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\begin{document}
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\title{Supplemental Information}
\maketitle
\section{Parameters settings for Voronoi Tessellation}
We used different cut-off thresholds for molecules and crystals datasets when considering neighbor atom from Voronoi polyhedra.
The cutoff thresholds $d_t$ and $w_t$ are applied to filter out neighboring atoms $a_j$ with large Euclidean distances ($d_{ij} > d_t$) and small weights ($w_{ij} < w_t$) corresponding to atom $a_i$. We set $d_{t} = 4.0 \AA$ for QM9, Fullerence-MD, and Pt/Graphene-MD datasets and $d_{t}=6.0 \AA$ for SmFe$_{12}$-CD dataset. Atoms with Voronroi spatial weight $w_{ij} < 0.2$ are not included in the neighbor list of the central atom.

\section{Model architecture}

\subsection{Embedding layers}\label{subsec:embedding_layer}
Given a material structure $S$ of interest with $M$ atoms ($\mathcal{A}_S=\{a_1,a_2,\cdots , a_M\}$), the intrinsic atomic information is initially embedded to form a representation of $M$ atoms in the structure $S$. The atomic number $\text{Z}$ of an atom $a_i$ ($1 \le i \le M $) is considered and represented using a $v$-dimensional embedding vector $\textbf{e}_{i}$ ($\textbf{e}_{i} \in \mathbb{R}^v$). Next, a linear function $F_e:\mathbb{R}^{v}\rightarrow \mathbb{R}^{h}$ is learned to project this information to provide a better representation of atom $a_i$. Consequently, atom $a_i$ is represented by an $h$-dimensional vector $\textbf{c}^0_{i}=F_e(\textbf{e}_{i}) $. The vector $\textbf{c}^0_{i} \in \mathbb{R}^h$ is used as the initialization of local structure $\{a_i, \mathcal{N}_i\}$ for the local attention layers computation in the next steps.

%Given a material structure $S$ of interest with $M$ atoms ($\mathcal{A}_S=\{a_1,a_2,\cdots , a_M\}$), the intrinsic atomic information is initially embedded to form a representation of $M$ atoms in the structure $S$. Assuming that $m$ intrinsic attributes of atom $a_i$ ($1 \le i \le M $) are considered and that each attribute is represented using an embedding vector $\textbf{e}_i^k\in \mathbb{R}^{v_k}$ ($1 \leq k\leq m $), all $m$ embedding vectors are concatenated into a $v$-dimensional vector $(v=\sum_{k=1}^{m}v_k)$: $\textbf{e}_{i}$ = $[\textbf{e}^1_{i},\textbf{e}^2_{i},\dots,\textbf{e}^m_{i}]$, which embeds all atomic information related to the atom $a_i$. 
%
%Next, a linear function $F_e:\mathbb{R}^{v}\rightarrow \mathbb{R}^{h}$ is learned to merge the information of all the $m$ atomic attributes to provide a better representation of atom $a_i$. Consequently, atom $a_i$ is represented by an $h$-dimensional vector $\textbf{c}^0_{i}=F_e(\textbf{e}_{i}) $. The vector $\textbf{c}^0_{i} \in \mathbb{R}^h$ is used as the initialization of local structure $\{a_i, \mathcal{N}_i\}$ for the local attention layers computation in the next steps.

\subsection{Geometrical influence}\label{subsec:geometrical}
The Euclidean distance $d_{ij}$ is expanded by using $k$ Gaussian basis functions  $\phi_i(x) = \text{exp}(-(x-\mu_k)/2\sigma^2)$ located at each centers $0\textup{~\AA}<\mu_k< d_{t}\textup{~\AA}$ for every $\sigma=0.5\textup{~\AA}$\cite{chen2019graph,schutt2018schnet}. Next, the distancce  embedding layer is defined as follows.
\begin{align}
	\textbf{DE}(d_{ij}) = F_d([\phi_1(d_{ij}),\phi_2(d_{ij}),\dots,\phi_k(d_{ij})])
\end{align} 
where $F_d:\mathbb{R}^{k}\rightarrow \mathbb{R}^{h}$ is a fully-connected layer with a Swish activation function \cite{Ramachandran2017SwishAS}. Subsequently,  the geometrical influence of a neighboring atom $a_j (\in \mathcal{N}_i)$ on atom $a_i$ ($1\le j \neq i \le M$) is represented by a vector $\textbf{g}_{ij} \in \mathbb{R}^h$, which is defined from the element-wise multiplication of Euclidean distance $d_{ij}$ and the Voronoi weight\cite{pham2017machine} $w_{ij}$ between the atoms as follows.
	\begin{align}
		\textbf{g}_{ij} & = \textbf{DE}(d_{ij}) \times w_{ij} 	\\ \nonumber
		& = \textbf{DE}(d_{ij}) \times\frac{\theta_{ij}}{\max_{\{a_k\in \mathcal{N}_i\}} \theta_{ik}},
	\end{align}
where the $\theta_{ij}$ is solid angle determined by the Voronoi polyhedron face between the atomic sites of atoms $a_i$ and $a_j$. 

\subsection{Local attention layers}\label{subsec:local_attention}
The representation $\textbf{c}^{l+1}_{i}$ $\in \mathbb{R}^{h}$ of local structure $\{a_i, \mathcal{N}_i\}$ at the ${(l+1)}^{th}$ local attention layer is derived from the representation vectors in the preceding layer of itself ($\textbf{c}^{l}_{i}$), its $N$ neighboring local structures ($\textbf{C}^{l}_{\mathcal{N}_i}=[\textbf{c}^{l}_{j}]_ {a_j \in \mathcal{N}_i}$), and the geometrical influence ($\textbf{G}_{\mathcal{N}_i}=[\textbf{g}_{ij}]_{a_j \in \mathcal{N}_i}$) as follows.
\begin{align}
	\textbf{c}_i^{l+1} & = \text{LocalAttention}^{l+1}(\textbf{c}^{l}_i, \textbf{C}^{l}_{\mathcal{N}_i}\times\textbf{G}_{\mathcal{N}_i}) \\ \nonumber
	&= \text{Attention}(\textbf{q}^{l}_i, \textbf{K}^{l}_{\mathcal{N}_i}) + \textbf{q}^{l}_i  \\ \nonumber
	& = \text{softmax} ({\textbf{q}^{l}_{i}}^\top \textbf{K}^{l}_{\mathcal{N}_i}) \textbf{K}^{l}_{\mathcal{N}_i} + \textbf{q}^{l}_i \\ \nonumber
	&= \sum_{a_j\in \mathcal{N}_i} \alpha^{l}_{ij} \textbf{k}^{l}_j + \textbf{q}^{l}_i, 
\end{align}
where $\textbf{K}^{l}_{\mathcal{N}_i}=[\textbf{C}_{\mathcal{N}_i}^{l} \times \textbf{G}_{\mathcal{N}_i}] \textbf{W}^{l}_k=[\textbf{k}^l_1,\textbf{k}^l_2,...,\textbf{k}^l_N]$  and $\textbf{q}^{l}_{i}=\textbf{c}^{l}_{i} \textbf{W}^{l}_q$ ($\textbf{k}^l_j, \textbf{q}^{l}_{i} \in \mathbb{R}^{h}$) are the degree each atom $a_j$ influencing the atom $a_i$ and degree atom $a_i$ accepting the influence from the neighboring atoms, respectively. In addition, $\textbf{W}^{l}_q$ and $\textbf{W}^{l}_k$ $\in \mathbb{R}^{h \times h}$ are the trainable weight parameters. The local attention score $\alpha^{l}_{ij}$ can be interpreted as the degree of attention to which the information of the local structure centered at atom $a_{j}$ should be referred to appropriately represent the local structure centered at atom $a_i$.
\begin{align}
	\alpha^l_{ij}  = \frac{e^{s_{ij}}}{ \sum_{a_k \in \mathcal{N}_i} e^{s_{ik}}}, \quad s_{ij} = {\textbf{q}^l_i}^\top \textbf{k}^l_j.
\end{align}

The local attention layers are learned to guarantee that the neighboring atoms $a_j$ exerting more influence on the central atom $a_i$ have higher $s{ij}$ scores. Herein, a softmax function is used to normalize these scores to the interval $[0, 1]$ to obtain the attention scores $\alpha^{l}_{ij}$. The sum of the attention scores of the neighboring atoms $a_{j}$ is 1 ($\sum_{a_j \in \mathcal{N}_i} \alpha^{l}_{ij} = 1$).  Consequently, we employed standard attention mechanism \cite{Vaswani2017att} to create the representation for local structure as follows.
\begin{align}
	\textbf{c}_i^{l+1} = \text{LayerNorm}(F_n(\textbf{c}_i^{l+1})+\textbf{c}_i^{l+1}),
\end{align}
where $F_n:\mathbb{R}^{h}\rightarrow \mathbb{R}^{h}$ is a fully-connected layer with a Swish activation function and LayerNorm is the layer normalization \cite{ba2016layer}.

\subsection{Global attention layer}\label{subsec:global_attention}
After L local-attention layers, we obtain the representations of local structures in S from the last local attention layer as $\textbf{C}^{L}={[\textbf{c}^{L}_{i}]}_{1 \le i \le M}$. SCANNet again utilizes the dot-product key-query attention \cite{Vaswani2017att} to coherently learn the representation of local structures and integrate them to representation of material structure in a target-dependent manner. We define $\textbf{Q}^g=\textbf{C}^{L}\textbf{W}_{\textbf{q}}^{g}=[\textbf{q}^g_1,\textbf{q}^g_2,...,\textbf{q}^g_M]$ and $\textbf{K}^g=\textbf{C}^{L}\textbf{W}_{\textbf{k}}^{g}=[\textbf{k}^g_1,\textbf{k}^g_2,...,\textbf{k}^g_M]$ as query and key matrix, where $\textbf{W}^{g}_{\textbf{k}}, \textbf{W}^{g}_{\textbf{q}}  \in \mathbb{R}^{h \times h} $ are learnable parameters of the global-attention layer. Then, we compute the attention matrix $\textbf{A} =\textbf{Q}^g{\textbf{K}^g}^\top \in \mathbb{R}^{M\times M}$ in the same manner as previous local attention layers. 

 In essence, a local structure that receives higher total attention scores from other local structures should be prioritized when representing the material structure $S$. As a result, the degree of attention to a local structure $\{a_i, \mathcal{N}_i\}$ in $S$ is quantitatively modeled by summing all the attention received from other atoms. We show here the details for calculating the Global attention (GA) score of local structure $\{a_i, \mathcal{N}_i\}$ as the sum of the attention from all query vectors $\textbf{q}^g_j$ ($j\neq i$) to the key vector $\textbf{k}^g_i$ as follows.
	\begin{align}	 	
		\text{s}_i  = \sum_{j=1}^M [\textbf{A}(1-\textbf{I})]_{j,i} =  \sum_{j, j \neq i}^M {\textbf{q}_j^g}^\top \textbf{k}^g_{i}
		\label{eq:norm},
	\end{align}
where $I$ is denoted for the identity matrix. In practice, these scores $\text{s}_i$ are normalized by using the $\boldsymbol\ell_2$ normalization to prevent the sum of them from becoming extremely high in structures that contain a significant number of atoms, as follows.
\begin{align}
	\hat{\text{s}}_i=\frac{\text{s}_i}{|| [ \text{s}_1,\cdots,\text{s}_M ] ||_2}
\end{align}
The function $\rho(.)$ is designed based on the hypothesis that greater attention should be given to a local structure if its representation is crucial for accurately representing other local structures to interpret the structure-property relationship in $S$ effectively, as follows.
\begin{align}
	\rho(\textbf{A}) &= \bm{\alpha}^g = \text{softmax}([\hat{\text{s}}_1,\hat{\text{s}}_2,...,\hat{\text{s}}_M])
	\end{align}
 The representation vector $\textbf{x}_S$ of the material structure $S$ is then formulated by aggregating the representations of  $M$ local structures according to the obtained global attention (GA) scores, as follows.
\begin{align}
	\textbf{x}_S &= \text{GlobalAttention}(\textbf{C}^{L}) \\ \nonumber
			     &= \rho({\textbf{Q}^g}^\top\textbf{K}^g)\textbf{K}^g = \rho(\textbf{A}) \textbf{K}^g  \\ \nonumber
			     &= \bm{\alpha}^g \textbf{K}^g = \sum_{i=1}^M \alpha^g_i \textbf{k}^g_{i}
		\label{eq:ga_score},
\end{align}
 The GA score $\bm{\alpha}^g = [\alpha^g_1,\alpha^g_2,...,\alpha^g_M]$, which describe the degrees of attention paid to each local structure in $S$, are used to reveal critical aspects to help interpret the structure-property relationship of $S$.
 
\section{Experimental design}\label{sec:experiments}
 The datasets used in the evaluation are summarized as follows. Hereinafter, a model derived from the SCANNet with parameters fitted from the dataset is referred as the SCANNet model.

\textbf{QM9} \cite{ramakrishnan2014quantum}: This computational dataset comprises of data of 133,885 drug-like organic molecules composed of C, H, O, N, and F. However, 3054 files were removed due to the questionable geometric stability \cite{anderson2019cormorant} that 130,831 molecules remained were used for the experiments. Each molecule contains up to nine heavy atoms, and its physical properties were calculated using DFT calculations. Five physical properties from the QM9 dataset were used as targets for the prediction tasks in the experiments evaluating the predictive capability of SCANNet models. These properties include the energy of highest occupied molecular orbital ($\mathbf{E}_{HOMO}$), the energy of lowest unoccupied molecular orbital ($\mathbf{E}_{LUMO}$), the gap between the energies ($\mathbf{E}_{gap}=\mathbf{E}_{LUMO}-\mathbf{E}_{HOMO}$), the isotropic polarizability ($\alpha$), and the heat capacity at $298$ $K$ ($\mathbf{C}_{v}$). 

\textbf{Fullerene-MD}\cite{FullerenePtMD}: This is an in-house-developed computational material dataset that comprises the data of three well-known fullerene molecules, viz. C$_{60}$ (I$_h$), C$_{70}$ (D$_{5h}$), and C$_{72}$ (D$_{6h}$). It includes optimized structures and 3000 deformed structures obtained from molecular dynamics simulations (1000 structures for each molecule). The HOMO ($\mathbf{E}_{HOMO}$) and LUMO ($\mathbf{E}_{LUMO}$) energies of these structures are determined using DFT calculations, similar to the approach used in the QM9 dataset. Experiments are performed on this dataset to evaluate the predictive capability of the SCANNet models for HOMO and LUMO energies and to assess the interpretability of the models’ predictions for these properties. A distinctive feature of all structures in this dataset is that they only contain carbon atoms. Furthermore, due to the symmetric nature of fullerene molecules, the local structures within each molecule are highly similar with only minor differences. Therefore, this dataset allows for a precise evaluation of the interpretability of the SCANNet model. In the evaluation experiment using this dataset, SCANNet models pre-trained on the QM9 dataset are applied to train the prediction models for the HOMO and LUMO energies of the fullerene molecules.

\textbf{Pt/graphene-MD}\cite{FullerenePtMD}: This dataset is also an in-house-developed computational material dataset representing a system composed of a platinum atom adsorbed on a graphene flake terminated by hydrogen atoms\cite{DamHC2006,DamHC2009}. It consists of data of approximately 21,000 optimized and deformed structures generated through molecular dynamics simulations. The adsorption energies of these structures are determined using DFT calculations, similar to the approach used in the QM9 dataset. The purpose of the experiments conducted on this dataset is twofold: to evaluate the predictive performance of the SCANNet models for deformation energies of the structures ($\Delta U$) and to assess the interpretability of the models’ predictions for these deformation energies. The unique structural characteristic of this dataset is the presence of a two-dimensional honeycomb network of carbon atoms forming the graphene flake. Although the local structures of each carbon atom in the system exhibit slight distortions from the ideal $sp^2$ hybridization structure~\cite{DamHC2009}, this dataset allows for the quantitative evaluation of the interpretability of the SCANNet models in terms of the distortion of the honeycomb network on the graphene surface.
	
\textbf{SmFe$_{12}$-CD \cite{nguyen2022}}: This dataset is an in-house-developed computational material dataset containing the data of crystalline magnetic materials. It comprises the data of 3307 optimized structures of SmFe$_{12}$-based compounds, along with their corresponding formation energies ($\Delta E$) as the target properties. The dataset was generated by introducing partial substitutions of Mo, Zn, Co, Cu, Ti, Al, and Ga into the iron sites of the original SmFe$_{12}$ structure, which exhibits notable magnetic properties. Subsequently, the structures were optimized, and their formation energies were assessed using DFT calculations. Further details regarding the DFT calculation method used to create this dataset can be found in a previous work\cite{nguyen2022}. Using this dataset, the predictive capability of the SCANNet models for the formation energies of the structures ($\Delta E$) and the interpretability of the models are quantitatively evaluated to investigate the structural stability of the SmFe$_{12}$-based structures.

\section{Evaluation of the predictive power}\label{sec:prediction}
The predictive capability of SCANNet is measured using three in-house-developed material datasets, viz. Fullerene-MD, Pt/graphene-MD, and SmFe$_{12}$-CD. Herein, a similar number of train--validation--test splits are applied as those used in the QM9 dataset experiments. In addition, to predict $\mathbf{E}_{HOMO}$ and $\mathbf{E}_{LUMO}$ for the fullerene structures in Fullerene-MD, the weights from the QM9 dataset are applied as the pre-train model to initialize weights of the SCANNet model in the learning process on the Fullerene-MD dataset. Details regarding the optimal SCANNet hyperparameters for these datasets are presented in \autoref{sec:model_hyperparameters}.

In the test for predicting the HOMO and LUMO energies with the Fullerene-MD dataset using pre-trained weights, the SCANNet models yield MAEs of 23 $meV$ and 27 $meV$, respectively, which are less than two-third of the "chemical accuracy" threshold. For the Pt/Graphene-MD dataset, the SCANNet model achieves an MAE of 0.16 $meV/atom$ in predicting the deformation energy $\Delta U$ of the system. Moreover, the testing error for predicting the formation energy of magnetic materials in the SmFe12-CD dataset is 0.01 $eV/atom$, exhibiting a slight improvement compared to the prediction model proposed in a previous study\cite{nguyen2022}.

\section{Model hyperparameters}\label{sec:model_hyperparameters}
The crucial hyperparameters for the SCANNet model include the dimension of local structure representation ($h$) and the number of local attention layers (L). The optimal hyper-parameters for each dataset of consideration are obtained using the $\textsc{KerasTuner}$\cite{kerastuner}, an extensible hyperparameter optimization framework. As a result, the optimal dimensional for representations of local and material structures is $128$. The details about the SCANNet architecture used in our work are shown in table \ref{Table:architec_model}. In the training process, all parameters of the SCANNet model are updated by minimizing a loss function using Adam optimization \cite{Adam} with a scheduled learning rate decay from $5.10^{–3}$ to $10^{-4}$. In addition, the Swish activation\cite{Ramachandran2017SwishAS} is used for all fully-connected layers. In contrast, each dataset has different numbers of local attention layers.

The QM9 dataset's overall molecule sizes are around 17 atoms/structure. We found that L=8 is enough to model the interaction between molecules in the local attention layers. On the other hand, in the Pt/graphene-MD dataset, which has complicated structures (102 atoms/structure), we found that L = 11 performs well, while more than 12 layers can confuse the converging of the model. For the SmFe$_{12}$-MD dataset, we used L=9 for training these crystalline structures. 

 
\begin{table} [h]
\newcolumntype{C}[1]{>{\centering\arraybackslash}p{#1}}
\renewcommand{\arraystretch}{1.5}
\centering
\caption {The SCANNet architecture: $d_{in}$ is the dimension of the input to each layers, $M$ is the number of atoms in the structure as mentioned and $s_{out}$ is the corresponding output shape.\\} 
\begin{tabular}{C{2cm}  C{5cm} C{2.5cm} C{2.5cm}}
\hline
\hline
No. Layers & Layer Name & $d_{in}$ & $s_{out}$ \\ 
%\hline
%\multirow{2}{*}{1x} & Input Atom Embedding & 1 & (M, 48) \\  & Input Ring-Aromatic (Optional) & 2 & (M,10) \\
%\hline
%\multirow{1}{*}{1x} & Fully-connected & 48 (58)*  & (M, 128)  \\
%\hline

\hline
\multirow{1}{*}{1x} & Input Atom Embedding & 1 & (M, 48) \\
\hline
\multirow{1}{*}{1x} & Fully-connected & 48 & (M, 128)  \\
\hline
\multirow{1}{*}{Lx} & \multirow{1}{*}{Local Attention} & 128 & (M,128) \\ 
\hline
\multirow{1}{*}{1x} & Fully-connected & 128 & (M, 128)  \\
\hline
\multirow{1}{*}{1x} & Global Attention & 128 & (1, 128) \\
\hline
\multirow{1}{*}{1x} & Fully-connected & 128 & (1, 128) \\
\hline
\multirow{1}{*}{1x} & Output Prediction & 128 &  (1,)  \\
\hline
\hline
\end{tabular}
%\begin{tablenotes}
%\item * For molecules datasets (QM9, Fullerene-MD, Pt/graphene-MD), the aromatic property (which indicates if an atom is part of an aromatic ring) is also used along with atomic number for creating embedding vectors. Meanwhile, only atomic number were used for creating embedding vectors in crystalline dataset SmFe$_{12}$-CD.
% \end{tablenotes}
\label{Table:architec_model} 
\end{table}

%Comparison in the predictive abilities of five physical properties when employing the QM9 dataset and either SCANNet \tcb{model} or five other state-of-the-art DL models. The bold numbers indicate the lowest MAEs among the six models.  accounting for real-world application scenarios and chemical accuracy thresholds as references, we also evaluate the models by 

\begin{table} [t]
		\renewcommand{\arraystretch}{1.5}
			\centering
			\caption {Comparison in the practical assessment of the SCANNet models and five other state-of-the-art DL models for predicting five physical properties in the QM9 dataset. For each property, the evaluation score of each model is assessed as either the ratio of prediction error to the chemical accuracy threshold (when the error exceeds the threshold) or $100\%$ (when the error is below the threshold). Performances of these models are then averaged across five target properties. }
			\begin{tabular}{l|ccccc|c}
				\hline
				\hline
				& $\mathbf{E}_{HOMO}$ & $\mathbf{E}_{LUMO}$  & $\mathbf{E}_{gap}$  & $\alpha$  & $C_v$  & Average \\
				&  $(\%)$ &  $(\%)$ &  $(\%)$ & $(\%)$ & $(\%)$ & $(\%)$ \\
				\hline		
				
				WaveScatt\cite{hirn2017wavelet} 		 & 198 & 177 & 274 & 160 & 100 & 182 \\
				SchNet\cite{schutt2018schnet}   		 & 100 & 100 & 146 & 235 & 100 & 136 \\
				MEGNet\cite{chen2019graph} 	   		 & 100 & 100& 142  & 100 & 100 & 108 \\
				Cormorant\cite{anderson2019cormorant}  & 100 & 100 & 142  & 100 & 100 & 108 \\
				SE(3)-Trans\cite{Fabianb2020se3}  						 & 100 & 100 & 123  & 142 & 108 & 115 \\
				SCANNet 								 & 100 & 100 & 142  & 141 & 100 & 116 \\
				\hline
				\hline
			\end{tabular}
			\begin{tablenotes}
			\item $\mathbf{E}_{HOMO}$: Energy of the highest occupied molecular orbital; $\mathbf{E}_{LUMO}$: Energy of the lowest unoccupied molecular orbital; $\mathbf{E}_{gap}$: Energy gap; $\alpha$: Isotropic polarizability; $C_v$: Heat capacity at 298 K.
  			\end{tablenotes}
			\label{table:practical_evaluation}
\end{table}

	\begin{figure}[t]
		\centering
		\includegraphics[width=\textwidth]{Sup_Fig1_C60.pdf}
		\caption{Visualizations of structure--property relationships obtained from SCANNet model for C$_{60}$. The left figure illustrates the electron density of the degenerate HOMO (bottom) and LUMO (top) orbitals, as calculated using DFT. The isosurfaces with positive and negative electron densities are represented by the blue and red bubbles, respectively. The right figure displays the GA scores of local structures obtained from the SCANNet model for the corresponding property.}
		
		\label{Fig:C60_HOMO_LUMO}
	\end{figure}

\section*{References}

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