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Predicting Activities in Aqueous Electrolyte Solutions with Hybrid Machine Learning

This paper introduces a hybrid Bromley-MCM model that combines the physics-based Bromley equation with a machine learning matrix completion method to predict ionic and osmotic activity coefficients for thousands of unstudied aqueous electrolytes by inferring missing parameters from a sparse experimental dataset.

Original authors: Zeno Romero, Maximilian Kohns, Fabian Jirasek

Published 2026-07-22

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CC BY 4.0

Original authors: Zeno Romero, Maximilian Kohns, Fabian Jirasek

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 the invisible world of chemistry as a bustling, chaotic dance floor. In this dance, water molecules are the floor, and dissolved salts are the dancers. When these salts dissolve, they split into positively charged "cations" and negatively charged "anions," spinning and swirling around each other. But they aren't just dancing freely; they are constantly pushing and pulling on one another, creating a complex social atmosphere that changes how the whole group behaves. Scientists call this behavior "activity." It's a fancy way of describing how "effective" a chemical is in a solution, which determines everything from how batteries store energy to how we purify seawater.

To predict this dance, scientists have long used rulebooks called "models." These models are like maps that tell us how the ions will interact at different concentrations. However, traditional maps have a major flaw: they are hand-drawn for specific pairs of dancers. If you want to know how a new pair of ions will dance, you have to measure it in a lab, which is slow, expensive, and often impossible for every single combination in the universe. For decades, the scientific community has been stuck trying to draw these maps one by one, leaving huge gaps where we simply don't know the rules.

This is where a team of researchers from Germany steps in with a clever new trick. They didn't throw away the old rulebooks; instead, they combined a classic physics map with a machine learning technique called "matrix completion." Think of it like a giant Sudoku puzzle. Imagine a spreadsheet where the rows are all the different positive ions and the columns are all the different negative ions. Each square in the grid represents a specific salt, and the number inside tells us how that salt behaves. The problem is that we only have numbers for a tiny fraction of the squares—most of the grid is empty because nobody has measured those specific combinations yet.

The researchers, led by Zeno Romero, Maximilian Kohns, and Fabian Jirasek, built a hybrid model they call "Bromley-MCM." They took the established "Bromley model," a trusted physics-based formula, and used it as the skeleton. Then, they plugged in a machine learning algorithm designed to fill in the missing Sudoku squares. The algorithm looks at the patterns in the few squares that are filled and uses them to guess the numbers for the thousands of empty ones. It's like a detective who, after seeing a few clues about how certain dancers interact, can predict how two strangers will dance together without ever having seen them meet before.

The results are impressive. By training this hybrid model on experimental data for 478 different salts, the team successfully predicted the behavior of 9,296 electrolytes at 298 K (about 25°C). This includes 8,818 salts for which no experimental data existed in the literature. When they tested their predictions against salts they deliberately hid from the model during training, the Bromley-MCM proved to be far more accurate than previous predictive methods, such as the "Ion-Specific Bromley model" and the "Simoes model."

The paper explicitly rules out the idea that purely data-driven machine learning (without physics) or old-school ion-specific formulas are the best way forward. The authors show that while those methods exist, they either lack accuracy or cover a much smaller chemical space. For instance, the Simoes model, which relies on ionic radii, failed to predict the behavior of many complex or organic ions that the new model handled easily. The authors also note that while their model is powerful, it isn't magic; it struggles slightly with weak electrolytes (like certain acids) that don't fully dissociate, because the underlying physics model they built upon wasn't designed for those specific cases.

Ultimately, the study demonstrates that by embedding a machine learning "guessing engine" inside a solid physics framework, we can fill in the missing pieces of the chemical puzzle. The authors provide a complete, ready-to-use set of parameters for nearly 9,300 salts, effectively handing engineers and scientists a much more complete map of the aqueous dance floor. This doesn't just solve a theoretical problem; it offers a practical tool for designing better batteries, desalination plants, and industrial processes without needing to run endless, costly experiments for every new chemical mixture.

Technical Summary: Predicting Activities in Aqueous Electrolyte Solutions with Hybrid Machine Learning

Problem Statement
Accurate prediction of activities in aqueous electrolyte solutions, specifically mean ionic activity coefficients ( $\tilde{\gamma}_{\pm}$ ) and osmotic coefficients ( $\phi$ ), is critical for industrial and natural processes such as energy storage and desalination. Established thermodynamic models, such as those by Pitzer and Bromley, rely on fitting electrolyte-specific parameters to experimental data. Consequently, these models cannot predict properties for unstudied systems where no experimental data exists. While some predictive approaches exist (e.g., the ion-specific Bromley model or the Simoes model), they often suffer from limited predictive accuracy, restricted applicability due to missing ion descriptors (e.g., ionic radii), or reliance on correlations rather than true predictions for unseen systems. There is a need for a method that can generalize across chemical spaces to predict activities for unstudied electrolytes while maintaining thermodynamic consistency.

Methodology
The authors propose a novel hybrid model, Bromley-MCM, which integrates the physics-based Bromley model with a Machine Learning Matrix Completion Method (MCM).

Data Curation: The study utilizes a consolidated database derived from the Dortmund Data Bank (DDB) containing 24,856 data points for 478 electrolytes at 298 K. The data includes mean ionic activity coefficients and osmotic coefficients. A rigorous filtering process was applied to ensure data consistency, including Gibbs-Duhem verification and the exclusion of weak electrolytes that do not fit the Bromley model framework.
Matrix Formulation: The electrolyte-specific Bromley parameter ( $B_{CA}$ ) is arranged in a matrix where rows represent cations ( $C$ ) and columns represent anions ( $A$ ). Due to the sparsity of experimental data, this matrix is largely empty (only 5.1% of possible cation-anion pairs have data in the full database).
Hybrid Architecture:
- Physics Component: The Bromley model equations (9–13) describe the relationship between ionic strength, temperature, and the activity coefficients, utilizing a single adjustable parameter $B_{CA}$ .
- Machine Learning Component: An MCM is employed to predict the missing $B_{CA}$ values. The method decomposes the parameter matrix into the product of two low-rank matrices, $\theta$ (cation features) and $\beta$ (anion features), such that $B_{CA} = \theta_C \cdot \beta_A$ .
- Training: The model is trained end-to-end in a Bayesian framework using automatic differentiation variational inference. The training objective minimizes the difference between predicted and experimental $\tilde{\gamma}_{\pm}$ and $\phi$ values. The ion-specific feature vectors have a length of $K=3$ .
Evaluation Strategy: Predictive performance is assessed using a "leave-one-electrolyte-out" strategy. The model is trained on all electrolytes except one, which serves as a test set for true prediction. This approach is applied to a "reduced" database (containing ions with at least two known pairs) to ensure robustness.

Key Contributions

Novel Hybrid Model: Introduction of Bromley-MCM, which successfully embeds a matrix completion method within a thermodynamic framework to predict electrolyte-specific parameters for unstudied systems.
Expanded Applicability: The model enables the prediction of activities for 9,296 electrolytes (83 cations $\times$ 112 anions) at 298 K. Of these, 8,818 electrolytes had no prior activity data in the literature under these conditions.
Comprehensive Benchmarking: The study provides a rigorous comparison against two established predictive models: the Ion-Specific Bromley Model (IBM) and the Simoes model (which predicts Pitzer parameters).
Data Release: The authors provide a complete set of predicted $B_{CA}$ parameters for the expanded chemical space, available in the Supporting Information.

Results

Superior Accuracy: On the reduced database, Bromley-MCM substantially outperforms the retrained IBM for both $\ln \tilde{\gamma}_{\pm}$ and $\phi$ . The prediction errors (Absolute Error) are significantly lower, with the model capturing the qualitative dependence of activities on ionic strength, including the characteristic minimum at low ionic strength.
Comparison with Simoes Model: When evaluated on the scope of the Simoes model, Bromley-MCM again demonstrates superior accuracy. The Simoes model exhibits larger outliers and a wider interquartile range. Notably, while the Simoes model results for some electrolytes are correlations (based on fitted parameters), Bromley-MCM provides true predictions for unseen systems.
Chemical Scope: Bromley-MCM successfully predicts activities for a broader chemical space than the Simoes model, including electrolytes with organic anions (e.g., tosylate, acetate), ions with charges greater than one (e.g., $Ca^{2+}$ , $Lu^{3+}$ ), and ionic complexes (e.g., $[Co(en)_3]^{3+}$ ).
Limitations: The model shows slightly higher errors for $\ln \tilde{\gamma}_{\pm}$ compared to $\phi$ , potentially due to inconsistencies in the experimental $\tilde{\gamma}_{\pm}$ data derived via Gibbs-Duhem integration. Additionally, predictions for electrolytes containing ions with multiple protonation states (e.g., phosphates) are less accurate, reflecting the inherent limitations of the underlying Bromley model for such complex acid-base behaviors.

Significance and Claims
The paper claims that the Bromley-MCM approach effectively addresses the limitation of classical electrolyte models by enabling the prediction of activities for unstudied systems without sacrificing thermodynamic consistency. By leveraging the sparsity of the parameter matrix and the physical constraints of the Bromley model, the hybrid approach "closes gaps" in established physical models. The authors assert that this method substantially extends the applicability of the Bromley model while maintaining high predictive accuracy, offering a robust tool for engineering practice where experimental data is unavailable. The work does not claim to solve the modeling of weak electrolytes or systems at temperatures other than 298 K, noting these as areas for future extension.

Technical Summary: Predicting Activities in Aqueous Electrolyte Solutions with Hybrid Machine Learning

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