🔬 materials science

Physics-informed Machine Learning Prediction of Hubbard Interaction Parameters

This study presents physics-informed machine learning models that accurately predict cRPA-derived Hubbard interaction parameters ( $U_{\rm eff}$ , $V$ , and $J$ ) for transition-metal oxides, offering both high-throughput screening capabilities and new physical insights into the electronic and structural factors governing these interactions.

Original authors: Jiyeon Kim, Indukuru Ramesh Reddy, Bongjae Kim, Sooran Kim

Published 2026-07-30

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CC BY 4.0

Original authors: Jiyeon Kim, Indukuru Ramesh Reddy, Bongjae Kim, Sooran Kim

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

Imagine you are trying to build a perfect model of a tiny, bustling city made of atoms. In this city, electrons are the citizens, zipping around and interacting with one another. Sometimes, these electrons are well-behaved and follow the standard rules of the road. But in certain materials, like transition-metal oxides, the electrons get rowdy. They start pushing and shoving each other so hard that the usual rules break down. This is called "strong electron correlation." To understand these materials—which are crucial for making better batteries, superconductors, and magnets—scientists use a special set of rules called "Hubbard parameters." Think of these parameters as the "traffic laws" for these rowdy electrons: how much they repel each other when they are on the same spot (on-site), how they interact with neighbors across the street (inter-site), and how they spin in sync (Hund's coupling).

For a long time, figuring out these traffic laws was a guessing game. Scientists would tweak the numbers until their computer models matched real-world experiments, but this made the models unreliable for new materials. Later, a more rigorous method called "constrained random-phase approximation" (cRPA) was developed. It's like having a super-accurate traffic camera that can calculate the exact rules based on physics, but it's so computationally expensive that it takes forever to run, making it impossible to check every material in the universe. This is where the story gets interesting: how do we get the accuracy of the super-camera without waiting years for the results?

The Paper's Story: Teaching Computers to Guess the Rules

In this study, researchers Jiyeon Kim, Indukuru Ramesh Reddy, and their team at Kyungpook National University decided to teach a computer to be a master traffic cop. They wanted to build a "physics-informed" machine learning model that could predict these tricky Hubbard parameters (specifically $U_{eff}$ , $V$ , and $J$ ) for transition-metal oxides. Instead of just letting the computer guess blindly, they gave it a cheat sheet of physical clues—things like how wide the electron's "highway" is (bandwidth) and how far apart the electron's energy levels are from the oxygen neighbors (band-center separation).

The team first built a massive dataset using the slow, expensive cRPA method to get the "true" answers for a variety of materials. Then, they trained three different types of machine learning detectives (called Random Forest, Gradient Boosting, and XGBoost) to learn the patterns. The results were impressive. The best detective, an XGBoost model, could predict the on-site repulsion ( $U_{eff}$ ) with an error of just 0.148 eV, the neighbor interaction ( $V$ ) with an error of 0.062 eV, and the spin alignment ( $J$ ) with a tiny error of 0.007 eV. To put that in perspective, the model is so good that it can tell the difference between materials almost as well as the slow, expensive method, but in a fraction of a second.

But the authors didn't stop at just making a "black box" that gives answers. They wanted to know why the computer was making those guesses. So, they used a second trick called a "regression-based brute-force search" (BFS). Imagine this as taking the computer's best guesses and forcing it to write out a simple math equation that explains the logic. This revealed some fascinating truths about the atomic world:

For the On-Site Repulsion ( $U_{eff}$ ): The model found that this value is a tug-of-war between how "localized" an electron is (how stuck it is to its atom) and how much it mixes with its oxygen neighbors. The math showed that if the electron is more localized (narrower bandwidth) and less mixed with oxygen (larger energy separation), the repulsion gets stronger. The derived equation directly linked these physical concepts to the final number.
For the Neighbor Interaction ( $V$ ): This one was all about structure. The model suggested that the closer the atoms are packed together (structural compactness) and the stronger the mixing with oxygen, the stronger the interaction between neighbors. Interestingly, the "localization" of the electron didn't matter as much here as it did for the on-site repulsion.
For the Spin Alignment ( $J$ ): This was the simplest of all. The model discovered that the spin alignment is almost entirely determined by the identity of the metal atom itself. It's like a fingerprint; if you know the element (its atomic number and group), you can predict the spin behavior with incredible accuracy, regardless of the complex structure around it. The model suggested that because this interaction is weakly screened by the environment, the atom's natural character shines through.

The authors emphasize that while these machine learning models are highly accurate, they are not magic. They are built on the foundation of the cRPA data, which serves as the "truth" for the training. The study suggests that by combining the speed of machine learning with the physical insights of these derived equations, scientists can now screen thousands of materials quickly without sacrificing the deep understanding of why they behave the way they do. It's a way to get the best of both worlds: the speed of a guess and the wisdom of a physicist.

Technical Summary: Physics-informed Machine Learning Prediction of Hubbard Interaction Parameters

Problem Statement
Accurate determination of Hubbard interaction parameters—specifically the effective on-site Coulomb interaction ( $U_{eff}$ ), inter-site Coulomb interaction ( $V$ ), and Hund's coupling ( $J$ )—is critical for applying beyond-DFT methods (such as DFT+U, DFT+DMFT, and DFT+U+V) to correlated materials. While empirical tuning of these parameters is common, it limits transferability. First-principles approaches like the constrained random-phase approximation (cRPA) offer a rigorous, transferable route by explicitly treating electronic screening. However, the high computational cost of cRPA calculations creates a bottleneck for large-scale materials screening. Although recent machine learning (ML) studies have predicted Hubbard parameters, the specific prediction of cRPA-derived $U_{eff}$ , $V$ , and $J$ using descriptors that capture electron localization and screening effects remains largely unexplored.

Methodology
The authors developed a physics-informed ML framework to predict cRPA-derived parameters for a diverse set of transition-metal oxides (TMOs) with perovskite, olivine, spinel, layered, and layered perovskite structures.

Data Generation: A dataset of $U_{eff}$ , $V$ , and $J$ was generated using cRPA calculations within VASP. The correlated subspace (TM-d manifold) was defined via Wannier function projections, and screening processes associated with these bands were excluded to compute the screened interaction.
Feature Construction: The study constructed 12 descriptors categorized into:
- Electronic-structure features: TM-d bandwidth ( $W$ ) and the energy separation between TM-d and O-p band centers ( $D_{d-p}$ ), representing localization and hybridization/screening, respectively.
- Atomic and Compositional features: Atomic number ( $Z_{TM}$ ), group ( $G_{TM}$ ), period ( $P_{TM}$ ), average d-electron count ( $n_d$ ), and average oxidation state ( $n_{ox}$ ).
- Structural features: Unit cell atom count ( $N_{u.c}$ ), TM ion count ( $TM_{u.c}$ ), shortest TM-O bond length ( $d_{TM-O}$ ), volume per formula unit ( $V_{f.u}$ ), and space group number ($SG$).
Modeling Strategy: The study employed a dual-approach strategy:
- Ensemble Learning: Random Forest (RF), Gradient Boosting Regression (GBR), and Extreme Gradient Boosting (XGB) models were trained using Leave-One-Out Cross-Validation (LOOCV) to identify key features and achieve high predictive accuracy.
- Regression-based Brute-Force Search (BFS): To obtain explicit analytical expressions, a BFS algorithm was applied. Primary features selected from ensemble analysis were transformed using 15 nonlinear functions (e.g., $x, 1/x, e^x, \ln(x+1)$ ) to generate prototypical features. These were multiplicatively combined into compound features to construct linear regression models (denoted as $mCnF$, where $m$ is the number of compound features and $n$ is the number of primary features).

Key Results

Prediction of $U_{eff}$ :
- Ensemble Models: XGB achieved the best performance with an RMSE of 0.148 eV and $R^2$ of 0.899 using six features ( $n_{ox}, n_d, N_{u.c}, V_{f.u}, Z_{TM}, G_{TM}$ ). A four-feature subset ( $n_{ox}, n_d, N_{u.c}, V_{f.u}$ ) yielded nearly saturated performance (RMSE 0.168 eV).
- Analytical Expressions: The BFS method produced explicit formulas. A model using the physically motivated set ( $W, D_{d-p}, n_d, V_{f.u}$ ) achieved an RMSE of 0.207 eV. The derived analytical form (Eq. 3) directly relates $U_{eff}$ to localization ( $W$ ) and hybridization ( $D_{d-p}$ ), showing that $U_{eff}$ increases with reduced hybridization (larger $D_{d-p}$ ) and exhibits a non-monotonic dependence on $n_d$ .
Prediction of Inter-site $V$ :
- Ensemble Models: XGB achieved an RMSE of 0.062 eV ( $R^2 = 0.784$ ) using five features ( $n_d, N_{u.c}, V_{f.u}, SG, Z_{TM}$ ).
- Analytical Expressions: The best BFS model (3C2F configuration) using the set ( $n_d, N_{u.c}, V_{f.u}, P_{TM}$ ) achieved an RMSE of 0.071 eV. The resulting formula (Eq. 4) indicates that $V$ is governed by structural compactness (smaller $V_{f.u}$ and larger $N_{u.c}$ increase $V$ ) and hybridization-related screening ( $D_{d-p}$ ), while $W$ (localization) plays a minimal role.
Prediction of Hund's Coupling $J$ :
- Ensemble Models: $J$ was predicted with exceptional accuracy using minimal descriptors. XGB achieved an RMSE of 0.007 eV ( $R^2 = 0.989$ ) using only three features ( $G_{TM}, P_{TM}, SG$ ). Even a single feature ( $Z_{TM}$ ) yielded an RMSE of ~0.015 eV.
- Analytical Expressions: The BFS analysis confirmed that $J$ is primarily governed by intrinsic atomic characteristics. The best analytical model (Eq. 6) depends only on elemental descriptors ( $G_{TM}$ and $P_{TM}$ ), reflecting the trend that $J$ increases with the TM series and decreases with the period, consistent with the preservation of bare atomic trends due to weak screening in cRPA.

Significance and Claims
The paper claims to provide an efficient approach for predicting cRPA-derived Hubbard parameters, thereby accelerating DFT+U and DFT+U+V workflows without relying on empirical fitting. By combining ensemble learning with BFS, the study achieves both high predictive accuracy and physical interpretability.

The authors emphasize that their framework offers specific physical insights:

$U_{eff}$ is directly linked to the competition between electron localization and TM-d/O-p hybridization.
$V$ is determined primarily by hybridization-related screening and structural compactness, rather than orbital localization.
$J$ is governed almost entirely by the intrinsic atomic properties of the transition metal ion, as cRPA screening is weak for this parameter.

The study concludes that this physics-informed ML framework provides a practical route for identifying descriptor-property relationships in correlated oxides and offers explicit analytical formulas that can be applied to new materials without requiring access to the original training data or complex cRPA calculations.

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