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Small-supercell and Small-dataset Training Strategy of Machine Learning Interatomic Potentials for Point Defects

This paper proposes a computationally efficient machine learning interatomic potential training strategy that utilizes a small dataset derived from limited DFT calculations on small supercells, combined with multi-size defect and pristine bulk data, to accurately predict point defect properties in large-scale systems with minimal error.

Original authors: Zhenxing Dai, Mingjue Ni, Xinpeng Li, Menglin Huang, Anderson Janotti, Shiyou Chen

Published 2026-09-22

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CC BY 4.0

Original authors: Zhenxing Dai, Mingjue Ni, Xinpeng Li, Menglin Huang, Anderson Janotti, Shiyou Chen

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

Inside the solid materials that power our modern world, from the silicon chips in our phones to the solar cells on our roofs, tiny imperfections dictate performance. These imperfections, known as point defects, are essentially missing atoms or misplaced ones within a crystal's orderly grid. While they are microscopic, their influence is macroscopic; they determine how well a semiconductor conducts electricity or how efficiently it converts light into energy. For decades, scientists have relied on complex computer simulations based on the laws of quantum mechanics to understand these defects. These simulations act like a digital microscope, allowing researchers to see how atoms shift and how energy changes when a defect is present. However, this digital microscope is incredibly slow and expensive to use. To get accurate results, especially for defects that carry an electric charge, the simulations must model a vast number of atoms—often hundreds or even thousands—to avoid artificial errors caused by the boundaries of the computer model. This requirement makes it nearly impossible to study these defects in the large, realistic systems where they actually function.

A newer approach has emerged to speed up these calculations: machine learning interatomic potentials. Think of these as smart shortcuts. Instead of solving the heavy quantum equations every time, a computer program learns the rules of how atoms interact from a small set of high-quality examples. Once trained, this program can predict how atoms will behave in massive systems with near-perfect accuracy but in a fraction of the time. The challenge, however, has been that training these programs usually requires a massive library of data, often involving thousands of expensive simulations. This is particularly difficult for charged defects, where the need for huge atomic models to avoid errors makes gathering data prohibitively costly. Researchers at Fudan University and the University of Delaware have now developed a way to train these machine learning models using a tiny fraction of the usual data, proving that a small, carefully chosen set of examples can teach the computer to predict the behavior of defects in much larger systems.

The team focused on a specific strategy to see if they could build a reliable model without the usual mountain of data. They chose three different materials—gallium nitride, silicon dioxide, and a copper-based compound used in solar cells—as their test subjects. For each, they looked at specific defects, such as a missing nitrogen atom or a misplaced copper atom, in various electric charge states. Their goal was to see if they could train a machine learning model using only a few small atomic models, containing fewer than one hundred atoms, and then have that model accurately predict what would happen in a much larger model, containing over two hundred atoms. They tested two different ways of feeding data to the computer. In the first method, they provided only data from the defective atoms themselves, taken from simulations of different small sizes. In the second method, they added data from perfect, defect-free chunks of the same material to help the computer understand the background environment.

The results revealed a clear lesson about how these digital shortcuts learn. When the researchers trained the model using data from just one small size of atomic model, the program failed miserably when asked to predict the behavior of larger systems. The errors in energy predictions were enormous, sometimes reaching tens of electronvolts, which is a massive mistake in the world of atomic physics. The computer had simply memorized the specific quirks of that small model rather than learning the underlying physics. However, when the training data included examples from several different small sizes, the model's ability to predict the larger systems improved dramatically. By seeing how the defect behaved in a small cluster, a medium cluster, and a slightly larger one, the computer learned to recognize the patterns that hold true regardless of the system's size. This allowed it to make accurate predictions for systems containing over two hundred atoms, with energy errors dropping to less than 0.3 electronvolts in many cases.

Adding perfect, defect-free material to the training data provided an extra boost, particularly for defects with a low electric charge. This extra information helped the model understand the stable, undisturbed environment that surrounds the defect, making its predictions even more reliable. Yet, the study also highlighted a limit to this efficiency. For defects with a very high electric charge, simply using small models and adding perfect material was not enough. The long-range electrical forces in these highly charged systems are so strong that the model still needed to see examples of larger, charged-defect systems to learn the correct behavior. Without these larger examples, the predictions remained unstable.

This work provides a practical roadmap for scientists who want to use machine learning to study defects without waiting years for supercomputers to finish their calculations. It shows that you do not need a massive dataset to get good results; you need a smart one. By carefully selecting a few examples from different sizes and mixing in data from perfect materials, researchers can train models that are accurate enough to simulate large-scale defects. This approach makes it feasible to study the microscopic flaws that control the performance of the electronic and optical devices that shape our daily lives, turning a computationally impossible task into a manageable one.

Technical Summary: Small-Supercell and Small-Dataset Training Strategy for MLIPs in Point Defect Simulations

Problem Statement
Machine Learning Interatomic Potentials (MLIPs) offer a pathway to simulate large-scale material systems with near first-principles accuracy, significantly accelerating point-defect studies. However, training reliable MLIPs for defects, particularly charged defects, traditionally requires extensive Density Functional Theory (DFT) datasets. This is computationally prohibitive because charged defects necessitate large supercells (often hundreds of atoms) to mitigate finite-size effects and long-range Coulomb interactions. Furthermore, standard MLIPs, which rely on local atomic environments within a finite cutoff radius, struggle to explicitly represent long-range electrostatics. Consequently, models trained on small supercells often fail to extrapolate accurately to larger systems, learning finite-size errors inherent in the reference DFT data rather than the true physical behavior. The challenge lies in developing a training strategy that utilizes limited DFT data and small supercells while maintaining predictive accuracy for larger defect systems.

Methodology
The authors propose an efficient training scheme for neutral and lowly charged point defects using small supercells (less than 100 atoms) and a minimal number of DFT calculations. The study focuses on three representative semiconductor systems with varying chemical complexity and bonding environments: Nitrogen vacancies ( $V_N$ ) in Gallium Nitride (GaN), Oxygen vacancies ( $V_O$ ) in Silicon Dioxide ( $SiO_2$ ), and Copper-on-Zinc antisites ( $Cu_{Zn}$ ) in Copper Zinc Tin Sulfide ( $Cu_2ZnSnS_4$ ).

Key methodological components include:

Data Generation: Training datasets are constructed from the relaxation trajectories of only four DFT structural optimizations. For neutral systems and substitutional defects, relaxations start from unrelaxed configurations. For charged vacancies, relaxations initiate from the optimized equilibrium structures of their corresponding neutral states to better mimic structural evolution.
Training Schemes: Two distinct strategies are compared:
1. Defect-Only Scheme: The training set contains only charged-defect configurations sampled from DFT trajectories, utilizing an incremental multi-size supercell approach (e.g., adding 15, 31, 63, and 95-atom supercells sequentially).
2. Defect+Host Scheme: This scheme retains the multi-size charged-defect data but adds size-matched neutral, defect-free bulk supercell configurations to constrain the local atomic environments.
Model Architecture: The Allegro package is used to train independent MLIPs for each specific defect type and charge state to avoid mapping conflicts between different potential energy surfaces. The models utilize a cutoff radius of 6 Å, spherical harmonic expansion up to $l_{max}=2$ , and full $O(3)$ parity symmetry.
Evaluation: The trained models are integrated into LAMMPS to evaluate their extrapolation capabilities. Performance is assessed by comparing MLIP-predicted total energies and structural relaxations (measured by Root-Mean-Square Displacement, RMSD) against DFT reference structures in larger supercells (up to 323 atoms).

Key Results

Neutral Defects: Standard MLIPs trained on limited small-supercell data demonstrate robust size-transferability for neutral defects. Structural predictions (RMSD) remain stable (often < 0.02 Å), and energy errors generally stay below 0.3 eV, even when extrapolating to significantly larger supercells.
Charged Defects (Defect-Only Scheme):
- Training exclusively on a single small supercell size leads to catastrophic failure when extrapolating to larger systems, with energy errors reaching tens of eV (e.g., for $Cu_{Zn}$ in $Cu_2ZnSnS_4$ ).
- Incorporating defect configurations from multiple small supercell sizes substantially improves performance. For example, including 15, 31, 63, and 95-atom configurations reduced energy errors for $V_N^+$ in GaN for a 255-atom test cell.
- However, the convergence of energy and structural errors is not always monotonic; adding intermediate sizes does not guarantee immediate improvement, and some systems (e.g., $V_O^{2+}$ in $SiO_2$ ) require the inclusion of the largest available training supercells to achieve low errors.
Charged Defects (Defect+Host Scheme):
- Adding size-matched neutral bulk data significantly reduces energy errors for low-charge defects (e.g., $V_N^+$ , $Cu_{Zn}^-$ ), often bringing errors down to the 0.1–0.3 eV range even with smaller training sets compared to the Defect-Only scheme.
- For highly charged defects (e.g., $V_N^{2+}$ , $V_N^{3+}$ in GaN, $V_O^{2+}$ in $SiO_2$ ), the addition of neutral bulk data alone is insufficient to ensure stable cross-size prediction. While it helps, reliable extrapolation for these high-charge states still requires the inclusion of larger charged-defect supercells in the training set for reliable results.

Significance and Claims
The paper claims to provide practical guidance for constructing reliable MLIP models for defect systems with low computational costs. The authors demonstrate that while standard MLIPs cannot explicitly model long-range electrostatics, their extrapolation capability can be significantly enhanced through careful dataset composition. Specifically, the study establishes that:

A training dataset can be constructed from as few as four DFT structural relaxations.
Multi-size training data is critical for charged defects; single-size training fails to generalize.
Including neutral pristine bulk configurations further refines accuracy for low-charge states.
Despite these improvements, the practical limits of standard MLIPs remain for highly charged defects, where finite-size effects are dominant, necessitating the inclusion of larger charged-defect supercells in the training set for reliable results.

This work clarifies the trade-offs between dataset size, composition, and model performance, offering a computationally efficient strategy for extending MLIP-based simulations to broader defect studies without the prohibitive cost of generating massive DFT datasets.

Technical Summary: Small-Supercell and Small-Dataset Training Strategy for MLIPs in Point Defect Simulations

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