This repository provides a reproducible workflow for predicting the pressure-dependent bulk modulus of elemental crystalline solids. It combines MatTen, an equivariant Graph Neural Network (GNN), with a suite of regression-based correction models to enhance its predictions. The primary goal is to offer a high-throughput, accurate, and physically meaningful approach for elastic property prediction across varying pressure conditions, directly supporting computational materials science and informatics.
- Overview
- Scientific Motivation
- Dataset Description
- Methodology
- Results and Analysis
- Limitations
- Dependencies
- Future Directions
- Contact
- References
- Acknowledgments
This project presents a data-driven approach to predict the bulk modulus—a fundamental mechanical property—of elemental crystals as a function of applied pressure. By integrating MatTen, an equivariant GNN designed for tensorial properties, with regression-based corrections, this workflow significantly reduces the computational cost compared to direct DFT calculations, while maintaining high accuracy and physical interpretability.
Elastic moduli, particularly the bulk modulus, are crucial for understanding the mechanical behavior and phase stability of solids under extreme conditions. While traditional quantum-mechanical simulations (e.g., Density Functional Theory - DFT) offer reliable results, they are computationally prohibitive for systematic studies across a wide range of compositions and pressure conditions.
Recent advancements in machine learning, especially physically-informed Graph Neural Networks, offer a promising avenue for rapid property prediction. However, pre-trained GNNs may exhibit systematic biases when applied to new domains or under extrapolation. This work addresses these challenges by:
- Quantifying the performance of the MatTen GNN for elemental systems under pressure.
- Applying machine learning regression models to correct systematic errors and calibrate predictions against DFT data.
- Evaluating the model's performance for both interpolation (within DFT-sampled pressure ranges) and extrapolation (beyond those ranges).
This project utilizes a custom dataset of elemental crystal structures evaluated at various pressures. This dataset is specifically designed for benchmarking and developing machine learning models for elastic property prediction, such as the bulk modulus.
- Systems: A diverse range of elemental crystals (e.g., Fe, Cr, Mn, Mo, Nb, Rh, Ru, Sr, Tc, Y, Zr, Rb), each evaluated under hydrostatic pressure.
- Pressure Range: Typically spans from 0 to 200 GPa, with data sampled at discrete intervals for each system.
- Properties per Entry: Each entry includes:
- POSCAR: Atomic structure in standard VASP format.
- target.json: DFT-computed physical properties, notably:
"bulk_modulus"(in GPa, the primary target property for ML)."elastic_tensor"(6x6 Voigt matrix).- Additional fields such as pressure, free energy, ELF descriptors, and lattice constants.
- MatTen Model Pre-training: The MatTen GNN model is pre-trained on a comprehensive dataset of 10,276 elasticity tensors from the Materials Project. This pre-training utilizes an 8:1:1 train/validation/test split, ensuring robust generalization.
This integrated dataset, combining quantum-computed properties with standardized structure files, forms a robust foundation for data-driven elastic property prediction and model evaluation.
The dataset should be organized into the following directory structure: gnn_dataset/ ├── Mo_0.0GPa/ │ ├── POSCAR │ └── target.json ├── Mo_10.0GPa/ │ ├── POSCAR │ └── target.json ├── Nb_0.0GPa/ │ ├── POSCAR │ └── target.json ...
- Naming Convention: Each subdirectory is named as
{ElementSymbol}_{Pressure}GPa(e.g.,Mo_0.0GPafor molybdenum at 0 GPa). - File Contents:
- POSCAR: VASP format crystal structure.
- target.json: DFT-computed properties and physical descriptors.
MatTen is utilized to predict the full elasticity tensor from the crystal structure.
- Feature Construction: Atomic positions are encoded as displacement vectors, and atomic species are represented as one-hot vectors.
- Network Architecture: The model employs multiple equivariant GNN layers to capture spatial symmetries and interactions. Radial basis functions and spherical harmonics ensure geometric expressiveness, while interaction blocks update atomic features through tensor operations and normalization.
- Output: The network directly outputs a full 6x6 Voigt matrix elasticity tensor, designed to inherently respect symmetry constraints. This ensures that predictions are structure-aware and physically plausible, even for systems not explicitly present in the training set.
The bulk modulus is derived from the predicted elasticity tensor.
-
Mathematical Expression: The bulk modulus (
$K$ ) here (for cubic system) is calculated as:$$K = \frac{C_{11} + 2C_{12} }{3}$$ -
Interpretation: The bulk modulus is a standard method for estimating the hardness of a system subjected to hydrostatic stress.
Following the MatTen prediction, several supervised machine learning regressors are trained to correct the predicted bulk moduli against the DFT reference data.
-
Regression Models Evaluated:
- Linear Regression: For capturing global linear relationships.
- Polynomial Regression: To capture systematic nonlinear deviations (tested with degrees 2–4).
- Isotonic Regression: Ideal for monotonic but potentially nonlinear relationships.
- k-Nearest Neighbors: To capture local data structures.
- Random Forest Regression: For modeling complex, nonlinear dependencies and reducing overfitting.
- Gaussian Process Regression: Provides both predictions and uncertainty quantification.
- Exponential Regression: To investigate potential log-scale or exponential relationships.
-
Features: The input features for these correction models include:
- The MatTen-predicted bulk modulus.
- Pressure.
- Tensor norm (characterizing the overall elastic response).
-
Model Selection and Validation: Hyperparameters are optimized using cross-validation and grid search. Models are rigorously evaluated based on
$R^2$ (coefficient of determination) and MAE (Mean Absolute Error).
Our initial MatTen predictions showed a strong correlation (
Figure 1: Initial MatTen Prediction vs. DFT Bulk Modulus
This scatter plot compares raw MatTen-predicted bulk modulus values against DFT-computed values, illustrating the initial correlation and systematic deviations from the ideal $y=x$ line.
Figure 2: 
This chart provides a direct comparison of the $R^2$ values across different correction models, showcasing their respective performance.
Performance metrics for all evaluated models are detailed in results_ml_ecs/bulkmod_correction_metrics.csv.
The application of machine learning correction models, particularly Random Forest (rf) or Polynomial Regression, significantly enhanced prediction accuracy. We achieved an impressive
Figure 3: ML-Corrected Prediction vs. DFT Bulk Modulus
This figure demonstrates the enhanced accuracy post-correction, showing data points significantly tighter around the $y=x$ line and a substantial reduction in prediction error.
Post-correction, residuals (DFT - prediction) were minimized and exhibited no significant pressure-dependent drift, confirming the effectiveness of the corrections in removing systematic biases.
Figure 4: Residual Comparison: Raw MatTen vs. Best Corrected
This plot clearly contrasts the residual distribution before and after correction, visually confirming the successful minimization of errors.
Within the observed DFT-sampled pressure ranges, the corrected models closely matched DFT data, exhibiting small residuals and no evidence of overfitting. This confirms our pipeline's robust interpolation capabilities.
Furthermore, the developed pipeline demonstrates the ability to extrapolate trends beyond the available DFT data. MatTen's inherent physical plausibility ensures reasonable physical behavior (e.g., monotonic increase or decrease in bulk modulus under compression) even in extrapolated regions.
Figure 5: Pressure-Dependent Bulk Modulus for Selected Elements (Interpolation & Extrapolation)
This plot effectively illustrates the model's ability to accurately track DFT data within the sampled range and extend these trends into unobserved pressure regimes, maintaining physical consistency.
While promising, it's crucial to validate extrapolated predictions with new DFT data or expert physical reasoning, as ML corrections may become less reliable far outside the training domain. Overall, the workflow successfully captures distinct pressure dependencies and structural characteristics across various elemental systems.
The primary source of residual error stemmed from the domain mismatch between MatTen's pre-training data and the specific pressure/structure combinations in the current dataset. The effectiveness of regression corrections indicates these errors were largely systematic rather than random.
This approach enables rapid screening of elastic properties for materials discovery and design under varying environmental conditions. It supports experimental prioritization and hypothesis generation by providing fast, accurate, and physically interpretable predictions.
- numpy, pandas, matplotlib
- scikit-learn
- pymatgen
- matten
Run the code with:
python3 matten_bulk_modulus_prediction.py
This script will perform the ML analysis, generate plots, and save results in the results_ml_ecs/ directory.
- The current dataset is limited to elemental solids; extending the methodology to multicomponent systems will require further validation and potentially retraining.
- The reliability of ML corrections is inherently tied to the quality and coverage of the underlying DFT reference data across the relevant phase space.
- The physical interpretability of ML corrections may be limited, especially for highly nonlinear relationships or outlier cases.
- Multi-property predictions (e.g., elastic anisotropy, higher-order moduli, ...)
- Transfer learning to alloys, compounds, or disordered systems
- Uncertainty quantification for risk-aware materials discovery
- Active learning and feedback with on-the-fly DFT
- Scaling up to larger, more diverse datasets
Abhiyan Pandit https://www.linkedin.com/in/abhiyan-pandit-a8838a138/ abhi99lucky@gmail.com
- Graph Neural Network: https://www.nature.com/articles/s43586-024-00294-7
- Scikit-learn: https://scikit-learn.org/stable/
- MatTen: https://github.com/wengroup/matten