers, which are composed of (16, 32, 32) neurons. For the training, regularization such as dropout was not employed due to the small datasets and small-scale architectures. We have employed Adam optimizer to update weights and bias during the training. To tune the hyperparameters, we have focused on learning rate, scheduler factor, epochs, weight decay while keeping other hyperparameters as default values. It turns out that the training is very sensitive to the learning rate. The results that we show below concern fixed values (epochs = 2500, weight decay = $10^{-5}$, scheduler factor = 0.85) versus learning rates for which the best results were achieved. Note that the training was carried out without batching since the datasets are not memory-consuming and the number of epochs is a stopping criterion. The input data having different units and scales were transformed by standardization to make all features equal and to stabilize the training.

In the following, we refer the ANN model trained without descriptor engineering, the ANN and the PINN-like models trained with descriptor engineering to M1, M2, M3, respectively.

### 3.2.1. *Model training on int defect dataset*

The training RMSE of the formation energy for the int dataset as a function of learning rate is shown in Fig. 6. The impact of the descriptor engineering on the ANN models is assessed by comparing the performance of M1 against M2 and M3, which include the features generated by vectorizing property matrices. Qualitatively, we found for all learning rates that the RMSE decreases drastically at the first stage of the training (about 300 epochs), then, it keeps going to decrease slowly until the quasi-plateau. Nevertheless, the training dynamics of the model M1 differs from the ones of M2, M3. Indeed, after 300–400 epochs, the RMSE of the models M2 and M3 are lower than M1, indicating the positive effect of the generated features on the model
<figure><img src="image_7.png" /><figcaption>Fig. 4. RMSE metrics of the training with the feature sets 1, 2 and the prediction on the test sets of the int dataset (a, b) and the ads dataset (c, d).</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Fig. 5. Model-induced feature importance of descriptors regarding the statistics-based model training of defect formation energy.</figcaption></figure>

training. Quantitatively, the RMSE at the end of the training is lower for the models M2, M3 as compared to M1, from 20% to 30% smaller when training with generated features for learning rate equal to $5 \times 10^{-4}$ (LR1) and $1 \times 10^{-3}$ (LR3), respectively. The lowest RMSE of 1.9 eV is found for the models M2 and M3 (LR3). However, a close inspection unveils a poor training of the model M3 with learning rate equal to $8 \times 10^{-4}$ (LR2) where the plateau was reached out earlier than the models M1 and M2. This is indicative of an over-regularization due to the physics-based regularizer used in the model M3. The physics regularization somehow penalizes the learning process, indicating an instability of the regularizer with respect to the learning rate hyperparameter. Therefore, a good compromise in selection of an appropriate learning rate could lead to an improved performance when employing regularization technique as shown in Fig. 6 (training with LR3). It is worth noting that the learning rate values (not shown) outside the considered range substantially downgrade the training performance, hence, they are not open to discussion.

### 3.2.2. Model training on ads defect dataset

In parallel, we assess the model training behaviour on the ads defect dataset under the same learning rate values (LR1, LR2 and LR3). The RMSE trends are shown in Fig. 7. As in the int dataset, we compare three model configurations: M1 (MLP without descriptor engineering), M2 (MLP with generated descriptors) and M3 (MLP with physical constraints or PINN-like with generated descriptors). We observe similar training dynamics: an abrupt decrease of the error and a slow decrease above 100 epochs. The results also show that the models trained with the full descriptor set (M2 and M3) consistently outperform as compared to the reduced-feature model (M1), confirming the importance of enriched feature representation. For LR1, the model M3 achieves the lowest RMSE of 1.2 eV following the stopping criterion (number of epoch), highlighting the benefit of physics-based regularizer into the training process. Interestingly, under LR2 and LR3, the model M2 slightly surpasses the M3, echoing the trend observed for the int dataset. This again suggests that, with appropriate learning rates and
<figure><img src="image_9.png" /><figcaption>Fig. 6. Training RMSE metrics obtained by ANN-based training on the int dataset with different learning rates (LR1, LR2, LR3).</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>Fig. 7. Training RMSE metrics obtained by ANN-based training on the ads dataset with different learning rates (LR1, LR2, LR3).</figcaption></figure>

informative features, conventional MLP architectures can offer competitive performance compared to more complex physics-informed models. Therefore, the learning rate, a pivotal hyperparameter should need an equal treatment with model architectures in the pursuit of optimal regressive model performance. As for the int dataset, several other learning rates, which gave poor outcomes, are not open to discussion.

### 3.2.3. Evaluation of the test sets (int and ads)

It is clearly shown that the ANN-based training using generated descriptors improves the learning performance. To assess the inference of ANN models on the unseen data, the best models in terms of RMSE score obtained with LR3 were used to make prediction on the test sets of the int and ads datasets. The results are shown together with the training data in Fig. 8. By analogy with the statistical models, it is first shown that the prediction errors are greatly reduced for the test sets by training with the generated descriptors (M2, M3 versus M1). In the best cases, a decrease of 12% and 24% in RMSE compared to the model M1 was obtained for the model M3 (int dataset) and M2 (ads dataset), respectively. This decrease corresponds to the lowest RMSE values of 2.77 eV and 1.57 eV for M3 (int test set) and M2 (ads test set), respectively. These results provide evidence for the universality of the improvement of the training-inference processes thanks to the descriptor engineering. The second remark is based on the difference in the formation energies between the training and the test sets (RMSE<sub>test</sub> - RMSE<sub>train</sub>). For the int dataset, we find 0.42 eV, 0.88 eV, 0.88 eV for M1, M2, M3, respectively. This indicates that the overfitting becomes particularly pronounced when using the ANN-based models with integration of the generated descriptors (M2, M3). Regarding the ads dataset, we observe an emergence of overfitting between the models but much less pronounced than the case of the int dataset. Indeed, we find an energy difference of 0.37 eV, 0.42 eV, 0.39 eV for M1, M2, M3, respectively. These values suggest that the descriptor engineering leads to a poor prediction with respect to the test set for both datasets within the framework of descriptor engineering.

These findings point out that when ANN models reach out the limit of training (low-quality metrics) for a ML cycle, data engineering could enable an improvement of training but the prediction on unseen data should be carefully considered.

### 3.3. Model evaluation and discussion

The metrics analysis above unveils the interplay between the datasets, the descriptor engineering and the model algorithms. Together these elements have impact on the model training, inference and model interpretability. Indeed, when we evaluate the entire training-inference process, the descriptor engineering was advantageous for the tree-based models (statistical models) in terms of training performance and reduction in the overfitting for the ads, but not for the int dataset. This indicates the dependence of the descriptor engineering on the dataset characteristics. On the other hand, the generated descriptors, which permit the ANN-based regression models (ANN models) to improve the underfitting, do not show any benefit regarding the overfitting for both int and ads datasets. This finding reveals that the data engineering greatly enhances the model training but has lower sensitivity to the ANN models than the statistical models with respect to the prediction on the test set.

From the point of view of model interpretability, the different degrees of impact of descriptor engineering on the statistical and ANN models can be explained by the fact that the pattern extraction from the statistical models is mostly based on the decision logical rules. These rules are essentially grounded in the input features and the output target, thus, the interaction between the features and the output target is expected to be very strong. Therefore, the use of features from the generated descriptors could have a significant impact on the pattern extraction. On the contrary, ANN models learn the patterns by tuning the weights with non linearity through activation functions to capture the relationship between the features and the output target. Due to the fixed hidden layers of the model architectures, the weight tuning is limited to a same number of neurons when embedding the new features, therefore, the margin of change in weights could not be large, making the learning process more difficult. In addition, we are aware of other potential factors contributing to the observations mentioned above: the data fed into the model training and the limit of the model performance. For the input data, the descriptor engineering would introduce additional noise into the input data that might be effectively captured more by the ANN models than the statistical models. Regarding the model performance, the model M1 (without
<figure><img src="image_11.png" /><figcaption>Fig. 8. RMSE of the training and test sets regarding the int and ads datasets using ANN-based models.</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>Fig. 9. R², RMSE-based figures of merit (FoM) of model performance with respect to the int (a) and ads (b) datasets. “ANN” and “stat.” stand for ANN and statistical models, respectively. The dashed and dashed-dotted lines help guide the eye.</figcaption></figure>

generated descriptors) could reach out the performance limit of the training (R² quite low: 0.6 for int dataset and 0.46 for ads dataset), the prediction on the test sets might also reach out the upper value range, resulting in a low difference in energy between the training and test sets, i.e. lower overfitting compared to the models M2 and M3 (with generated descriptors) (see Fig. 8).

In order to take into account the aforementioned points (trade-off training, overfitting) for model evaluation and selection in the framework of data engineering, we introduce figure of merit (FoM) for each model with respect to the R² and RMSE metrics. The $FoM_{R^2}$ is derived from the proportional relationship owing to the normalization character of R²: $FoM_{R^2} = \frac{R_{train}^2 * R_{test}^2}{|R_{train}^2 - R_{test}^2|}$ whereas the $FoM_{RMSE}$ is

given as a linear combination of metrics owing to its non-normalized value: $FoM_{RMSE} = \frac{1}{RMSE_{train}} + \frac{1}{RMSE_{test}} + \frac{1}{|RMSE_{train} - RMSE_{test}|}$ (see Supplementary Material for other definitions which all converge to a same trend). In this manner, an increased FoM results in a better compromise for model selection. The results are shown in Fig. 9 for the int and ads datasets. For the int dataset (Fig. 9a), the highest FoM indices are observed for the RF and XGB models regarding R² and RMSE metrics, respectively. This constitutes a valuable indicator for the search of the most suitable model based on either the variance explained (R²) or the average magnitude of error (RMSE). Furthermore, it turns out that the ANN models exhibit lower FoM (for both two metrics) compared to the statistical models, indicating a superiority

of the statistical models. As discussed in the previous paragraph, the model interpretability could be one of the key explanations for the superiority of the statistical models in the context of data engineering. Regarding the ads dataset, Fig. 9b shows an inverse tendency of the FoM. Indeed, the FoM of the ANN models are higher than those of the statistical models for both metrics R² and RMSE, except for the XGB model, indicating a superior performance of the ANN models with respect to the ads dataset. For a holistic model evaluation and selection, combining two metrics is desirable. In this way, the ANN models are able to generate a great compromise compared to the statistical models. This finding clearly reveals the role of the dataset characteristics in the assessment of the ML process. A close inspection of the ads dataset indicates that it has a bigger amount of data (almost double), a value range of the target energy greater than the one of the int dataset and a broad distribution (see Supplementary Material). This would impact the generalized prediction for the statistical models. These results shed lights on the critical role of the dataset characteristics on the model selection in the context of the data engineering.

Finally, it is worth noting that the $FoM_{RMSE}$ of the XGB model has a highest value for both datasets but the $FoM_{R^2}$ is among the lowest for the ads dataset. This is because of an extreme high-variance fitting of the XGB model to the training set while it fails to predict the test set (Fig. 4b,d), resulting in high values of $FoM_{RMSE}$. The excessive fitting of the XGB model was fuelled by the features generated by the descriptor engineering (high-dimensional input), leading to an out-performance regarding the training process. If this effect emerges, the $FoM_{R^2}$ value becomes more reliable when selecting models with
<caption>**Table 2**</caption>

Overview of descriptor engineering and different models in regressive ML training with defect-related datasets. ASPH stands for atom-specific persistent homology and CGCNN, MEGNET, ALIGNN, GAT refer to different types of graph neural network models.

<table><thead><tr><th>Material</th><th>Model</th><th>Defect type</th><th>Training metrics</th><th>Database</th><th>Ref.</th></tr></thead><tbody><tr><td rowspan="2">Graphene MoS<sub>2</sub></td><td rowspan="2">CNN</td><td rowspan="2">Large-scale defects</td><td>RMSE = 1.36 eV</td><td rowspan="2">In-house dataset</td><td rowspan="2">[24]</td></tr><tr><td>RMSE = 2.43 eV</td></tr><tr><td rowspan="4">Doped monolayer structures</td><td>Tree based models</td><td>Interstitial</td><td>RMSE (with added descriptors) &lt; 1.4 eV</td><td rowspan="4">IMP2D</td><td rowspan="4">This work</td></tr><tr><td rowspan="2">ANN</td><td>Adsorbed</td><td>RMSE (with added descriptors) &lt; 1.1 eV</td></tr><tr><td>Interstitial</td><td>RMSE (with added descriptors) = 2.1 eV</td></tr><tr><td>LightGBM</td><td>Adsorbed</td><td>RMSE (with added descriptors) = 1.3 eV</td><td></td></tr><tr><td rowspan="3">Zinc blende semiconductors</td><td>CGCNN</td><td rowspan="3">Quasi-all point defects</td><td>RMSE (with structural features) = 0.84 eV</td><td rowspan="3">In-house dataset</td><td rowspan="3">[43]</td></tr><tr><td>MEGNET</td><td>RMSE = 0.19 eV</td></tr><tr><td>ALIGNN</td><td>RMSE = 0.36 eV</td></tr><tr><td rowspan="2">Cd-based II-VI semiconductors</td><td rowspan="2">RF</td><td rowspan="2">Interstitial</td><td>RMSE = 0.03 eV</td><td rowspan="2">In-house dataset</td><td rowspan="2">[45]</td></tr><tr><td>RMSE = 2.43 eV</td></tr><tr><td rowspan="3">Oxide-based perovskite</td><td>CGCNN</td><td rowspan="3">Monovacancies</td><td>MAE: w/o ASPH = 1.79 eV, w/ASPH = 0.98 eV</td><td rowspan="3">In-house dataset</td><td rowspan="3">[46]</td></tr><tr><td>GAT</td><td>MAE: w/o ASPH = 1.67 eV, w/ASPH = 0.94 eV</td></tr><tr><td>Transformer network</td><td>MAE: w/o ASPH = 1.55 eV, w/ASPH = 0.72 eV</td></tr></tbody></table>

a good compromise. Therefore, a combinatorial evaluation of at least 2 metrics-based FoM, i.e. R² and RMSE, is necessary for a fair and equitable selection of models for ML regression of materials properties.

For an overview, Table 2 presents representative performance metrics from the literature on ML-based prediction of defect-related property across different materials and defect types. Since differences in datasets, descriptors, defect definitions and training protocols prevent a strict head-to-head evaluation and a fair comparison of model performance, the table indicates typical error ranges reported in prior works, thus, providing a benchmark of our work.

## 4. Conclusion

In conclusion, we have shown an advanced study of descriptor engineering and model selection within the scope of ML regression of defect energy formation in 2D materials. The model evaluation and selection were assessed by considering two distinct learning algorithms on the same datasets: statistical and ANN models. The combinatorial investigation of data engineering and model selection was also conducted by considering two datasets with different characteristics (int and ads). This study highlights again the potential of the descriptor engineering using vectorized property matrices for the model training on small datasets. We have also introduced figures of merit with respect to the model metrics (R², RMSE) to assess the model performance by considering both the training and the test outcomes. Our results obtained throughout the ML workflow provide insights into the model selection in the framework of data engineering: additional input data could be beneficial to the training-inference ML cycles regarding the statistical models whereas the ANN models would be a reasonable choice with more stability in terms of data scaling and noise introduction. This finding underscores the importance of model algorithms and the dataset characteristics when investigating data engineering. Finally, our study offers a comprehensive picture on the key components of a ML workflow (data engineering, model selection) and their interplay with datasets in a regressive training and inference of the physical, chemical properties of 2D materials and other materials families such as III-V semiconductors and perovskites.

### CRediT authorship contribution statement

A. El Alouani: Writing – review & editing, Writing – original draft, Visualization, Methodology, Investigation, Formal analysis. M. Al Khalfioui: Writing – review & editing, Visualization, Investigation. A.

**Michon:** Writing – review & editing, Visualization, Investigation, Funding acquisition. **S. Vézian:** Visualization, Investigation. **P. Boucaud:** Writing – review & editing, Visualization, Project administration, Investigation, Funding acquisition, Conceptualization. **M.T. Dau:** Writing – review & editing, Writing – original draft, Supervision, Methodology, Investigation, Funding acquisition, Formal analysis, Conceptualization.

### 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.

### Acknowledgements

The authors acknowledge the financial support from NANOFUTUR project handled by French National Research Agency (ANR) in the framework of Programme d'Investissement d'Avenir (ANR-21-ESRE-0012). This work was also supported by the ANR through the VanaSiC project (ANR-22-CE24-0022-01) and by a France 2030 government grant managed by the ANR in the framework of the PEPR Electronique through the ADICT project (ANR-22-PEEL-0011). MTD thanks the Université Côte d'Azur for supporting this work from UCA-JEDI Investments in the Future project managed by ANR (ANR-15-IDEX-01) and Emergence-CNRS 2025.

## Appendix A. Supplementary data

See Supplementary Material for description of descriptors, metrics, grid search, feature importance and basic principles of the PINN-like model. The code for ANN training-prediction that supports the study is available at http://gitlab.com/tuan2dplus/paper-DE-MS upon reasonable request from the corresponding author (mtd@crhea.cnrs.fr). For the code related to the descriptor generator and tree-based models, please refer to our previous work [31].

Supplementary material related to this article can be found online at https://doi.org/10.1016/j.mtphys.2025.102006.

### Data availability

Data will be made available on request.
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