Source-linked AI summary
Applying machine learning techniques to predict the properties of energetic materials
Daniel C. Elton, Zois Boukouvalas, Mark S. Butrico, Mark D. Fuge, Peter W. Chung
TL;DR
Energetic-material candidates are commonly screened with expensive quantum simulations and thermochemical codes, while prior machine-learning evidence is limited by small or narrow datasets. The paper compares models and molecular featurizations on diverse energetic molecules, finding that sum over bonds and kernel ridge regression perform best, with acceptable out-of-sample predictions despite limited data. The authors also identify scope limitations in some featurizations and show that additional data can reduce error slowly.
Problem
Energetic-material discovery relies on expensive computational screening, while prior machine-learning studies often use narrow molecular domains or large datasets.
Method
The study compares machine-learning models and molecular featurizations on 109 energetic compounds spanning ten classes, using molecular structures to predict energetic properties.
Results
Sum over bonds was the best base featurization, kernel ridge or ridge regression achieved the highest accuracy across tested properties, and out-of-sample predictions showed acceptable errors and correlations.
Takeaways & Limitations
Small but diverse datasets can support out-of-sample prediction of multiple energetic properties, providing a proof of concept for machine learning in this domain.
Takeaways & Limitations
Oxygen balance neglects variation in bond strengths and energy contributions from nitrogen recombination, halogen reactions, and strain release.
Abstract
from arXiv · showhide
We present a proof of concept that machine learning techniques can be used to predict the properties of CNOHF energetic molecules from their molecular structures. We focus on a small but diverse dataset consisting of 109 molecular structures spread across ten compound classes. Up until now, candidate molecules for energetic materials have been screened using predictions from expensive quantum simulations and thermochemical codes. We present a comprehensive comparison of machine learning models and several molecular featurization methods - sum over bonds, custom descriptors, Coulomb matrices, bag of bonds, and fingerprints. The best featurization was sum over bonds (bond counting), and the best model was kernel ridge regression. Despite having a small data set, we obtain acceptable errors and Pearson correlations for the prediction of detonation pressure, detonation velocity, explosive energy, heat of formation, density, and other properties out of sample. By including another dataset with 309 additional molecules in our training we show how the error can be pushed lower, although the convergence with number of molecules is slow. Our work paves the way for future applications of machine learning in this domain, including automated lead generation and interpreting machine learning models to obtain novel chemical insights.
1 Introduction
Energetic-material discovery has produced many synthesized compounds but few industrial replacements, while machine learning offers a potential alternative to expensive screening methods. This study addresses limited, narrow prior applications by testing diverse energetic molecules with small-data machine learning.
- Motivation: Decades of research have sought energetic materials with improved performance, stability, and safety to replace established compounds including HMX, RDX, TNT, PETN, and TATB.Hundreds of materials have been synthesized, but very few have reached industrial production; CL-20 is one exception.
- Prior machine-learning applications: Machine learning has predicted properties across molecular, materials, pharmaceutical, and polymer applications, but published molecular studies typically use 10,000−100,000 examples.With sufficiently large datasets, ML has reproduced DFT-calculated properties with deviations smaller than DFT’s deviation from experiment.
- Gap: Energetic-material applications of machine learning remain limited, with little prior work predicting explosive energy, detonation velocity, or detonation pressure.Existing studies often focus on narrow molecular families and combine hand-picked descriptors with linear models.
- Study objective: The study compares machine-learning methods and featurizations using 109 energetic compounds from ten distinct classes to test performance with small, diverse data.The authors later add data from Mathieu but restrict much of the work to the Huang & Massa dataset to characterize small-data performance.
- Data and properties: The underlying dataset contains nine properties, including density, solid heat of formation, explosive energy, shock velocity, detonation pressure, and TNT equivalent per cubic centimeter.Huang & Massa derived energetic properties using calculated heats of formation and densities as inputs to a thermochemistry code under Chapman-Jouguet detonation theory, with experimental validation.
2 Featurization methods
The paper compares chemically informed and general-purpose molecular featurizations, emphasizing that feature design is especially important for small datasets. The methods include custom descriptors, bond counts, Coulomb-matrix representations, bag-of-bonds, and fingerprints.
- Featurization methods: Small-data models benefit more from carefully chosen featurizations than from model selection alone.Feature vectors should have dimensionality d much smaller than the number of training molecules, while chemically meaningful hand-selected features can improve accuracy.
- Custom descriptor set: Custom descriptors combine oxygen balance, nitrogen-to-carbon ratio, functional-group and bond-environment counts, and raw elemental counts.The descriptors were selected using physical intuition and computational efficiency, without requiring physics computations.
- Sum over bonds: Sum over bonds represents each molecule by counts of the 20 bond types occurring in the dataset.Bond types include single, double, triple, and aromatic bonds expressed using SMARTS nomenclature.
- Coulomb matrices: Coulomb-matrix features derive from nuclear coordinates and charges, using either sorted eigenvalues or vectorized matrix elements.Eigenvalue spectra remove sensitivity to atom-index permutations, whereas raw matrix elements can have different physical meanings across molecules.
- Bag of Bonds: Bag of Bonds groups Coulomb-like bond information into bond-type-specific bags, sorts each bag by magnitude, pads them, and concatenates them.The representation adapts the bag-of-words idea from natural-language processing to molecular bond types.
- Fingerprinting: Fingerprints transform molecular graphs into bit or integer vectors encoding atom-level and local-environment information, but the studied fingerprints contain only 2D graph information.Fingerprint length can be adjusted through folding, and the study compares E-state, Avalon, RDKit graph, ErG, and physicochemical-property fingerprints.
3 Results
The results compare featurizations and machine-learning models across energetic-material properties, finding strong performance from bond-counting representations and kernel-based regression. Dimensionality reduction generally did not improve predictive error, while concatenated features improved correlation for explosive energy.
- Fingerprint comparison: The E-state fingerprint had the lowest fingerprint explosive-energy test error, with Avalon nearly matching it.
- Featurization comparison: Concatenating E-state and custom descriptors with sum over bonds increased the correlation coefficient r from 0.65 to 0.78.
- Featurization comparison: Sum over bonds was the best featurization for all target properties except speed of sound, where bag of bonds performed slightly better.
- Model comparison: Kernel ridge regression and ridge regression performed best among the compared machine-learning models.
- Model comparison: A gap between training and test mean absolute errors indicated overfitting in the evaluated models.
- Dimensionality reduction: Dimensionality reduction did not improve error, although Coulomb-matrix, E-state, and combined features converged by D = 15.
4 Analysis & Discussion
Analysis shows that model behavior depends on chemical diversity, applicability-domain coverage, and training-set size. Residuals identify molecule-specific failures, while learning curves indicate that additional data helps but yields slow improvement.
- Training data quantity & diversity: 25 pyrazole-based molecules produced very high accuracy, but the models could not generalize beyond that molecular class.The narrow dataset mainly varies functional groups attached to a common pyrazole backbone.
- Applicability domain: The applicability domain can contain internal holes where predictions perform badly, and practical distance cutoffs require trial and error.Convex-hull, principal-component, leverage, and nearest-neighbor heuristics each have limitations.
- Residual analysis: Cubane-derived molecules expose a feature limitation because strain energy is not explicitly represented in the featurizations.The model overestimates cubane energy and underestimates nitrocubane energy.
- Residual analysis: Removing cubanes left explosive-energy MAE at 0.36 kJ/cc but reduced Pearson r from 0.76 to 0.68.The result indicates that the apparent correlation depends partly on how these structurally separated molecules are represented.
- Learning curves: The combined 418-molecule learning curves use log-log plots for detonation velocity and pressure, with shaded 5-fold cross-validation error deviations.The gap between training and test curves indicates variance, while curve height indicates bias.
- Learning curves: 0.15−0.30 was the observed learning-curve exponent range, compared with 1 < β < 2 typically reported for neural-network training.Random forests showed similar curves with β ≈0.20.
5 Conclusion & Future Directions
The study concludes that machine learning can predict energetic properties from a small, diverse molecular dataset, with sum over bonds and ridge-based models performing best. Future work targets limited data, model interpretation, and coupling prediction with molecular generation.
- Conclusion: Ntrain = 87 and Ntest = 22 supported out-of-sample prediction of energetic properties from a small, diverse training set.Kernel ridge or ridge regression had the highest accuracy across tested properties.
- Conclusion: Sum over bonds was the best base featurization across tested properties, while concatenating featurizations sometimes produced small improvements.The best reported r values were 0.94 for heat of formation, 0.74 for density, 0.79 for explosive energy, and 0.78 for shock velocity.
- Conclusion: 11% mean percentage error was obtained for explosive energy and detonation pressure, versus 4% for density and detonation velocity.These values were obtained with kernel ridge regression and sum over bonds.
- Conclusion: Approximately 300 additional molecules lowered mean absolute errors, although convergence with molecule count was slow.The result indicates that expanding the training set improves performance gradually rather than rapidly.
- Future directions: Transfer learning, non-energetic CNOHF molecules, interpretable models, and generative models are proposed as future research directions.Generative models could be coupled with property predictors for molecular generation and optimization.
6 Methods
The study evaluates energetic-material property models using molecular representations derived from connectivity graphs and several error and correlation metrics. It cautions that Q2 should not be used alone to assess model quality.
- Evaluation metrics: Model performance is evaluated with mean absolute error, which measures the average absolute prediction error.The paper describes MAE as the simplest scoring function.
- Evaluation metrics: Root mean squared error is more sensitive to outliers than mean absolute error.RMSE is also called the standard error of prediction.
- Evaluation metrics: Mean average percent error is reported, although it can be misleadingly inflated when true values approach zero.
- Evaluation metrics: Pearson correlation and the coefficient of determination are also reported to assess agreement between predictions and targets.For test or validation data, the coefficient of determination is often called Q2.
- Evaluation metrics: A good Q2 does not necessarily indicate a good model, so models should not be evaluated using Q2 in isolation.Q2 can range from -∞ to 1, whereas a bad Q2 indicates a bad model.
- Molecular representation: Molecular connectivity graphs, encoded as SMILES strings, provide the main input for most featurizations.Coulomb-matrix and bag-of-bonds featurizations are exceptions; SMILES encode molecular graphs as ASCII strings but are non-unique.