🔬 materials science

Efficient Grand Canonical Global Optimization with On-the-fly-trained Machine-learning Interatomic Potentials

This paper presents an efficient grand canonical global optimization algorithm that utilizes on-the-fly trained machine-learning interatomic potentials and *ab initio* thermodynamics to predict stable structures and chemical states of nanostructured materials under reactive environments while significantly reducing the computational cost of first-principles calculations.

Original authors: Jon Eunan Quinlivan Dominguez, Mads-Peter Verner Christiansen, Konstantin M. Neyman, Bøjrk Hammer, Albert Bruix

Published 2026-06-30

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Jon Eunan Quinlivan Dominguez, Mads-Peter Verner Christiansen, Konstantin M. Neyman, Bøjrk Hammer, Albert Bruix

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 find the perfect outfit for a party, but the "perfect" outfit depends entirely on the weather outside. If it's sunny, you want shorts; if it's raining, you need a coat. But here's the catch: you don't know exactly what the weather will be, and you have millions of possible combinations of clothes (shirts, pants, hats, shoes) to try on.

In the world of chemistry, scientists face a similar problem when studying nanomaterials (tiny materials used in things like car exhaust filters or fuel cells). They need to figure out the most stable shape and chemical makeup of these tiny particles under specific conditions, like high heat or high pressure.

This paper introduces a new, super-smart computer program designed to solve this "outfit search" problem much faster than before. Here is how it works, broken down into simple concepts:

1. The Problem: A Needle in a Haystack

Traditionally, to find the best structure, scientists had to check every single possible combination of atoms one by one.

The Old Way: Imagine trying to find the best outfit by testing every single shirt, then every single pair of pants, then every single hat, one by one. If you want to see how the outfit changes if you add a scarf (an extra atom), you have to start the whole process over again. This takes a massive amount of computer power and time.
The Challenge: The number of possibilities is so huge that it's like trying to find a specific grain of sand on a beach while blindfolded.

2. The Solution: A "Smart Guessing" Assistant

The authors created a new algorithm called Grand Canonical Global Optimization (GCGO). Think of this algorithm as a very clever assistant who doesn't try every single outfit. Instead, they use a "smart guesser" (a Machine Learning model) to predict which outfits look good without actually putting them on.

The "Smart Guesser" (Machine Learning): This is a computer program trained on a few examples. Once it sees a few outfits, it gets really good at guessing how comfortable or stable a new outfit would be without needing to test it physically.
Learning on the Fly: The best part is that this assistant learns while it works. Every time the computer actually tests a promising outfit (using expensive, high-precision calculations), the assistant learns from it and gets even better at guessing the next one.

3. The "Grand Canonical" Twist: Changing the Rules

Most old methods forced the computer to pick a specific number of atoms (a fixed "outfit size") and then search for the best shape. If you wanted to see what happened if you added or removed an atom, you had to run a completely new search.

The new GCGO method is different. It allows the "outfit" to change size during the search.

The Metaphor: Imagine you are searching for the best outfit, but the weather is changing. The old method would say, "Okay, let's find the best outfit for a 70-degree day," and then stop. If the weather drops to 50 degrees, you have to start over.
The New Method: The GCGO algorithm says, "Let's search for the best outfit right now, and if the weather changes, we'll instantly add a jacket or take off a hat and keep searching." It searches for the best shape and the best number of atoms simultaneously.

4. How They Tested It

The team tested this new "smart assistant" on three different scenarios to prove it works:

Floating Clusters: Tiny clusters of Iridium and Oxygen floating in space.
Supported Clusters: Tiny Platinum clusters sitting on a Cerium Oxide surface (like a car catalyst).
Surface Layers: A flat sheet of Palladium metal reacting with oxygen.

In all three cases, the new algorithm found the most stable structures faster than the old methods. It was able to find the "global minimum" (the absolute best, most stable state) while also discovering other stable states nearby, all in a single run.

5. The Result: Faster and Smarter

The paper claims that this new approach is:

More Efficient: It needs far fewer expensive computer calculations to find the answer.
More Flexible: It can handle changing conditions (like temperature and pressure) and changing chemical compositions (adding or removing atoms) in one go.
Accurate: It successfully reproduced results from previous studies and even found better (more stable) structures than were known before in some cases.

In short: The authors built a computer program that acts like a seasoned fashion stylist who learns as they go. Instead of trying on every single outfit in the world, it uses smart predictions to quickly zero in on the perfect look for any weather condition, saving a tremendous amount of time and effort.

Technical Summary: Efficient Grand Canonical Global Optimization with On-the-fly-trained Machine-learning Interatomic Potentials

Problem Statement
The characterization of nanostructured materials under reactive environments is hindered by the complexity of structural motifs and their chemical transformations. While global optimization (GO) methods can predict stable structures, addressing both configurational and compositional search spaces simultaneously is computationally prohibitive when relying on first-principles calculations. Traditional approaches typically optimize a single stoichiometry at a time, requiring separate searches for every possible chemical state (e.g., varying oxygen content). This scaling is inefficient, as the number of required stoichiometries grows exponentially with the number of constituent elements. Furthermore, many evaluated stoichiometries are unstable and irrelevant to the target reaction conditions, wasting computational resources.

Methodology
The authors implement a Grand Canonical Global Optimization (GCGO) algorithm within the Atomistic Global Optimization X (AGOX) Python library. This approach integrates on-the-fly trained Machine Learning Interatomic Potentials (MLIPs) with an ab initio thermodynamics framework to simultaneously explore structural and compositional spaces.

Key methodological components include:

Surrogate Model: The algorithm utilizes a Sparse Gaussian Process Regression (GPR) model based on the Smooth Overlap of Atomic Positions (SOAP) descriptor. This local GPR model is trained on-the-fly during the search. It is "size-extensive," meaning it can predict energies for structures with varying numbers of atoms (different stoichiometries) by decomposing total energy into atomic contributions.
Thermodynamic Framework: Instead of optimizing potential energy alone, the algorithm evaluates the Gibbs energy of formation ( $\Delta G_c$ ) for each candidate. This is calculated using the ab initio thermodynamics approximation:
$\Delta G_c = E_c - E_r - \sum_{v} \Delta n_v (E_v + \Delta \mu_v)$
where $E$ represents potential energies at 0 K, $\Delta n_v$ is the change in atom count, and $\Delta \mu_v$ accounts for temperature and pressure-dependent chemical potentials of reservoirs (e.g., $O_2$ gas). Thermal contributions to the candidate structures are neglected to maintain computational efficiency, a simplification validated by the authors to yield qualitatively correct phase diagrams.
Active Learning Loop: The algorithm follows an iterative cycle:
1. Generation: New candidates are generated via randomization, lattice rattling, or by adding/removing atoms (Addition/Removal Generator) from seed structures.
2. Selection: A Sampler module selects seeds based on Gibbs energy of formation. Two specific samplers are introduced: a standard Gibbs energy sampler and a K-Means Gibbs energy sampler that clusters candidates by structural similarity (SOAP) to ensure diverse exploration.
3. Relaxation & Evaluation: New candidates are relaxed using the MLIP. An Acquisitor module selects candidates for high-level ab initio evaluation based on an acquisition function: $F = \Delta G^M_c - \kappa\sigma$ . This function balances the predicted stability ( $\Delta G^M_c$ ) and the model's uncertainty ( $\sigma$ ), favoring candidates that are predicted to be stable but structurally distinct from the training set.
4. Training: Selected candidates are evaluated at the target level of theory (DFT or a high-fidelity MLIP) and added to the training set to update the GPR model.

Key Contributions

Algorithmic Implementation: The development of a GCGO algorithm capable of handling multiple stoichiometries simultaneously, overcoming the limitations of fixed-stoichiometry GO methods.
Efficiency via On-the-fly Training: The demonstration that GPR-based MLIPs, trained dynamically during the search, significantly reduce the number of expensive first-principles evaluations required to find global minima compared to traditional methods.
Size-Extensive Modeling: The successful application of SOAP-GPR models to systems with varying compositions, enabling the direct comparison of Gibbs energies across different chemical states without retraining separate models for each stoichiometry.
Uncertainty-Guided Exploration: The integration of GPR-derived uncertainty estimates into the acquisition function to guide the search toward unexplored regions of the potential energy surface.

Results
The algorithm was tested on three systems with varying complexity and target levels of theory:

Free-standing $Ir_3O_x$ clusters: Using a MACE MLIP as the target theory, the GCGO algorithm successfully identified global minima across various oxygen chemical potentials ( $\Delta \mu_O$ ). The cumulative success rates were comparable to, and in some cases exceeded, those of single-stoichiometry searches. Notably, a single GCGO run identified global minima for multiple stoichiometries (e.g., $Ir_3$ , $Ir_3O_2$ through $Ir_3O_9$ ), whereas single-stoichiometry runs would only find the minimum for their specific composition.
$Pt_3O_x$ on $CeO_2(111)$ : Using DFT as the target theory, the method reproduced and improved upon previous literature results. The GCGO runs found global minima for $Pt_3O_6$ and $Pt_4O_4$ that were 0.70 eV and 0.56 eV more stable, respectively, than those found in prior single-stoichiometry GOFEE studies. The algorithm efficiently navigated the phase diagram, identifying stable states for different $\Delta \mu_O$ values with fewer total DFT evaluations than performing independent single-stoichiometry searches.
$Pd(100)$ Surface: The algorithm successfully identified surface reconstructions, including the transition from a pristine metallic surface to oxygen-decorated states and finally to a surface-oxide phase ( $Pd_4O_4$ ) under oxidizing conditions. The resulting phase diagram agreed well with previous experimental and theoretical findings.

Significance and Claims
The paper claims that the GCGO algorithm constitutes a promising alternative to existing methods for characterizing solid catalysts under reaction conditions. Its primary significance lies in its ability to:

Identify stable structures and chemical states with higher efficiency than canonical approaches that explore stoichiometries independently.
Reveal regions of phase diagrams beyond the immediate target conditions within a single computational run.
Provide a practical framework for studying complex nanomaterials where the stable phase is unknown a priori.

The authors modestly note that while the algorithm is highly efficient, it is not a formal operando approach due to the neglect of vibrational contributions to the Gibbs energy of the candidates. However, they assert that the simplification yields reliable phase boundaries and that the method is particularly effective when a single stoichiometry is significantly more stable than others under the target conditions. The work paves the way for tackling larger, more complex nanomaterials in reactive environments by reducing the computational bottleneck of first-principles evaluations.