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Machine-Learning Exploration of Defect Topologies and Thermodynamic Stability in Graphene with Atomic Vacancies

This study combines semiempirical atomistic thermodynamics with interpretable machine learning and symbolic regression to map the thermodynamic stability of graphene with atomic vacancies, successfully condensing the complex structure-stability relationship into a compact, high-accuracy analytical law that identifies defect concentration and inter-vacancy distance as the dominant factors.

Original authors: Marcos V. N. da Costa, José R. da M. Lima Filho, Kádila R. de S. Oliveira, Raphael M. Tromer, Marcelo L. Pereira Junior

Published 2026-09-18

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CC BY 4.0

Original authors: Marcos V. N. da Costa, José R. da M. Lima Filho, Kádila R. de S. Oliveira, Raphael M. Tromer, Marcelo L. Pereira Junior

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Graphene is a sheet of carbon atoms so thin it is essentially two-dimensional, yet it is incredibly strong and conducts electricity with remarkable ease. In the real world, however, these perfect sheets are never flawless. They contain gaps where atoms are missing, known as vacancies. These missing pieces are not just flaws; they fundamentally change how the material behaves, altering its magnetic properties, its strength, and how it interacts with other chemicals. Scientists have long known that the stability of a piece of graphene depends heavily on how many of these gaps exist and how far apart they are from one another. When gaps are close together, the surrounding atoms rearrange themselves to fill the space, often making the structure more stable than if the gaps were far apart. But trying to map out exactly how these gaps interact across every possible arrangement is a task too vast for traditional computer simulations, which struggle to handle the sheer number of possibilities.

A team of researchers from the University of Brasília has tackled this problem by combining computer simulations with a new kind of artificial intelligence to create a clear, understandable rule for how these defects behave. Instead of trying to calculate every single possibility from scratch, they built a library of hundreds of different graphene sheets, each with a specific number of missing atoms and specific distances between those missing spots. They used a fast, semi-empirical computer method to relax the structures, allowing the atoms to settle into their most comfortable positions, and then measured the energy required to create each specific arrangement. This energy value, known as the heat of formation, serves as a direct measure of stability: the lower the value, the more stable the structure.

To make sense of this massive amount of data, the researchers turned to machine learning. They fed the computer a special digital description of each graphene sheet, a code that captures the shape and connectivity of the holes without needing to know the exact coordinates of every atom. They then trained a powerful computer model to predict the stability of any new arrangement based on this code. The model learned with high precision, accurately predicting the stability of sheets it had never seen before, proving that it had truly understood the underlying physics rather than just memorizing the data.

The most significant finding came when the researchers asked the computer to explain its own reasoning. By analyzing which factors the model relied on most, they discovered that two things dominate the stability of the material: the total number of missing atoms and the distance between them. The number of missing atoms is the primary driver; more missing atoms generally mean a less stable sheet. However, the distance between them is the critical second factor. When missing atoms are close together, they tend to merge and reconstruct the surrounding lattice into new shapes, effectively healing the damage and lowering the energy cost. When they are far apart, they act as independent weak points, leaving the material in a higher energy, less stable state.

The researchers did not stop at a complex computer model. They used a technique called symbolic regression to distill their findings into a single, simple mathematical law. This law expresses the stability of the graphene sheet using only the number of missing atoms and the distance between them. The resulting formula is compact and easy to use, yet it reproduces the complex simulation results with nearly perfect accuracy. This work demonstrates that the chaotic and high-dimensional world of defects in two-dimensional materials can be reduced to a clear, predictable rule. It provides a practical tool for engineers and scientists who wish to design new materials by intentionally introducing specific defects, allowing them to predict the stability of their designs without needing to run expensive and time-consuming simulations for every single variation.

Technical Summary: Machine-Learning Exploration of Defect Topologies and Thermodynamic Stability in Graphene with Atomic Vacancies

Problem Statement
The thermodynamic stability and functional response of graphene are fundamentally controlled by atomic vacancies and their aggregates. While the introduction of defects is a known strategy for tuning material properties, the configurational space spanned by multiple vacancies at variable concentrations and separations is too vast to be mapped exhaustively using first-principles methods like Density Functional Theory (DFT). A critical gap exists in understanding how vacancy concentration and inter-vacancy distance jointly control the heat of formation, particularly because the stability of multivacancy systems is non-additive; nearby vacancies coalesce into reconstructed rings (pentagons and heptagons) that significantly alter stability compared to isolated defects. Existing studies often focus on isolated vacancies or specific pairs, lacking a systematic account of the cooperative, non-additive coupling between defects across a broad range of configurations.

Methodology
The authors developed a workflow integrating semiempirical atomistic thermodynamics, interpretable machine learning (ML), and symbolic regression to address this high-dimensional structure–stability problem.

Dataset Generation: Several hundred defective supercells were constructed from a 72-atom pristine graphene cell. The dataset was systematically varied by two control variables:
- Defect Concentration: 2, 4, and 7 vacancies per supercell.
- Inter-vacancy Distance: Separation ranging from first to fourth neighbors.
- Relaxation: All structures were fully relaxed using the PM7 Hamiltonian within the MOPAC package. The heat of formation (HOF) was calculated as the primary metric for thermodynamic stability, alongside enthalpy, entropy, and vibrational frequencies.
Feature Encoding: Each relaxed structure was encoded using the Dynamic Collision Fingerprint (DCF). This descriptor simulates virtual probe particles colliding with the atomic framework, converting collision statistics into a translationally and rotationally invariant feature vector. The DCF captures local topology through transport-like quantities (e.g., effective diffusivity, mean free path, mean relaxation time) that reflect the openness and connectivity of the defective network.
Machine Learning Model: A Gradient Boosted Decision Tree (XGBoost) model was trained to predict the HOF. The model's hyperparameters (e.g., tree depth, learning rate, regularization) were optimized using Bayesian hyperparameter search (via the Optuna framework) to prevent overfitting. The dataset was split 80/20 for training and independent testing.
Interpretability and Symbolic Regression:
- SHAP Analysis: SHapley Additive exPlanations (rooted in cooperative game theory) were used to attribute prediction contributions to specific variables, revealing the global importance ranking and non-linear interactions.
- Symbolic Regression: Based on the hierarchy identified by SHAP, genetic programming was employed to distill the learned mapping into a compact, closed-form analytical expression.

Key Results

Predictive Accuracy: The optimized XGBoost model reproduced the heat of formation of an independent test set with a Root Mean Squared Error (RMSE) of approximately 23.5 kcal/mol, showing no evidence of overfitting.
Variable Hierarchy: SHAP analysis identified defect concentration as the dominant variable governing stability, followed by inter-vacancy distance. Other descriptors (enthalpy, entropy, vibrational frequencies) contributed to fine-tuning but were secondary.
- Physical Interpretation: Higher concentrations increase the energetic penalty of bond breaking. Larger inter-vacancy distances lead to higher HOFs because defects behave as independent monovacancies with dangling bonds, whereas closer vacancies coalesce into stable reconstructed rings (pentagon-heptagon pairs) that eliminate dangling bonds.
Analytical Law: Symbolic regression successfully condensed the complex, non-linear mapping into a closed-form equation:
$HOF = (conc - 1.06) e^{dist + \sin(dist + 0.15) + 2.25} + 460.20$
This expression achieved a coefficient of determination ( $R^2$ ) of 0.9966, a Mean Absolute Error (MAE) of 15.51 kcal/mol, and an RMSE of 24.18 kcal/mol on the test set.

Significance and Claims
The paper claims to provide a "transferable route to rational defect engineering in two-dimensional materials." By converting a high-dimensional structure–stability problem into an interpretable analytical law, the workflow bridges the gap between computationally expensive atomistic simulations and explicit physical modeling. The authors assert that this approach demonstrates that the essential physics of vacancy-induced destabilization can be captured by a low-dimensional, strongly non-linear relation dependent primarily on concentration and separation. The work validates that semiempirical methods, when coupled with interpretable ML and symbolic regression, can efficiently explore configurational spaces that are impractical for first-principles methods, yielding predictive, physically transparent, and mathematically explicit structure–stability relationships.

Technical Summary: Machine-Learning Exploration of Defect Topologies and Thermodynamic Stability in Graphene with Atomic Vacancies

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