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Page 2 of 15 Wu et al. J. Mater. Inf. 2025, 5, 14 https://dx.doi.org/10.20517/jmi.2024.77

INTRODUCTION
In the realms of surface science and catalysis, the interactions between surfaces and adsorbates are
[1-6]
foundational for understanding catalytic processes and crucial for catalyst design and discovery . With the
rapid advancement of computational and theoretical chemistry, particularly the extensive application of
density functional theory (DFT), researchers can now calculate the adsorption energy of adsorbates on
surfaces with unprecedented accuracy, playing an indispensable role in catalyzing the design and innovation
process [6-12] .

The study of lateral interactions among adsorbates and their impact on catalytic activity, selectivity, and
surface stability, such as in carbon dioxide reduction reactions (CO RR) [13-17] , nitrogen oxide reduction
2
reactions (NO RR) [18-20] . and the Fischer-Tropsch synthesis [21-23] , has gained increasing attention. These
x
interactions, especially those modulated by varying adsorbate coverage, are vital for precisely controlling the
catalytic reaction process, making a deep understanding of lateral adsorbate interactions and coverage
effects crucial for optimizing catalyst design and enhancing performance. However, exhaustively calculating
coverage-dependent adsorption energies using DFT alone often proves impractical due to the combinatorial
growth of adsorption configuration spaces with coverage and site types, and the substantial computational
cost of a comprehensive analysis. Various methods have been proposed to address this, including cluster
[24]
expansion , multi-order lateral interaction models , graph theory [18,25,26] , and machine-learning
[14]
approaches . The combination of graph theory and machine learning has been regarded as one of the most
[27]
effective means to analyze coverage-dependent adsorption energies: graph theory tools for automating the
enumeration of vast adsorption configurations, and machine learning models for predicting adsorption
energies across the entire configuration space based on a limited DFT dataset [18,26] . However, graph-based
enumeration algorithms encounter significant computational bottlenecks at the stage of isomorphism
comparison of configurations, as graph isomorphism comparison is an extremely time-consuming Non-
deterministic Polynomial (NP) problem, exponentially growing with the number of atoms in adsorption
configurations . Since the differences between adsorption configurations mainly depend on the occupation
[28]
of different adsorbate sites, symmetry-based methods often offer a more efficient evaluation of adsorption
configurations with multiple site occupations. Machine-learning approaches, especially deep learning
methods based on neural networks (NNs), have been primarily divided into two strategies for accelerating
the prediction of adsorption energies in vast configuration spaces: (1) direct prediction of stable state
adsorption energies from initial configuration guesses using NNs; and (2) acceleration of the geometry
optimization process using machine-learning force fields (MLFF) [29,30] . While NN methods are highly
efficient in prediction with negligible computational cost, their accuracy depends not only on the model
[31]
quality but also on the size of the training dataset . In contrast, MLFF methods require significantly less
DFT computational data for training and can obtain both stable configurations and more accurate
adsorption energy predictions. However, the optimization time needed for MLFF methods far exceeds the
prediction time of NN methods, though it still represents a substantial saving in computational cost
compared to direct DFT calculations. Each strategy has its strengths, and they contribute from different
perspectives to the exploration of coverage-dependent adsorption energies. Yet, all these methods can only
consider the optimization of a limited number of configurations, facing insurmountable computational
burdens when confronted with potentially hundreds of thousands or millions of configurations.

In this work, we have developed a “structure enumeration + MLFF + NNs” approach for efficiently
exploring adsorbate-adsorbate interactions, enabling the rapid exploration of nearly ten million
configurations on various copper facets with different CO adsorption coverage. By rapidly enumerating
*
and deduplicating adsorption configurations through a “geometry + graph theory + symmetry” approach,
we generated an approximately 7 million configuration guess space. Then, using a designated sampling