# Local environment-based machine learning for molecular adsorption energy
prediction

Yifan Li  Department of Mechanical Engineering, National University of
Singapore, Singapore 117542, Singapore  Yihan Wu  Department of Mechanical
Engineering, National University of Singapore, Singapore 117542, Singapore
Yuhang Han  Department of Mechanical Engineering, National University of
Singapore, Singapore 117542, Singapore  Qiujie Lyu  Department of Mechanical
Engineering, National University of Singapore, Singapore 117542, Singapore
Hao Wu  Department of Mechanical Engineering, National University of
Singapore, Singapore 117542, Singapore  Xiuying Zhang  Department of
Mechanical Engineering, National University of Singapore, Singapore 117542,
Singapore  Lei Shen<EMAIL_ADDRESS>Department of Mechanical Engineering,
National University of Singapore, Singapore 117542, Singapore

###### Abstract

Most machine learning (ML) models in Materials Science are developed by global
geometric features, often falling short in describing localized
characteristics, like molecular adsorption on materials. In this study, we
introduce a local environment framework that extracts local features from
crystal structures to portray the environment surrounding specific adsorption
sites. Upon OC20 database ( 20,000 3D entries), we apply our local environment
framework on several ML models, such as random forest, convolutional neural
network, and graph neural network. It is found that our framework achieves
remarkable prediction accuracy in predicting molecular adsorption energy,
significantly outperforming other examined global-environment-based models.
Moreover, the employment of this framework reduces data requirements and
augments computational speed, specifically for deep learning algorithms.
Finally, we directly apply our Local Environment ResNet (LERN) on a small
2DMatPedia database ( 2,000 2D entries), which also achieves highly accurate
prediction, demonstrating the model transferability and remarkable data
efficiency. Overall, the prediction accuracy, data-utilization efficiency, and
transferability of our local-environment-based ML framework hold a promising
high applicability across a broad molecular adsorption field, such as
catalysis and sensor technologies.

##  I Introduction

Machine learning (ML) has found widespread applications in the field of
Materials Science and Engineering[1, 2]. ML techniques, such as SchNet[3],
CGCNN[4], and Alignn[5], have been employed to establish relationships between
atomic structures and their properties. These methods have been utilized to
predict more than fifty different characteristics of crystals and molecular
materials, including formation energy and electronic band gaps. Moreover,
atomic neural networks have been instrumental in developing interatomic
potentials[6] or Hamiltonian[7] instead of direct predictions of physical
properties. Additionally, deep learning techniques have been utilized in
various applications to identify chemically feasible spaces. For instance,
Bayesian optimization methods, in conjunction with MEGNet, have been employed
as energy evaluators for direct structure relaxation[8]. To further enhance
the performance, BOWSR incorporates band symmetry relaxation alongside
Bayesian optimization[9].

Accurate characterization of localized physicochemical properties is of
paramount importance in numerous scientific and engineering disciplines.
Ranging from electrochemical catalysis, sensors, carbon capture, energy
storage and conversion, to drug delivery, exploring and exploiting the
intricacies of local environments are at the heart of many frontier
investigations. For instance, adsorption is a pervasive surface phenomenon in
areas like electrocatalysis, with its understanding rooted in foundational
theories such as bonding and adsorption thermodynamics[10]. Crucially, the
influence of neighboring atoms on adsorption sites must be fully accounted
for, as factors including atomic electronic structures, spatial constraints,
surface stoichiometry, and surface defects can all impinge on the behavior of
adsorbates on surfaces[11].

For such localized concerns, earlier studies proposed numerous feature
engineering descriptors to enhance the prediction of ML models for adsorption
energy, encompassing atomic number, ionization energy, electronegativity,
ionic radius, and inter-atomic interactions[12, 13, 14]. These descriptors are
equally pertinent to the field of electrocatalysis. Yet, adequately
representing metal compound adsorption sites remains a major challenge. Tran
and Ulissi introduced the use of geometric fingerprints to depict the local
region around each atom, which were restricted to rudimentary elemental
properties[15]. Recently, a variety of end-to-end graph neural network models
have been developed in the OCP challenge, encompassing equivariant e3nn
networks, SCN, and eSCN, among others[16, 17, 18, 19, 20, 21, 22, 23, 24, 25,
7]. While these models showcase stellar performance on the data-rich OC20
database[26], their computational complexity often leads to overfitting on
smaller datasets. Specifically, for certain complex adsorption systems like
large organic molecules and certain transition metal oxides, accurate density
functional theory (DFT) computations are challenging, resulting in a paucity
of reliable data[27, 28]. The Adsorbate Chemical Environment-based Graph
Convolution Neural Network (ACE-GCN) endeavors to capture the local
information of adsorbates using molecular graphs, introducing incremental
improvements to address this issue. Additionally, employing ML approaches
boasts advantages such as heightened interpretability, reduced data
dependency, flexibility in designing and selecting features tailored to
specific problems, and capabilities in prediction interpretation and error
analysis [29, 30, 31, 32, 33, 34]. Very recently, feature engineering based on
central environments has demonstrated efficacy in describing local
environments[35], but facing generality and transferability issues and proving
not universally applicable to surfaces[36].

In light of the above backdrop, this work embarks on exploring a novel local
environment-based approach for adsorption-energy prediction. We not only
explore deep into the detailed description of the local chemical environment,
but also strive to overcome the limitations of existing methodologies. The
method we introduce marries the robust performance of deep learning with the
flexibility and interpretability of machine learning, aiming to accelerate the
training process and enhance predictive accuracy across multiple domains.
Ultimately, this work paves the way for new possibilities in understanding and
manipulating the complexity of local environments.

Figure 1: The general framework of the local environment model. The method
provides a descriptor-driven solution for adsorption calculations, especially
for large surfaces, without the need for extensive structure-driven training
resources. Given the generalizability of the descriptor extraction method,
trained models are expected to be able to predict the sorption energy of other
sorbents on a variety of complex surfaces. The method is also highly
interpretable and retains the advantages of traditional machine learning
algorithms.

##  II Methods

The concept of local environment, for machine learning-based prediction of
adsorption, is primarily rooted in the feature extraction of the surrounding
environment of the adsorbate. We first locate the position of the adsorbate
within the three-dimensional structure, and then sequentially extract the
atoms neighboring the adsorbate at each layer (Fig. 1 Step 1). Through feature
processing of the surrounding atoms at each layer, we obtain descriptors that
describe the local environment (Fig. 1 Step 2). This approach allows us to
capture the essential information about the adsorbate’s immediate surroundings
and characterize its local environment effectively. Figure 1 (Step 1) shows an
optimized Voronoi Tessellation (VT) technique, which is used in this work, to
the surface adsorption system (see more details of the optimized VT in SI).
The optimized VT method offers several advantages, including freedom from
parameter tuning, symmetry and composition independence, transferability, and
reproducibility.

After transforming the original material structure into a Voronoi diagram, we
extract improved local environment fingerprint information. The new features,
different from conventional fingerprint methods, are highlighted in red in
Fig. 1 (Step 2). Our final results prove that only six descriptors make the
conventional ML models reach a significantly high accuracy (Table 1). The
feature details and comparison with the conventional VT model can be found in
Supporting Information.

Besides for conventional ML algorithms, our local environment features can
also be transformed into a 6*11 matrix, which meets all the necessary criteria
for neural network utilization. By doing so, we can leverage the advantages of
deep learning in the context of adsorption energy calculations. In this work,
we use the ResNet-34 model as our backbone neural network (Fig. 1 Step 3).
ResNet-34 consists of 33 convolutional layers, providing depth and capability
to the model, enabling it to be applied to larger databases, thereby enhancing
the model’s accuracy and robustness. Overall, by constructing a local
environment convolutional network, we integrate the high-importance features
of the three nearest neighbor atoms surrounding the adsorbate and learn the
relationships between each layer’s elements. This local environment framework
is general for both conventional ML algorithms and deep learning neural
networks, for both large datasets and small datasets, and for both 3D and 2D
materials. Performance metrics for all methods used in this article are Median
Absolute Error (MDAE), Mean Absolute Error (MAE), Root Mean Squared Error
(RMSE), Mean Absolute Relative Percent Difference (MARPD). Our local
environment-based models show significant accuracy and efficiency compared
with conventional global environment ML models.

##  III Results

###  III.1 Datasets

This work utilize two computational material databases for training,
comparison, validation and generalization, a big-data 3D material-based
adsorption energy dataset (from OC20) and a small 2D material-based adsorption
energy dataset (from 2DMatPedia). The 3D material datasets comprise a
remarkable 47,279 DFT-calculated adsorption energy values. These calculations
were conducted using the Generalized Adsorption Simulator for Python[37][15].
The dataset includes 21,269 adsorption energies concerning hydrogen atoms,
which are the central dataset of this study. Additionally, there are 26,010
adsorption energies pertaining to other atoms. This extensive dataset covers a
wide range of 52 chemical elements and 1,952 bulk materials, thereby enhancing
its relevance and applicability. Furthermore, it is enriched with 9,102
symmetrically distinct surfaces and 29,843 distinct coordination environments,
all carefully characterized based on the surface and the adsorbate
neighbors.The 2D material database contains a substantial collection of 2,472
DFT calculations for hydrogen-atom adsorption energy[38]. These calculations
are performed on surfaces obtained from our developed 2Dmatpedia database[39],
which currently encompasses over 10,000 distinct 2D materials. The utilization
of these comprehensive and diverse databases ensures that our study’s findings
are both robust and pertinent, paving the way for significant contributions to
the field. The data distribution is largely normal and is therefore deemed
suitable for machine learning methods. The training set comprises 80% of the
dataset, while the test and validation sets account for 10% each.

###  III.2 Local Environment-based Conventional ML

Though the low accuracy compared to DL, conventional machine learning
algorithms can output weights for descriptors and have high interpretability.
Thus, they are often used to evaluate the accuracy of the improved
descriptors[40]. Here, we first applied our local environment features on
three widely-used ML algorithms, including in Gradient Boosting (GB)[41],
K-Nearest Neighbors (KNN)[42], and Random Forest (RF) algorithms[43],
following data filtering. We employed the RF algorithm initially to determine
the feature importance in the training process, refer to Appendix A.3 for
specific results upon further analysis.

Tablereftab:ML summarises the training results. The random forest model
incorporating local environment descriptors produces the lowest MAE value of
0.13 eV. The RF algorithm outperforms the other two machine learning methods
thanks to its ability to integrate decision trees and capture complex inter-
atomic relationships. In addition, the same ML algorithms equipped with our
local environment framework outperform those with the Voronoi Tessellation
(VT) method due to its inability to incorporate information about adsorbed
atoms into the model. As a result, the local environment descriptor methods
showed reliable performance in predicting the adsorption energy processes with
significantly lower error rates.

Table 1: Performance comparison of machine learning models using different descriptors | Modeling
---
method
| Learning
---
method
MDAE | MAE | RMSE | MARPD
Voronoi Tessellations | GB | 0.20 | 0.28 | 0.42 | 108%
KNN | 0.20 | 0.29 | 0.44 | 105%
RF | 0.19 | 0.27 | 0.41 | 103%
Improved Fingerprint | GB | 0.16 | 0.20 | 0.27 | 89%
KNN | 0.11 | 0.17 | 0.25 | 72%
RF | 0.08 | 0.13 | 0.22 | 61%

###  III.3 Local Environment-based CNN

####  III.3.1 Molecular CGCNN

Besides the accuracy, we next will discuss the data efficiency of our local
environment framework. Deep learning algorithms, especially convolutional
neural network (CNN), have shown their great advances and applications in
materials science. One of the famous DL methods in materials science, Crystal
Graph Convolutional Neural Network (CGCNN) deep learning framework, has
demonstrated remarkable performance in several applications. In this study, we
propose the use of our modified VT[44] method to optimize the original
structure of CGCNN into a Voronoi structure input, which we name the Molecular
Crystal Graph Convolutional Neural Network (MolCGCNN) as shown in Fig.2(a).
This approach shows that our modified VT method gives a significant develop in
the field of small data limit.

(a)

(b)

Figure 2: (a) Illustration of MolCGCNN crystal graph. (b)The heatmaps of two
models of MAE(eV) corresponding to different training sizes and Epochs.

Our results, as depicted in Fig.2(b), show that MolCGCNN achieves superior
convergence performance compared to the original CGCNN. The proposed model
efficiently captures the necessary information for calculating the adsorption
energy with fewer iterations. We attribute this to the enhanced filtering of
features by extracting local information, thus facilitating the identification
of information localized to the adsorption site. The faster training speed of
MolCGCNN is particularly relevant for large datasets, given that deep learning
algorithms typically require long training cycles due to the high number of
hidden layers in neural networks.In Appendix A.2, we present a detailed
account of our findings and provide supporting evidence for our
claims.However, there are some inherent limitations of crystal graph networks
in adsorption energy calculations, which we discuss in the Methods section,
and therefore we consider the application of feature engineering in this area.

####  III.3.2 Local Environment ResNet

Finally, to achieve both accuracy and efficiency, we utilize a convolutional
neural network with a 3$\times$3 convolutional kernel to further process this
graph, mapping the matrix to predicted parameters, specifically adsorption
energy. To enable the use of deeper networks, we introduce a residual network
(ResNet)[45] into our local environment model. The ResNet has excellent
tunability and fast training speed, allowing for greater versatility in our
model’s application to a wider range of adsorbates and scenarios.
Additionally, the ResNet’s strong generalization performance facilitates easy
portability of our model to other fields. In this model, we use mean absolute
error as the loss function and Adam as the optimization algorithm to train the
network.

The detailed structure is shown in Table.2.

Table 2: The structure of Local Environment ResNet Layer Name | Output Size | 34-Layer
---|---|---
Conv1 | $6\times 11$ |  $1\times 1,32$, stride 1
Conv2_x | $6\times 11$ |  $\left[\begin{matrix}3\times 3,32\\\ 3\times 3,32\end{matrix}\right]\times 3$, stride 1
Conv3_x | $3\times 6$ |  $\left[\begin{matrix}3\times 3,64\\\ 3\times 3,64\end{matrix}\right]\times 4$, stride 2
Conv4_x | $2\times 3$ |  $\left[\begin{matrix}3\times 3,128\\\ 3\times 3,128\end{matrix}\right]\times 6$, stride 2
Conv5_x | $1\times 2$ |  $\left[\begin{matrix}3\times 3,256\\\ 3\times 3,256\end{matrix}\right]\times 3$, stride 2
$1\times 1$ | AdaptiveAvgPool

The prediction results of the LERN model are shown in Fig.3. The Residual plot
is beneficial for analyzing the distribution of prediction errors. In the
scatter plot on the left, the x-axis and y-axis represent the LERN training
process and the residuals between predicted and actual values on both the
training and test sets, respectively. The overall residual regression line is
also calculated and displayed in the plot. The red and blue parts represent
the training and test sets, respectively. It can be observed that the LERN
prediction results are highly consistent with the actual values on both the
training and test sets, as evidenced by the residuals’ regression lines with
slopes close to zero. On the right, the overall distribution of residuals on
the training and test sets is displayed. The results show that the
distribution of residuals on both the training and test sets is close to
normal distribution, indicating that the model is well-suited for learning
from this type of data. Moreover, the distributions of residuals on the
training and test sets are very similar, indicating that the model does not
suffer from obvious overfitting. Therefore, the LERN prediction results are
highly consistent with the actual DFT calculation results within the allowable
error range, suggesting that LERN can be used to replace DFT calculations.

Figure 3: Residual plot of LERN models. The left-hand side is a scatter plot,
where the x-axis is the predicted value of the model on the training and test
sets, the y-axis is the corresponding predicted value minus the true value,
and the middle corresponds to its regression line respectively. On the right-
hand side are the residual distributions for the training and test sets,
respectively. Figure 4: (a)H adsorbed on 3D materials (b) and (c)Performance
comparison of LERN with other representative models on 3D
materials.(*op:Outlier percentage)

To benchmark the performance of our proposed model in a materials database, we
use other state-of-the-art neural networks. We compare our Local Environment
ResNet (LERN) with the original Crystal Graph Convolutional Neural Network
(CGCNN) and SchNet[46] and Atomistic Line Graph Neural Network(AliGNN)[5], as
well as our improved versions of the Molecular CGCNN. To elaborate further,
the robustness of LERN in dealing with outliers is a highly desirable trait in
machine learning models. Outliers are data points that deviate significantly
from the normal distribution of the dataset, and they can occur due to various
reasons such as measurement errors or anomalous samples. The presence of
outliers can negatively impact the performance of a model, especially if it is
not designed to handle them properly. In contrast, as shown in Fig.4, the
orange points represent the distribution of outliers for each model. Of these,
LERN has only 86 outliers, while the other models are all around 110. This
suggests that LERN has demonstrated an impressive ability to handle outliers
effectively, which is a crucial advantage in real-world applications where
data quality is often suboptimal. By being insensitive to outliers, LERN can
deliver reliable predictions even in the presence of noisy data. This is
particularly relevant in the field of materials science, where experimental
data can be scarce, noisy, or incomplete, making it challenging to develop
accurate models. Moreover, the performance of LERN on limited training data
samples is noteworthy. Discrete errors often arise from inadequate training
data or inherent low similarity in the training set, and the error
distribution of LERN is more concentrated, indicating higher prediction
accuracy and consistency. The ability to learn from a small amount of data is
an important aspect of machine learning, as it allows for the development of
models that can be trained with fewer computational resources and time. This
is particularly important in fields such as materials science, where
experiments can be both expensive and time-consuming. The ability of LERN to
effectively learn from limited data suggests that it has the potential to
significantly accelerate the discovery and design of new materials.

Figure 4 shows that LERN surpasses other models in predicting the adsorption
energy of HER. During our training, we discovered that, akin to MolCGCNN,
LERN’s accuracy remains stable even with fewer iterations and smaller sample
sizes, whereas other models show significant deterioration under low data
conditions. This finding reiterates that local environment descriptors are
more fitting for limited data scenarios, reflecting the current state of most
catalytic databases. Such high data efficiency can be attributed to the
model’s ability to extract vital local distance information and atomic
properties from the structure, embedding system knowledge and input-output
correlations. LERN is able to rapidly concentrate on the features surrounding
the adsorption site using the training dataset, whereas other models must
laboriously learn all atomic correlations without the benefit of system
knowledge.

Figure 5: (a)H adsorbed on 2D materials (b) and (c)Performance comparison of
LERN with other representative models on 2D materials.(*op:Outlier percentage)

The LERN model still shows stable performance superiority on two-dimensional
materials as showed in Figure 5. This is due to the fact that the feature
engineering of the model only focuses on local information, independent of the
material scale and thickness. The model also performs well on small datasets
due to the introduction of a deep residual structure and the sharing of
parameters. This also leads to the training time of LERN is faster than all
other neural networks by an order of magnitude. It is noteworthy that all the
aforementioned models were trained on a single RXT3080. When iterating 200
times, with the exception of LERN, the training time for all models exceeded
two hours, while LERN required only approximately 26 minutes. This is
considerably faster than other conventional graph neural networks while the
accuracy is greatly optimized.

##  IV Discussion

In this study, we have pioneered a cutting-edge local environment description
method to refine the feature input of machine learning models. Leveraging an
enhanced Voronoi Tessellation, we successfully extract geometric
characteristics of 3D structures. This is further complemented by an
innovative fingerprinting method that imparts localized information of
adsorption sites to the model. This proposed descriptor has been applied to
predict the adsorption energy of hydrogen atoms in catalytic processes, and
our findings underscore that the local environment description method
substantially outclasses prior models in prediction accuracy.

We have also broadened the scope of the proposed method by adapting it to deep
learning algorithms, specifically by incorporating the MolCGCNN and LERN
models. The experimental results affirm that our enhanced model offers
superior prediction accuracy for smaller datasets and demands fewer
computational resources. This underscores the efficacy of the local
environment description approach. A notable innovation in our work is the
employment of descriptors as graph inputs for the convolutional neural
network, leading to a local environment residual network that markedly
improves the final prediction accuracy by approximately 0.12 eV.

The marked advancement in prediction accuracy in our approach can be traced to
the meticulous attention to and extraction of the local information of
adsorption sites. By generating and predicting the adsorption energies of
1,283 adsorption sites for hydrogen atoms, based on the 2Dmatpedia database,
we were able to identify 272 predicted adsorption energies within the range of
plus and minus 0.5 eV after screening. Among these, seven have been previously
reported in experiments, 69 in other computational studies, and 196 remain
unprecedented in public literature or open databases. A significant highlight
of our work is its superior speed compared to other models, made possible by
extracting only the environment around the adsorbed atoms for training, thus
notably enhancing screening efficiency. We are confident that this methodology
can be expanded to a more diverse array of catalytic material discovery,
fortifying the capability of machine learning models to predict the adsorption
energy of various catalytic materials.

##  V Methods

###  V.1 Local environment fingerprint descriptors

Using crystal graph to construct network frameworks has certain limitations.
Firstly, the crystal graph structure only utilizes physical information such
as atomic distances and angles, which makes it difficult to reflect the
chemical properties between atoms. Additionally, the computational complexity
of crystal graph networks is extremely high, with state-of-the-art models
requiring single GPU computation times of up to hundreds of days. On the other
hand, the arrangement and properties of atoms in the vicinity of the
adsorption point are crucial factors influencing the energy of the hydrogen
adsorption process[47]. Therefore, we propose to use faster machine learning
algorithms to construct more effective descriptors for adsorption energy
prediction, not only greatly increasing training speed but also achieving
superior performance and accuracy. By simplifying the input and utilizing the
powerful performance of neural networks, we can further improve the accuracy
of adsorption energy prediction.

We have proposed a comprehensive framework for predicting adsorption energies,
which involves several key steps, as shown in Fig.1. Firstly, we transform the
original material structure into a Voronoi diagram, and extract improved local
environment fingerprint information. Next, we further optimize this descriptor
using machine learning techniques. Finally, we utilize the powerful
performance of deep learning to construct, train, and predict using the model.
This approach enables us to capture important physical features of the
adsorption process and achieve high prediction accuracy.

We first improved the Crystallography Neural Network, but in practice we found
that the construction of the Fingerprint draws on many previously successful
methods for modelling chemical crystals. The atomic radius is a feature that
better describes in vitro steric effects [14]. However, atomic radii may also
change due to changes in the environment. In the present work we use the
atomic number instead of this feature as it is simple and
deterministic.Pauling electronegativity has been shown to be a good feature of
electron affinity[48]. To account for steric and ambient electron effects, the
coordination number has been shown to be a successful feature[49]. Crude
estimates of the properties have proved successful and can improve predictive
power, so we use the average adsorption energy as a description.

For the neighboring elements at each adsorption site, we utilized four
elemental properties as follows: the atomic number of the element (Z), the
Pauling electronegativity of the element ($\chi$), the number of neighboring
atoms to which the element is coordinated to the adsorbate (CN), and the
average adsorption energy of the element ($\Delta E$). The $\chi$ value was
obtained from the Mendeleev database [29], and the $\Delta E$ value was
calculated from the adsorption energy database, which represents the average
of the adsorption energies of all catalysts containing this element.
Additionally, we included the atom distance (D) to the adsorbate H, a
parameter that is directly related to the adsorption energy magnitude in
adsorption. Finally, we added the valence number (VN), which is calculated as
the average of the elements within all the layers.

In addition, we note that the features generated by the general method all use
the missing value filling method, where the individual vectors are the same
for different elements and therefore do not reflect the information for each
case. We therefore processed this by modifying the values of atomic number and
Bowling electronegativity for each layer to the average of the layers. In this
way, the fill values are different for each case. The relevant chemical
properties of the cases can be better represented.

Thus, for the improved Fingerprint method, the two layers contain up to 7
elements, each corresponding to six parameters. Thus, a maximum of 42 feature
inputs can be generated after modelling each case. In addition, due to the
presence of edge adsorption, we consider three layers of nearest neighbours,
and experimentally, in surface adsorption, this method is able to adequately
contain the information needed to calculate the adsorption energy. In
addition, we use tools such as matminer to enrich the selection space of our
features, which are finally filtered and validated using machine learning
algorithms. The final result proves that we only need six descriptors to make
the model reach the highest accuracy.

###  V.2 Local environment descriptors for convolutional networks

In lateset adsorption energy calculations, graph neural networks are typically
utilized. Nevertheless, our local environment features can also be transformed
into a 6*11 matrix, which meets all the necessary criteria for neural network
utilization. As a result, we suggest a creative approach to feeding local
environment descriptors into a convolutional neural network in matrix form. By
doing so, we can leverage the advantages of deep learning in the context of
adsorption energy calculations.

In this model, for each element E in a layer, there are six features f
associated with it, which can be represented as:

$E=[f_{1},f_{2},f_{3},f_{4},f_{5},f_{6}]$ (1)

where each f corresponds to a descriptor mentioned earlier.

For a local environment with three layers and eleven elements, the input X can
be represented as:

$L=[E_{i}|i\in(1,11)]^{T}$ (2)

Here, L is a 6x11 matrix that serves as the input to a convolutional neural
network, denoted as O(L), consisting of 32 residual blocks. Each block
comprises two convolutional layers and a skip connection, which is used to
construct the layers of ResNet. ResNet consists of multiple residual blocks,
with four stages, each comprising 3, 4, 6, and 3 residual blocks,
respectively, followed by a skip connection that adds the input X to the
output of the second convolutional layer. The output of the n-th residual
block can be written as:

$Z_{n}=L+F_{n}(O_{n-1}(L))$ (3)

where $F_{n}$ represents the residual function of the n-th block, and
$O_{n-1}(L)$ is the output of the (n-1)-th block. Finally, global average
pooling and a fully connected layer are used to transform the feature map of
the last layer into a scalar output:

$O(L)=W_{out}Z_{n}+b_{out}$ (4)

where $W_{out}$ and $b_{out}$ are the weights and biases of the output layer,
respectively.

The residual function $F_{n}$ can be defined as:

$F_{n}(L)=\sigma(W_{2,n}\delta(W_{1,n}X+b_{1,n})+b_{2,n})$ (5)

where $W_{1,n}$ and $W_{2,n}$ are the weights of two convolutional layers,
$b_{1,n}$ and $b_{2,n}$ are their biases, $\delta$ represents the convolution
operation, and $\sigma$ is the ReLU activation function.

Our framework possesses two key features to ensure the robustness and high
performance of the model in adsorption tasks. Firstly, we adopt the mean
absolute error as the loss function and train the network with the Adam
optimization algorithm, ensuring the robustness and adaptability of the model.
This choice not only adapts to adsorption tasks with different complex
structures but also ensures that the model performs exceptionally well in
various data scenarios. Secondly, we use the ResNet-34 model as our backbone
network. ResNet-34 consists of 33 convolutional layers, providing depth and
capability to the model, enabling it to be applied to larger databases,
thereby enhancing the model’s accuracy and robustness. ResNet-34 is a widely
used deep learning framework, whose outstanding performance has been proven in
numerous fields. Our choice also provides robust support for adsorption tasks.
Overall, by constructing a local environment convolutional network, we
integrate the high-importance features of the three nearest neighbor atoms
surrounding the adsorbate and learn the relationships between each layer’s
elements. Then, we learn from a large dataset’s data based on the
convolutional network. Introducing residual networks reduces the model’s
learning difficulty and enhances its generalizability.

###  V.3 Regression result evaluation

To train the model, several regression outcome evaluation methods are
employed. The true and predicted values in the experiment are represented as
follows:

True value:

$\hat{y}=\left\\{\hat{y}_{1}\,{\hat{y}}_{2}\cdots,\hat{y}_{n}\right\\}$ (6)

Predicted value:

$y=\left\\{y_{1},y_{2},\cdots,y_{n}\right\\}$ (7)

Performance metrics for all methods used in this article, which include:
Median Absolute Error (MDAE), Mean Absolute Error (MAE), Root Mean Squared
Error (RMSE), Mean Absolute Relative Percent Difference (MARPD).

The median absolute error is particularly interesting because it is robust to
outliers, unit of this indicator is eV . The median absolute error estimated
over n samples is defined as follows:

${\rm MDAE}(y,\hat{y})=median(|y_{1}-\hat{y}_{1}|,\cdots|y_{n}-\hat{y}_{n}|)$
(8)

Mean Absolute Relative Percent Difference (MARPD) is used because it provides
normalized measures of accuracy that may be more interpretable for those
unfamiliar with adsorption energy measurements in eV .

${\rm
MARPD}=\frac{1}{n}\sum_{i=1}^{n}\left|\frac{y_{i}-\hat{y}_{i}}{|y_{i}|+|\hat{y}_{i}|}\cdot
100\%\right|$ (9)

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boundary value problem $Y^{\prime\prime}-\tfrac{1}{2}\eta Y^{\prime}-Y=0$,
$Y(0)=0$, $Y(\infty)=1$ which does not possess a bounded solution.

One does obtain the correct boundary value problem if one realizes that from
the perspective of someone moving with the accelerating gas flow, the
laboratory frame of reference is a non-inertial frame of reference where there
is an inertial acceleration corresponding to $a_{i}=dU_{g}/dt=U_{g}/\tau$,
that is, Eq. (104) needs to be replaced by

$\frac{\partial u}{\partial t}=\nu\frac{\partial^{2}u}{\partial
z^{2}}+\frac{U_{g}}{\tau}\,.$ (118)

Starting from the above equation in the derivation leads to the boundary value
problem (117).

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