Machine-learning octet $AB$-type binary compounds across chemical space with domain knowledge of the interatomic bond

Technical Summary: Machine-learning octet AB-type binary compounds across chemical space with domain knowledge of the interatomic bond

Problem Statement
The prediction of structural stability for octet AB-type binary compounds (compounds with two elements and eight valence electrons per formula unit) is a classical challenge in materials informatics. These compounds crystallize in either the zincblende (ZB, 4-fold tetrahedral coordination) or rocksalt (RS, 6-fold octahedral) structures. The stability is governed by a complex interplay of charge transfer, atomic size differences, and the transition from covalent to ionic bonding. While previous machine learning (ML) approaches have utilized atomic features such as valence electron counts, ionization potentials, and atomic radii, or employed symbolic regression to find descriptors, there remains a need to systematically incorporate explicit domain knowledge regarding interatomic bonds to improve predictive accuracy for the formation energy difference ($\Delta E_{AB}$) between these phases.

Methodology
The authors propose a methodology that encodes domain knowledge of interatomic bonds directly into the feature space used for ML models. The core of this approach involves:

1. Bond-Informed Recursion Features: Instead of relying solely on static atomic properties, the authors compute local electronic structure descriptors using a tight-binding (TB) bond model. Pairwise Hamiltonians ($H_{i\alpha j\beta}$) are derived from the downfolded eigenstates of density-functional theory (DFT) calculations for diatomic molecules, ensuring the features contain specific chemical knowledge of A–B pairs.
2. Lanczos Recursion: These Hamiltonians are transformed into a tri-diagonal form using the Lanczos algorithm, generating recursion coefficients ($a_n$ and $b_n$). These coefficients represent the local density of states (DOS) and capture the electronic environment up to the $n$-th shell of neighbors.
3. Feature Construction:
   • AtomF: A baseline set of fundamental atomic properties (electronegativity, orbital radii, ionization potential, etc.).
   • RecN: Recursion-based features computed for $n \leq 8$. To handle the structural dependence, features are calculated for idealized RS and ZB unit cells (scaled by Vegard's law) and compared using operators (difference $\Delta$ and ratio $r$) between the two structures.
   • Combined Vectors: Feature vectors are constructed by concatenating AtomF with increasing orders of recursion features (Rec1 to Rec8).
4. Machine Learning Architecture:
   • Ensemble Strategy: Given the small dataset size (82 compounds, reduced to 78 after outlier removal), an ensemble approach is used to ensure robustness. The workflow involves 150 repeated ShuffleSplit cross-validations.
   • Sequential Feature Selection (SFS): A greedy forward search with conditional exclusion is used to select the optimal subset of descriptors for each feature set.
   • Models: Two types of models are trained and combined: Kernel Ridge Regression (KRR) with polynomial kernels and the Sure Independence Screening and Sparsifying Operator (SISSO) for symbolic regression. The final ensemble prediction is a weighted average based on individual model performance.
5. Dataset: The study uses the same 82 octet binary compounds as a previous reference study [3], predicting the formation energy difference $\Delta E_{AB} = \Delta H_{F,AB}^{RS} - \Delta H_{F,AB}^{ZB}$. Outliers (BN, C2, LiBr, LiCl) were identified and analyzed, leading to two dataset versions: D1 (all 82) and D2 (78, outliers removed).

Key Results
The study demonstrates that incorporating bond-informed recursion features significantly improves the prediction of structural stability compared to models using only atomic features or previous symbolic regression benchmarks.

• Performance on Full Dataset (D1): The ensemble model trained on atomic features (AtomF) alone outperformed the reference model from Ghiringhelli et al. [3]. However, the best performance was achieved by combining AtomF with recursion features up to the 4th order (AtomF + Rec4). This configuration yielded a mean Root Mean Square Error (RMSE) of $0.015 \pm 0.002$ eV on the training set and $0.046 \pm 0.026$ eV on the test set, significantly lower than the reference model's test RMSE of $0.139 \pm 0.088$ eV.
• Performance on Refined Dataset (D2): On the dataset excluding outliers, the improvement was even more pronounced. The AtomF + Rec6 model achieved a mean test RMSE of $0.029 \pm 0.010$ eV. This outperformed both the reference model and the 3D SISSO ensemble, which plateaued at a higher error range ($0.062–0.074$ eV).
• Robustness: The inclusion of recursion features not only reduced mean errors but also significantly improved model stability. The maximum Mean Absolute Error (MAE) across 150 splits dropped from $0.382$ eV (reference model) to $0.101$ eV (AtomF + Rec4) on D1, and from $0.155$ eV to $0.045$ eV on D2.
• Order of Recursion: The results indicate that predictive accuracy improves with the inclusion of higher-order recursion features up to a certain point (4th order for D1, 6th order for D2), after which performance converges or slightly degrades, suggesting that the most critical bonding information is captured in the first few recursion levels.

Significance and Claims
The paper claims that explicitly incorporating domain knowledge of interatomic bonds through a coarse-grained representation of local electronic structure (recursion coefficients from TB models) systematically and significantly improves the prediction of phase stability in octet binaries.

The authors emphasize that this approach utilizes pre-computed, bond-specific Hamiltonians derived from DFT eigenstates of diatomic molecules to generate features that capture the essential physics of coordination and charge transfer. By leveraging these domain-informed descriptors, the study demonstrates that ML models can achieve higher accuracy and robustness than previous state-of-the-art symbolic regression models using the same training data, validating the utility of embedding physical domain knowledge directly into the feature engineering process for materials informatics.