# 6G comprehensive intelligence: network operations and optimization based on
Large Language Models

Sifan Long, Fengxiao Tang, Yangfan Li, Tiao Tan, Zhengjie Jin, Ming Zhao, and
Nei Kato Sifan Long, Fengxiao Tang (Corresponding author), Yangfan Li, Tiao
Tan, Zhengjie Jin and Ming Zhao are with Central South University, China; Nei
Kato is with Tohoku University, Japan.

###### Abstract

The sixth generation mobile communication standard (6G) can promote the
development of Industrial Internet and Internet of Things (IoT). To achieve
comprehensive intelligent development of the network and provide customers
with higher quality personalized services. This paper proposes a network
performance optimization and intelligent operation network architecture based
on Large Language Model (LLM), aiming to build a comprehensive intelligent 6G
network system. The Large Language Model, with more parameters and stronger
learning ability, can more accurately capture patterns and features in data,
which can achieve more accurate content output and high intelligence and
provide strong support for related research such as network data security,
privacy protection, and health assessment. This paper also presents the design
framework of a network health assessment system based on LLM and focuses on
its potential application value, through the case of network health management
system, it is fully demonstrated that the 6G intelligent network system based
on LLM has important practical significance for the comprehensive realization
of intelligence.

###### Index Terms:

6G, Large Language Model, Network health assessment, Network performance
optimization

##  I Introduction

The intelligence of 6G networks can enhance the network’s perception and
cognitive abilities, ensure network security, and drive application innovation
[1]. It has developed for many years since the maturity of Deep Learning
technology. However, due to the limited capacity of small models in handling
specific tasks, and even their difficulty in dealing with complex scenarios,
these limitations have led to lower network intelligence. On the one hand,
with the further development of Artificial Intelligence (AI) technology, Large
Language Models (LLMs) based on Deep Learning have proven to possess powerful
generalization and reasoning capabilities. These models are able to learn and
extract language patterns and rules from vast amounts of textual data,
enabling them to deeply understand and generate natural language. Their robust
“emergent” abilities guarantee their performance in providing powerful general
task-solving and complex task-reasoning capabilities within complex systems.
As shown in Figure 1, networks based on Large Language Models have achieved
numerous applications, allowing for convenient handling of traditional issues
faced in network operations and performance optimization, such as network
monitoring, network health assessment, network analysis, and resource
scheduling.

Figure 1: Application of Large Language Model in intelligent network
operations and performance optimization.

On the other hand, thanks to the advantages of the Transformer model [2] in
parallelized computation, capturing long-distance dependencies, adapting to
various tasks, and superior performance, research on Large Language Models [3]
based on Transformer has rapidly developed in various application research
fields, and it achieved remarkable progress in the field of Natural Language
Processing. As shown in Figure 2, various multimodal tasks based on processing
text, images, and videos have emerged one after another in the past ten years,
and it has sparked great interest in the Artificial Intelligence community and
the entire world. Currently, the main participants in the Artificial
Intelligence industry are competing to develop their own proprietary LLM
frameworks, so that they can be applied to their respective fields.
Representative examples include OpenAI’s GPT-3, Google’s PALM, and Meta’s
LLaMA, whose parameters have exceeded billions, enabling them to adapt to
fitting modeling in complex application scenarios. At the same time, these LLM
frameworks are trained on a wide range of data sets from the Internet. When
the size of the model increases significantly, they show a strong
generalization ability, enabling it to easily handle multimodal tasks. In
recent years, driven by the research on 6G networks, there have been greater
challenges in designing and optimizing communication protocols and general
methods for 6G networks [4], such as low latency, high frequency band, high
bandwidth, high transmission rate, and intelligence. Meanwhile, the network
application architecture based on Large Language Models encompasses multiple
crucial stages, including data preprocessing, model training, building and
deployment, network application integration, as well as model optimization and
iteration. These stages collectively form the complete lifecycle of Large
Language Models in network applications, and drive the development of network
intelligence and application innovation. The LLM framework based on
Transformer can effectively leverage its unique advantages in complex 6G
network research, making 6G networks more intelligent.

Figure 2: The development timeline of the typical representative LLM
framework.

##  II The background of Large Language Models and related applications in the
network

###  II-A Model

Large Language Models (LLM) have demonstrated extraordinary capabilities in
Natural Language Processing (NLP) [5], including tasks such as translation,
question answering, and text generation. From the initial representative GPT
model to the latest Sora model as shown in Figure 2, the capabilities of LLM
have been greatly improved. For example, Sora [6] can even generate video
images that are difficult to distinguish between real and fake, pushing the
research on applying LLM to video generation to another climax.

Currently, there are three main types of LLM architectures that are widely
used: encoder-only, decoder-only, and encoder-decoder. The first category is
the “encoder-only” language model, which is good at text understanding because
it allows information to flow in both directions in the text. The second
category is the “decoder-only” language model, which is good at text
generation because information can only flow from the left to the right of the
text and effectively generate new words in an autoregressive manner. Finally,
the “encoder-decoder” group combines the first two models to complete tasks
that require understanding input and generating output, such as real-time
translation. However, they are all built using the Transformer architecture as
a basic component, the core of Transformer is the (self-)attention mechanism,
which can capture long-term context information more effectively than
recurrence and convolution mechanisms using GPUs. Therefore, the advent of the
Transformer architecture marks the beginning of the era of modern Large
Language Models. The LLM has achieved many successful applications in the
fields of fault diagnosis, image processing, protein structure prediction,
drug research and development, autonomous driving technology, financial
analysis, and climate research.

###  II-B Related work of LLM in the network

Artificial intelligence agents based on multimodal Large Language Models are
expected to revolutionize the research on traditional network applications,
especially the deployment of LLM agents in 6G networks, which holds greater
significance for the design of network structures and performance
optimization. Therefore, intelligence based on multimodal LLMs has become a
more prominent feature of 6G networks, and as a result, intelligent agents
based on LLMs have achieved many important applications in various fields of
network application research. For instance, Lai et al. [7] proposed a general
framework for resource-efficient generation incentive mechanisms in resource-
constrained scenarios. They fully demonstrated their feasible solutions for
handling issues related to network overhead, resource allocation, and task
offloading in generative mobile edge networks through application cases.
Furthermore, in the design of collaborative Cloud-Edge-Device network
communication structures, the collaborative Cloud-Edge-Device methodology
exhibits unique capabilities in generating personalized services. Chen et al.
[8] introduced NetGPT, which can handle cloud collaboration, providing end-
users with personalized and inclusive network intelligence, while granting
them privileged operations for real-time access to generative services.
Additionally, in terms of 6G network performance optimization, Xu et al. [9]
proposed a split learning system for LLM agents in 6G networks to reduce
interaction latency and better protect user privacy. They leveraged the
collaboration between mobile devices and edge servers, distributing multiple
LLMs with different roles across mobile devices and edge servers to
efficiently collaborate in executing user-agent interaction tasks. This system
can offload complex tasks to global LLMs running on edge servers, effectively
improving the communication efficiency of the entire network. As we discussed
earlier, the Large Language Models have demonstrated remarkable capabilities
in areas such as Natural Language Processing and Computer Vision (CV),
revolutionizing the development concept of artificial intelligence and
potentially reshaping our future. However, given the multimodal nature and
structural complexity of the cloud, cloud-based deployments may face critical
challenges, including long response times, high bandwidth costs, and
violations of data privacy. Nevertheless, 6G Mobile Edge Computing (MEC)
systems can effectively address these pressing issues [10]. Taking into
account the inherent resource constraints at the edge, Lin et al. [11] have
discussed various cutting-edge techniques, including split learning/inference,
parameter-efficient fine-tuning, quantization, and parameter-sharing
inference, to facilitate the efficient deployment of LLMs. It is believed that
with the further development of AI, the LLM-based network applications will
become more widespread, further driving human society towards comprehensive
intelligence.

##  III Intelligent network architecture

Figure 3: Overall design scheme for network health assessment and performance
optimization architecture based on LLM.

Network operation and optimization face multiple technical barriers, which
involve multiple levels, making the work of operation and optimization complex
and challenging. Network operation and optimization require continuous
learning and adaptation to new technical standards and protocols, such as the
5G transition to 6G applications requiring updates to corresponding protocols.
In addition, network performance monitoring and fault diagnosis are also
important technical barriers faced by network operation and optimization. In a
complex network environment, how to effectively monitor network performance,
detect and locate faults in a timely manner is a huge challenge. In order to
validate the application of LLM in network health assessment and performance
optimization, as shown in Figure 3, this paper proposes a framework based on
LLM for network health assessment and performance optimization. It mainly
consists of several components, namely the Data layer, Large Language Model,
Function pool, Logical layer, and Task and scene modeling.

These components serve as data, platform, logic layer, function pool, and
model support to provide fault diagnosis services for the entire network
health assessment system, such as centralized global diagnosis, distributed
inference and fault diagnosis, interpretable performance evaluation and key
factor traceability, and task oriented health assessment. The functions and
interrelationships of these components are as follows:

  * •

Data layer: Data resources serve as the cornerstone for Large Language Models
to efficiently diagnose network faults. These resources can be primarily
divided into three parts: 1) real data, 2) generated data, and 3) data from
other fields. The construction and maintenance of the data layer involve data
storage, management, and manipulation, providing effective data support for
model training and inference. During model training, real data refers to
network fault data collected from real-world scenarios, such as common issues
like physical layer faults and data link layer faults. These data undergo
standardized processing procedures before being inputted into the LLM for
effective model training. However, it is difficult to obtain a large amount of
real data for training in real-world scenes. Therefore, efficient data
generation methods become an important research focus. These methods utilize
specific algorithms and techniques to simulate and generate data. For
instance, Generative Adversarial Networks (GANs) [12] can produce high-quality
text and highly realistic image data. Such algorithms can also generate data
that closely resembles real-world fault data in network fault diagnosis,
partially addressing the issue of insufficient training data. Additionally, as
an important supplement for data acquisition, Hardware in the Loop (HIL)
simulation is another effective data generation method. This approach combines
key components and algorithms without fully deploying the entire system,
simulating faults to obtain data close to real ones. This data generation
method offers real-time performance, scalability, and flexibility, making it
adaptable to various simulation requirements and scenarios. It plays a crucial
role in generating efficient 6G network fault data.

  * •

Large Language Model: This module is primarily used to provide network fault
detection algorithms and performance analysis tools in support of network
health assessments. Based on differences in scale, number of parameters,
computing power, adaptation scenarios, and functional flexibility, the model
base can be further subdivided into small model bases and large model bases.
However, while the small model base can be flexibly and conveniently deployed
on cloud-edge computing platforms, its network health assessment capabilities
are limited by factors such as model accuracy, storage, data volume, and
transmission rates. This makes it challenging to meet the service requirements
of 6G networks, which demand high-speed, low-latency, high-traffic density,
and comprehensive intelligence. Therefore, to achieve more intelligent and
personalized network services, it is necessary to deploy a Large Language
Model base to support efficient network operation and intelligent network
health assessment services. Given the superior capabilities of LLM in
processing multi-modal data such as text, images, and videos, they can easily
learn and understand vast amounts of network fault data, including fault
types, causes, and impacts. This enables them to identify various fault
patterns. By analyzing real-time data from the operation of 6G networks, using
LLM as an agent, it can also quickly identify potential or ongoing faults and
provide timely warnings. When a fault occurs in the 6G network, the LLM can
analyze relevant network logs, configuration information, performance data,
and combine its learned knowledge to infer the possible cause of the fault.
This helps to rapidly locate the problem and reduce the time required for
fault investigation. Furthermore, based on a vast library of fault handling
cases and expert experience, the LLM can recommend tailored solutions for
specific faults. This not only provides timely assistance to network
administrators but also optimizes solutions based on historical data,
enhancing the efficiency of fault handling. Additionally, by analyzing
historical network operational data and trends, LLM can predict potential
future faults. This allows network administrators to take proactive measures
to prevent faults and ensure the stable operation of 6G networks. By
leveraging LLM to build a more robust network fault diagnosis system and
continuously learning and updating its knowledge base through real-time data
collection and analysis, LLM can improve the accuracy and efficiency of fault
diagnosis.

  * •

Function pool: The functional pool mainly consists of three parts: 1)
functional programs, 2) small models, and 3) knowledge bases. Among them,
functional programs are primarily responsible for data reading and
preprocessing tasks. For example, network fault data fed back by LLM needs to
be output in a standardized format to the corresponding interface for
invocation, making the architectural design of the entire network health
evaluation system more modular and intelligent. Secondly, the small model base
mainly includes Fault Tree Analysis (FTA) and Failure Mode and Effects
Analysis (FMEA) [13]. The FTA method can establish a set of top-down network
fault diagnosis methods. Through fault tree analysis, it can understand the
causes of various faults or failures in the network system and find the best
way to reduce these risks. However, unlike the FTA method, the FMEA method
starts with the reasons for the occurrence of events and identifies the
potential consequences that may result from these events. For example,
physical layer faults in a network may lead to issues such as data
transmission interruptions, connection device failures, and mismatched
transmission rates. In network performance optimization, the hybrid model of
Transformer+Long Short-Term Memory (LSTM) [14] is highly suitable for
processing and predicting important events with extremely long time series
intervals and delays. This holds significant practical implications for
safeguarding the efficient communication services of 6G networks over extended
periods of time. Given the self-attention mechanism of Transformer, it is
capable of capturing global dependencies when processing sequential data and
exhibits excellent parallel computing capabilities. Meanwhile, LSTM, as a
variant of Recurrent Neural Networks (RNNs), is particularly adept at handling
sequential data with long-term dependencies. By combining Transformer and
LSTM, a more robust network fault diagnosis model can be constructed to
support the efficient provision of services by 6G networks. In addition,
another important function of the functional pool is to establish a
comprehensive knowledge base, including the construction of fault tree
diagnosis and analysis, a robust knowledge graph structure, and rich expert
prior knowledge. For example, the knowledge graph provides a powerful semantic
network by building comprehensive associations between network system failure
modes, failure causes, failure handling measures, and other information. This
allows the system to intelligently understand users’ colloquial questions and
return accurate answers. Fault information retrieval based on the knowledge
graph can greatly improve the accuracy and efficiency of retrieval, helping
operation and maintenance personnel quickly locate and resolve network
failures.

Figure 4: Overall architecture diagram of an intelligent Cloud-Edge-Device
collaborative Network Health Assessment System based on LLM.

On the other hand, expert knowledge also plays an important role in network
fault diagnosis. Expert system technology is applied to network fault
diagnosis, providing accurate fault diagnosis and maintenance solutions. This
system can not only complete fault diagnosis within a specified time, but the
expression of fault phenomena can usually guide the system builder to
construct reasonable thinking more accurately, thereby improving the accuracy
of fault diagnosis. In addition, the expert system can intelligently adjust
parameters during system identification, achieving the goal of automated
debugging.

  * •

Logic layer: To design an efficient and modular network health assessment
system, the logic layer plays a crucial role in the process of network fault
troubleshooting, serving as an abstraction of the underlying structure. The
logic layer structure comprises three core modules: observation module,
thinking module, and action module. Among them, the thinking module is
primarily responsible for integrating decision-making information from the
underlying layers, such as network fault diagnosis information provided by
large models and their corresponding solutions. This module plays a vital role
in fault localization within the network health assessment system. Secondly,
the action module translates the decisions made by the thinking module into
actual action information. For instance, it provides data processing functions
to the functional pool, enhances the expert prior knowledge base, and supplies
action inputs to the LLM agent as feedback. Additionally, it can recommend
appropriate solutions or emergency measures based on the type and severity of
the fault, assisting users in rapidly restoring normal network operations.
Consequently, the role of the logic layer in the network health assessment
system is to analyze, diagnose, and process network fault information fed back
from the underlying layers, while providing decision support and fault
records. This helps users better manage and maintain the network, ensuring its
stability and security. Furthermore, the logic layer provides reasoning
services to upper-level application scenarios, thereby safeguarding the
integrity of the entire health assessment system’s functional structure.

  * •

Tasks and scenarios: Tasks and scenarios are two important core modules of the
application layer. Scenarios are mainly divided into online and offline
states, which are interconnected with LLMs. The task module, on the other
hand, is primarily responsible for fault diagnosis and health assessment of
the entire network. It accomplishes network health assessment tasks by
invoking relevant decisions from the logic layer. Meanwhile, network fault
scenarios are diverse, potentially involving hardware failures, software
glitches, configuration errors, security attacks, and various other
situations. These scenarios can lead to issues such as network outages,
increased latency, and elevated packet loss rates, ultimately affecting the
normal provision of network services. Furthermore, in network fault
assessment, modularization and intelligentization can better handle the
relationship between scenarios and tasks. Modularization breaks down the
complex process of network fault assessment into multiple independent modules,
with each module responsible for handling specific tasks, thereby improving
the efficiency and accuracy of the assessment. Intelligentization, on the
other hand, leverages LLMs to automatically identify and diagnose network
faults, providing intelligent decision support for task issuance and
execution.

Integrating the above functions, as depicted in Figure 4, we designed a Cloud-
Edge-Device intelligent network health evaluation system based on LLM, which
is a novel intelligent network system that integrates LLM technology, cloud
computing, edge computing, and terminal devices. Leveraging the strengths of
LLMs in text understanding and generation, this system enables precise
evaluation and effective management of network health status. Firstly, task
assignments are issued at the edge of the network, where LLMs undergo pruning,
distillation, and other optimization techniques. Various terminal devices,
such as smartphones, cameras, and drones, collect real-time data related to
network health, including network latency, packet loss rate, and bandwidth
utilization. These raw data undergo preliminary processing at the edge devices
according to task requirements, including data cleaning, format conversion,
and initial pre-training. This lays the foundation for subsequent analysis and
evaluation, such as pre-training lightweight LLMs and establishing local
knowledge bases. Secondly, the cloud-based LLM also participates in the data
processing, extracting relevant information related to network health status
by understanding the linguistic patterns and structural features within the
textual data. It establishes a vast knowledge repository of network health
data, enabling the training of LLMs specifically tailored for network health
conditions. For instance, it can diagnose the location of faults based on
network link and protocol failure information and archive these details. The
cloud-based LLM, trained on vast amounts of textual data, possesses the
capability to understand, generate, and process natural language. It receives
data from edge devices, combines it with historical and real-time data, and
conducts a thorough analysis of network health status. By comprehending the
inherent patterns and potential semantic information within the network health
data, the LLM can accurately identify potential issues, predict changes in
network performance trends, and provide corresponding evaluation results.
Furthermore, the analysis and evaluation results of the LLM are presented to
users through a visual interface. Users can view real-time information about
the network’s health status, historical trends, and potential risks. When the
system detects abnormalities in the network health status, the LLM generates
corresponding warning messages and provides decision support. These warnings
and suggestions assist users in quickly locating issues and developing
solutions, thereby enhancing the efficiency and accuracy of network
management.

##  IV Our demo and case study

Figure 5: Schematic diagram of network health management system architecture
based on textual proposition learning and LLM.

This section mainly demonstrates some of our current progress in the
application of LLM in network health assessment. Nodes in highly dynamic
heterogeneous networks are characterized by strong mobility and heterogeneity,
resulting in frequent and complex network failures. To ensure the reliability
and availability of network communications, as shown in Figure 5, this paper
proposes a network health management scheme based on intelligent network
information textualization, proposition learning, and LLM. Firstly, we extract
a large number of labeled network failure datasets from the digital twin
network, which are highly similar to real-world network conditions, to address
the issue of insufficient fault samples. Then, by introducing a tree-
structured semantic generation model, structured data is first converted into
the required text sentences, mapping the information status of key network
nodes to words representing the current indicator status. After collecting
abnormal node information for a specific node, a fault report is generated
through an enhanced retrieval and generation architecture. Private domain data
is used to improve the input of model, generating descriptions with abnormal
information and diagnostic methods. Finally, the output of the Large Language
Model is used to update the tree structure, improving the model’s accuracy and
inference efficiency, and ultimately generating high-quality network health
reports.

##  V Several open issues

With the continuous development and improvement of 6G networks, the
application of Large Language Models within them will become more extensive
and profound. However, we also face numerous challenges. On the one hand, as
the scale and complexity of models increase, the costs of training and
inference will also rise. On the other hand, it is an urgent issue to achieve
efficient model operation and performance optimization while ensuring data
privacy and security. Therefore, we need to continuously explore new
technologies and methods to address these challenges. In summary, the
efficient network operation and performance optimization of LLM in 6G networks
are a research area with significant importance. Through in-depth research and
exploration, we hope to provide more intelligent and efficient LLM support for
the development and application of 6G networks, thus driving the process of
comprehensive intelligence in human society. However, important issues will
also be faced in the research of intelligent networks and performance
optimization.

###  V-A Scaling problem

The model has high requirements for computing and storage resources, making it
difficult to deploy collaboratively at the edge. How to set up a Cloud-Edge-
Device collaborative framework to properly allocate resources and coordinate
capabilities has become a new challenge. In addition, model lightweighting has
become a solution to successfully deploying at the edge, and exploring
lightweight models has become another important research topic.

###  V-B Multimodal alignment problem

6G network pursues three-dimensional coverage. It needs to build a more
complex and flexible network topology to meet the coverage requirements in
different scenarios. However, the traditional Large Language Model is
dominated by single text or image mode, which will face great difficulties in
realizing multimodal fusion and alignment in 6G heterogeneous complex network.

###  V-C Selection and equilibrium of LLM and small model

In the 6G network, how to efficiently select between Large Language Models and
small models is a crucial task that requires a careful balance of factors such
as model performance, resource requirements, application scenarios, as well as
data privacy and security. Determining the appropriate model type based on
specific needs and scenarios, and optimizing the deployment and inference
processes of these models, has become an important research focus in order to
fully leverage the advantages of 6G networks and enhance overall performance.

###  V-D Model self evolution problem

The issue of model self-evolution in 6G networks combined with reinforcement
learning and multi-agent systems represents a challenging and promising
research direction. A crucial question lies in how to fully leverage the
advantages of 6G networks, as well as the unique characteristics of
reinforcement learning and multi-agent technology, to achieve efficient self-
evolution of models. For instance, by integrating reinforcement learning with
multi-agent systems, models can take advantage of the synergistic effects
among multiple agents, enabling them to achieve more efficient self-evolution.

###  V-E Data collection and real-time performance of the system

Large-scale data collection faces a series of issues, including the costs of
data transmission and storage, as well as concerns regarding data accuracy,
reliability, privacy, and security protection. At the same time, data security
issues should not be ignored. For example, Mišić et al. [15] discussed in
detail a large number of existing security and privacy related restrictions on
the network, which reflected the importance of network security in the context
of Internet of things applications. To address these challenges, it is
necessary to adopt advanced data compression and encryption technologies while
leveraging cloud computing and edge computing techniques to achieve efficient
data processing and utilization. However, due to the complexity and
uncertainty of network transmission, as well as the limitations of device
processing capabilities, achieving system real-time performance is not an easy
task. Therefore, one reliable approach is to incorporate data simulation and
future prediction from digital twin networks to tackle this difficulty.

###  V-F Security and interpretability of LLM in networks

The issues of security and interpretability of LLMs in 6G networks are
significant challenges currently faced and expected to persist in the field of
communication technologies. Due to their extensive parameters and complex
structures, Large Language Models are susceptible to exploitation by
attackers. Addressing data privacy and leakage concerns is paramount,
necessitating enhanced security measures for these models, including the
adoption of advanced encryption technologies to safeguard the secure
transmission and storage of model parameters and data. On the other hand, the
application of LLMs in 6G networks also poses certain challenges. Their
complexity and black-box nature often render their decision-making processes
and output results difficult for humans to comprehend and interpret. To
enhance the interpretability of LLMs in 6G networks, technical approaches can
be employed to increase model transparency. For instance, visualization
techniques can be utilized to demonstrate the internal structure and decision-
making processes of the models, enabling users to gain a more intuitive
understanding of their operational mechanisms.

##  VI Conclusion

The LLM have broad application prospects in 6G network performance
optimization and operation, which also face many challenges, such as model
performance, data quality, privacy security, and interpretability issues. To
overcome these challenges, this paper proposes a network intelligent operation
and performance optimization framework based on LLM. By analyzing their role
in network health assessment, fault detection, and other aspects, it is found
that LLM have great potential for enhancing the intelligence level of 6G
networks. Although the performance of LLM may be limited when processing
complex network health assessment tasks, their advantage in processing natural
language provides a new perspective for network state analysis. By learning
and analyzing historical data, the model can predict the future trend of
network traffic changes, providing decision-making basis for network resource
allocation and service quality improvement. In the future outlook, with the
continuous development of 6G technology, LLM will play a more important role
in network performance optimization, fault prevention, and resource
scheduling, providing strong support for building a more intelligent,
efficient, and secure 6G network.

##  VII Acknowledgments

This work was supported by the Postdoctoral Fellowship Program of CPSF (Grant
No.GZC20233160), Changsha Municipal Natural Science Foundation (Grant
no.kq2208284), Hunan Provincial Natural Science Foundation (Grant
no.2023jj40774), National Natural Science Foundation of China (Grant
no.62302527), and the High Performance Computing Center of Central South
University.

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