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npj 2D Materials and Applications volume 10, Article number: 44 (2026)
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Two-dimensional (2D) materials such as graphene, phosphorene, and MoS₂ offer transformative opportunities but are often impaired by synthesis-induced vacancies. Existing characterization tools—atomic-resolution microscopy and molecular dynamics—are slow, costly, and unsuitable for large-scale screening. We introduce an AI-driven image-processing framework that combines automated grayscale image analysis with machine learning to rapidly and accurately detect and quantify vacancies. By converting structural information into tailored pixel-based descriptors, our method regresses defect coordinates, radii, and densities without atomic-scale input. Validated on thousands of simulated images with controlled defects, the framework attains over 96% prediction accuracy and surpasses thermal vibration analysis (~90%) while removing the need for specialized experimental setups. The approach is generalizable across material systems, enabling high-throughput screening and standardized defect analysis for nanoelectronics, sensing, and quantum technology applications. This work accelerates the AI-guided design and optimization of defect-engineered 2D materials.
The emergence of two-dimensional (2D) materials has revolutionized materials science, offering unparalleled electronic, optical, and mechanical properties that defy their atomic-scale thickness. Graphene’s exceptional conductivity1,2,3,4, phosphorene’s anisotropic behavior5,6,7,8,9, and molybdenum disulfide’s (MoS₂) tunable bandgap10 and other structures11,12 exemplify the transformative potential of these ultrathin systems for applications ranging from flexible electronics to quantum technologies. However, a critical bottleneck persists: the near-ubiquitous presence of defects during synthesis, which can drastically degrade performance in nanodevices13,14,15,16,17,18,19,20,21,22,23,24,25,26. Defects are not inherently detrimental; they can be engineered to tailor quantum states or enable sensing applications27,28,29,30,31. However, their atomic-scale variability and unpredictable nature make precise characterization and control formidable challenges.
However, the enormous variety of defect structures and host materials renders experimental screening slow and inefficient. This challenge motivates the use of computational methods to systematically explore the defect space, design optimal defects, and uncover the underlying physics32,33,34. Current strategies for defect analysis, such as atomic-resolution transmission electron microscopy (TEM)35,36,37,38, demand laborious sample preparation and specialized instrumentation, limiting their scalability. Computational approaches like molecular dynamics (MD) or Monte Carlo-based finite element methods (MC-FEM)39,40,41,42 offer alternatives but suffer from prohibitive computational costs and reliance on idealized defect models. However, even innovative methods like thermal vibration analysis39 are limited to ~90% accuracy and depend on complex models of defect-vibration coupling. This creates a pressing demand for scalable, high-throughput defect mapping techniques that bypass the need for atomic-resolution imaging or computationally expensive simulations.
Machine learning (ML) has recently emerged as a paradigm-shifting tool for materials discovery, enabling rapid prediction of properties from electronic structure to mechanical behavior43,44,45,46,47,48,49,50,51,52,53,54. For defect engineering, some groups introduced the ML approach for predicting defects in materials. Fu-Xiang Rikudo Chen et al. propose a deep learning-based atomic defect detection framework (DL-ADD) for STM-based detection in TMDs (MoS₂/WS₂). This method is robust to noise (F2-score: 0.86–0.89) but has narrow generalizability beyond TMDs55. A hybrid machine learning–density functional theory (ML-DFT) framework, developed for zinc blende semiconductors like GaN and SiC, enabled the rapid screening of 12,000 potential defects. However, this approach’s computational cost remains high because it still depends heavily on DFT calculations56. Meanwhile, complementary approaches such as physics-informed neural networks (PINNs) and other DL architectures address defect resolution in broader material systems by integrating physical laws or leveraging noise-robust feature extraction, offering versatility for complex geometries and multi-scale challenges but often requiring extensive training data or domain-specific optimization57,58.
Here, we introduce an artificial intelligence (AI) framework that synergizes image processing and ML to automate defect and hole characterization in 2D materials with unprecedented speed and accuracy. Our method eliminates the need for atomic-resolution probes or MD simulations by converting grayscale images of graphene, phosphorene, and MoS₂ into pixel-based descriptors that encode defect geometry. By training ML models on these features, we achieve about 96% accuracy in predicting defect coordinates, radii, and densities—surpassing state-of-the-art vibration-based methods39,40,41,42. This workflow (Fig. 1) is computationally efficient, non-destructive, and generalizable to arbitrary 2D grayscale systems, enabling rapid screening of defect-engineered materials for targeted applications.
From the bottom left, the code starts by creating images from the three types of 2D materials with random holes, converting the images to a grayscale image and digital table of pixel values, and then extracting descriptors, which ends with ML prediction results.
To standardize defect analysis across materials, we modeled graphene, phosphorene, and molybdenum disulfide (MoS₂) as 5 nm × 5 nm rectangular sheets, a size chosen to balance computational efficiency with sufficient spatial resolution for defect characterization. These sheets contain 979 atoms (graphene), 709 atoms (phosphorene), and 882 atoms (MoS₂), calculated based on their respective lattice constants (graphene: 0.246 nm59, phosphorene: 0.328 nm60, MoS₂: 0.316 nm61). Initial images of these materials were generated at 949 × 966 pixels but irregular borders were cropped to a uniform 5-pixel-wide margin, resulting in final dimensions of 778 × 786 (graphene), 771 × 786 (phosphorene), and 756 × 786 (MoS₂) pixels (Fig. 2b). Crucially, we avoided resizing images to preset dimensions to preserve the atomic arrangement and bond lengths, which are critical for accurate defect simulation. Artificial holes were introduced using a uniform random generator that assigned hole centers (X, Y) and radii (({{rm{R}}}_{{rm{h}}})) with 0.001 nm precision, ensuring no spatial overlap between defects. Hole positions spanned 0.3–4.7 nm (excluding a 0.3 nm margin to prevent edge artifacts), while radii ranged from 0.1 nm (minimum physically viable size, below atomic spacing) to 0.7 nm (≤6% of sheet area).
a The random values generated (Eq. 2, “Methods” section), the color range varies from purple to yellow based on the radius of the defect. b The colorful MoS2 image with the green 5-pixel wide border created around the image, and c the heatmap of MoS2 image with a yellow hole visible in it.
To ensure statistical robustness and minimize bias from overlapping or morphologically similar defects, we generated 1000 distinct images per material category (graphene, phosphorene, MoS₂). While random hole generation occasionally produced visually similar shapes (e.g., two holes with different radii appearing alike due to atomic bond disruptions), our dataset ensured sufficient variability to train models on subtle structural differences. Hole morphology was found to depend critically on defect position relative to the atomic lattice. For example, a 0.1 nm radius hole in graphene exhibited two distinct shapes (Fig. 3a) depending on whether it was centered on a carbon atom or between bonds, whereas MoS₂ holes remained uniform due to its symmetric trigonal structure (Fig. 3c). Phosphorene, with its puckered orthorhombic lattice, showed intermediate variability so as the graphene (Fig. 3b). Hole density (ρ) was calculated as the ratio of total hole area to sheet area (({rm{rho }}={rm{pi }}{{{rm{R}}}_{{rm{h}}}}^{2}/(dtimes d))), where d = 5 nm (sheet size). This metric quantifies defect impact on material properties, such as electrical conductivity, and ensures material comparability. By constraining ρ ≤ 6%, we avoided unrealistic defect densities that could compromise structural integrity.
Simulated hole structures (0.1 nm radius) reveal how atomic lattice geometry influences morphological variability. a In graphene’s hexagonal lattice, the hole shape is bimodal, differing based on placement relative to a carbon atom or bond center. b Phosphorene’s puckered orthorhombic structure shows intermediate variability. c In contrast, MoS₂‘s high-symmetry trigonal prismatic structure produces a single, uniform hole morphology.
Although for creating holes a circular cut is made on the materials as clear in Fig. 3, some atom–atom connections that are in the incision just removed from the structure and the final shape may be a different form or a secondary connection may take place in which change the form of the holes but the approximate area be equal to the circular one. So, in the boundary region where some parts of the holes may be out of the 2D material, the position of the holes in these lateral areas is determined. A value of 0.3 nm side margin is created to avoid the out-of-shape hole creation. In this case, the holes create an area between 0.3 and 4.7 nm in both directions. Figure 4 shows the four extracted features from a graphene sheet, where an orthogonal arrow vector shows the origin of coordinates in creating the positions and extracting features. The exact position of the hole (left({rm{X}}=3.899,mathrm{nm},,{rm{Y}}=3.974,mathrm{nm},,{{rm{R}}}_{{rm{h}}}=0.43,mathrm{nm}right)) in a 5 nm graphene layer, illustrated in Fig. 4a, the irregular shape of the hole covers a density of about 2.3% of the total shape of graphene. In the table of pixel values, the summation of y (Eq. 3) and x (Eq. 4) pixels is depicted at the right and top side of the grayscale image with brown and blue colors. At the hole position in which the colors return to white, the density of 255 value of the pixel is more than in other areas, so the summation of pixels in that row or column shows some uplift in diagrams (see Supplementary Table S1 for complete feature definitions).
a Representative sample image containing a void defect. b Illustration of image processing workflow: (i) original image conversion to grayscale, (ii) dimensional reduction through summation algorithms, and (iii) four sample feature maps highlighting defect characteristics used for machine learning analysis.
The blue and brown vectors are tangent to the maximum point of the graphs so that the summation and values of pixels around the charts depict the changes at the hole position. The maximum values of pixel summation in the x and y directions are 183,159 and 187,365, respectively. The first position where maxx appears is at pixel number 605, and the last is at 643, so two positions have this maximum value. In the y-direction, nine positions have the maximum summation of 187,365 between the locations of 128 and 197. The average of these nine positions and their median are the other descriptors; further, the difference between the first and the last location (197–128 = 69) illustrates the range of locations in this sample.
Based on SHAP (SHapley Additive exPlanations) analysis and performance metrics, the machine learning model demonstrates high reliability in predicting defect properties in graphene, with clear identification of the most influential features (Fig. 5a, b). The analysis of the Y position prediction reveals that max_y_pos_mean is the most important feature, while the high R² values across all targets confirm the model’s strong predictive power (see other models in Supplementary Fig. S1). The SHAP analysis for the Y position prediction in graphene identified max_y_pos_mean as the most impactful feature, with a substantially higher mean absolute SHAP value (0.4506) than other descriptors, indicating it is the primary driver of the model’s predictions (see Supplementary Table S2). This feature, along with hole_centroid_y, exerts a positive influence on the predicted Y value. In contrast, features such as hole_eccentricity and max_sum_y have a negative impact, meaning higher values for these descriptors decrease the predicted Y position. The high performance of the model is validated by the regression metrics, which show excellent agreement between predicted and actual values. This is evidenced by the high R² scores for the Y coordinate (Test R² = 0.929) (Fig. 5c) and the particularly strong results for the radius R (Test R² = 0.98) (Fig. 5d). The low MAE values further confirm the model’s accuracy across all predicted properties (see Supplementary Table S3).
a, b SHAP (SHapley Additive exPlanations) summary plots illustrating the impact of the top ten most important features on the model’s predictions for the vacancy’s Y-coordinate. c, d Scatter plots of predicted versus actual values from the test dataset demonstrate the model’s high predictive accuracy for the c Y-coordinate and d effective radius (R) of the vacancies. The dashed lines represent perfect agreement.
To evaluate our approach, we apply this method to a sample of phosphorene sheet with three vacancies and show that the information extracted from the model makes the appropriate results (Fig. 6). In multiple vacancy layers, the layers are divided into the number of holes, and each one’s descriptors are put as unseen test data. The three grayscale images are used to calculate the features from them, and the results provide the predicted holes (Fig. 6c). Hole number 2 is predicted to be quite similar to itself, and in the other two, hopping sites in phosphorene at the borders change their positions.
a Original atomic structure with three vacancies. b The image is segmented into three individual grayscale sub-images for separate analysis. c Model predictions for each vacancy, derived from their feature descriptors. Prediction accuracy is high, with vacancy 2 reproduced precisely, while vacancies 1 and 3 show slight positional shifts at the layer borders.
Table 1 shows the original values and the predicted result in this sample. The phosphorene image contains the five nearest hopping connections between the phosphors, and by changing the position of the hole by an accuracy of 0.01 nm, the other sites may be affected, and the hopping connections may be removed. However, the predicted image located the hole positions very well; nevertheless, in larger holes, the form of the boundary links may vary slightly due to more connections between the boundary atoms. The density of samples in the image, with attention to the predicted ({{rm{R}}}_{{rm{h}}}) shows the satisfying results with the highest amount of MAE = 0.004 in the larger hole (The comparative performance metrics for models are detailed in Supplementary Figs. S2, S3, S4).
The AI method is more advantageous than the experimental and theoretical methods for multiple reasons. First, this method has a low cost and high efficiency compared to experimental methods. Second, this method can process numerous holes in one computational action and cover all 2D materials. Last, it has a cheaper computational cost and higher efficiency than the MD models, computational methods, and faster training speed. Of course, in AI models, important things return to the size of the dataset, not only the number, but also the quality of images and the diversity of data. In this work, more images with the hole in image processing and tuning the hyperparameters or using the combined methods, may lead to higher predictions. Still, our approach with high accuracy promises a low imperfection and leads to a model for discovering new materials.
This work demonstrates a robust AI framework that synergizes image processing and machine learning to achieve high-accuracy quantitative characterization of vacancies in diverse 2D materials. By converting grayscale images into pixel-based descriptors, our method provides a rapid, non-destructive, and scalable alternative to resource-intensive simulations, delivering precise measurements of defect coordinates and radii with nanometer-scale precision. The novelty of our approach lies in its specific focus on quantitative characterization, moving beyond the binary detection tasks of existing deep learning frameworks. Unlike methods that require computationally expensive ab initio simulations, our model operates directly on standard image data, making it applicable to a wider range of characterization techniques like STM and TEM with minimal computational overhead. This positions our framework as a complementary tool within the materials science toolkit, specifically designed for high-throughput, quantitative analysis where rapid defect sizing and locating are prioritized. Looking forward, this methodology establishes a foundation for several impactful directions. Future work will expand the framework to predict defect-induced property changes, such as electronic bandgap shifts and mechanical stress concentrations. Furthermore, the approach is readily integrable with autonomous experimental systems, paving the way for real-time defect correction during synthesis. This study primarily validates the framework through simulated data, with experimental STM analysis serving as a qualitative proof-of-concept. To transition this into a quantitative tool, future work will focus on creating a comprehensive database of experimental images and establishing rigorous benchmarking protocols. This will transform the method into a community-wide resource for high-throughput defect characterization.
To create images of 2D materials (like graphene, phosphorene, and MoS₂), we used three software tools: VMD62, VESTA63, and the pybinding library64. In these images, we generate holes that are characterized by,
X and Y are the exact positions of the hole centers with a radius of ({{rm{R}}}_{{rm{h}}}), and the position in the 2D material’s sheet is indicated by i and j, where i and j represent the ith row and jth column, respectively. Three parameters X, Y, and ({R}_{h}) are created randomly with an accuracy of 0.001 nm, in which,
and (d) show the size of 2D materials (Fig. 2a). The radius of the holes is arbitrary, but it is selected in the program so that it covers up to the maximum density of about 6% of the total area of the sheet. It can be more than this, but it is much easier to position the holes at higher values than at smaller dimensions, even without the need for theoretical and experimental methods. To ensure realistic defect modeling, we excluded holes with radii smaller than 0.1 nm, as this threshold is below the typical atomic spacing in 2D materials (e.g., ~0.14 nm in graphene), making such “holes” physically implausible. After generating images with randomly positioned holes (radius: 0.1–0.7 nm), each image was saved in standard image format at a resolution of 949 × 966 pixels (Fig. 2b). To prepare the images for quantitative analysis, a multi-step preprocessing pipeline was implemented. All images were converted to grayscale, where pixel intensities range from 0 (black) to 255 (white), to mimic the contrast profile of experimental STM data and simplify subsequent analysis. Irregular borders from the original images were removed and replaced with a uniform 5-pixel-wide green border. This critical step ensured that the subsequent analysis focused solely on the material structure, eliminating edge artifacts and creating a consistent, high-contrast background where defect regions (bright, 200–255) are clearly distinguishable from the atomic lattice (dark, 50–100), as shown in Fig. 2b.
Each processed grayscale image was treated as a numerical matrix I, with each element I(i,j) representing the pixel intensity at position (i,j). To capture the spatial distribution of brightness indicative of defects, we computed two fundamental 1D projections by summing pixel intensities along each axis. The column-wise summation (({{rm{X}}}_{{rm{array}}})) and row-wise summation (({{rm{Y}}}_{{rm{array}}})) are given by:
where dx and dy are the image dimensions. These projections act as a signature for the defect; the presence of a vacancy produces a pronounced peak in both arrays due to the cluster of high-intensity pixels, as visualized in the heatmap of Fig. 2c.
The core of our analysis involved engineering a comprehensive set of selected 25 features from these arrays and the original image to feed into machine learning models for predicting defect parameters (X, Y, R). These features were designed to quantify the defect’s properties from multiple perspectives:
Positional Features: Metrics such as max_x_pos_mean (the average horizontal position of the peak in ({{rm{X}}}_{{rm{array}}})) and hole_centroid_y (the vertical center of the segmented defect) directly provide coordinates that correlate with the physical center of the vacancy. These are derived from the peak locations in Eqs. (3) and (4).
Intensity Distribution Features: Statistical measures of the projection arrays (sum_x_iqr, sum_y_std) capture the spread and concentration of brightness. A larger spread or a higher quartile value often indicates a larger or more intense defect region, directly informing the defect’s radius (R) and prominence.
Morphological Features: By thresholding the image to isolate the defect, we calculated geometric properties like hole_perimeter and hole_solidity. The perimeter directly measures the boundary length of the defect, scaling with its size and irregularity, while solidity quantifies its compactness, distinguishing between circular and elongated vacancies.
Global Statistical & Texture Features: Descriptors such as entropy (a measure of randomness in pixel intensities) and mean_gradient (average edge strength) capture the overall image texture. Defects create more structured patterns and sharper edges, reducing entropy and increasing gradient magnitudes. The entropy is calculated as:
where p(i) is the probability of pixel intensity ‘i’ occurring in the image.
The complete set of 25 engineered features, along with their mathematical formulas and a description of their impact on predicting X, Y, and R, provides a holistic numerical representation of each defect’s appearance. This multi-faceted feature set captures the essential characteristics of the defects. Rather than relying on pre-conceived assumptions, we allowed the machine learning model to discover the complex, non-linear relationships between these quantitative descriptors—such as the span of a peak or the standard deviation of gradients—and the target physical parameters. This data-driven approach ensures the predictive model is informed by the most salient characteristics of the defect’s structure and context.
To evaluate the predictive capability of our engineered descriptors, we employed eight machine learning (ML) regression models from Python’s scikit-learn library65. The models spanned linear (LSVR, SVR), tree-based (RFR, ETR, GBR, ABR), k-nearest neighbor (KNR), and neural network (MLP) architectures to predict the defect coordinates (X, Y) and effective radius (({{rm{R}}}_{{rm{h}}})). All features were scaled to address unit disparities prior to training. To ensure a fair, unbiased comparison of the models’ inherent predictive power, we retained default hyperparameters and employed k-fold cross-validation to guard against overfitting. Model performance was assessed using the mean absolute error (MAE) and the coefficient of determination (R²). Among the tested algorithms, the Random Forest Regressor (RFR) was selected for all subsequent analyses due to its superior balance of predictive accuracy, computational efficiency, and interpretability. Tree-based ensembles like RFR are particularly well-suited for materials data as they naturally capture non-linear relationships and high-dimensional feature interactions without requiring complex parameter tuning66. A comprehensive evaluation includes hyperparameter optimization via grid search, which significantly improved performance (e.g., achieving cross-validation R² values of 0.93 for X, 0.90 for Y, and 0.98 for ({{rm{R}}}_{{rm{h}}}) in graphene). We further established model robustness through permutation feature importance analysis and tests against introduced Gaussian noise, which showed maintained high performance even at significant noise levels. Benchmarking confirmed that our optimized RFR approach consistently outperformed or matched alternative methods, including Support Vector Regression, Gaussian Process Regression, and Neural Networks. Finally, a detailed error analysis (reporting MSE, MAE, and R²) confirms the model’s high accuracy and minimal prediction bias across all target variables.
The author confirm that the main data supporting the findings of this study are available within the paper and its supplementary informaiton. Furthermore, other relevant data can be made available from the corresponding author upon reasonable request.
The source code used in the current study is available at GitHub (https://github.com/ehsanab550/AI-vacancy-detection/.
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E.A. acknowledges the research funding provided by the Iran National Science Foundation (INSF) under project No. 40400449.
School of Quantum Physics and Matter, Institute for Research in Fundamental Sciences (IPM), Tehran, Iran
Ehsan Alibagheri
Department of Physics, University of Tehran, Tehran, Iran
Ehsan Alibagheri
PubMed Google Scholar
E.A.: design, conceptualize, analyzing data, and write and edit the manuscript.
Correspondence to Ehsan Alibagheri.
The author declares no competing interests.
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Alibagheri, E. AI-driven image processing framework for high-accuracy detection and characterization of vacancies in 2D materials. npj 2D Mater Appl 10, 44 (2026). https://doi.org/10.1038/s41699-026-00667-4
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