GLDiTalker: Speech-Driven 3D Facial Animation with Graph Latent Diffusion Transformer

1 Introduction

Speech-driven talking head generation aims to synthesize realistic and synchronized facial motion from speech, with applications in assistive technologies, such as virtual reality and augmented reality [Ping et al., 2013; Wohlgenannt et al., 2020]. Despite its potential, the task is challenging due to the many-to-many mapping between audio and facial motion, requiring a balance between accurate lip synchronization and natural expression generation.

To achieve realistic speech-driven talking head generation, various methods have been proposed. Traditional approaches [Edwards et al., 2016; Taylor et al., 2012; Xu et al., 2013]

rely on manual or rule-based designs, which are often costly and inflexible. In contrast, deep learning methods [Karras et al., 2017; Peng et al., 2023a] demonstrate stronger adaptability and have made significant progress. VOCA [Cudeiro et al., 2019] employs a CNN-based framework to generate cross-identity facial animations, while MeshTalk [Richard et al., 2021] leverages U-Net to enhance expression diversity by modeling latent facial expression spaces. Transformerbased models such as FaceFormer [Fan et al., 2022] improve temporal and cross-modal alignment through mechanisms like Biased Cross-Modal Multi-Head Attention. Similarly, CodeTalker [Xing et al., 2023] introduces a VQ-VAEbased motion prior to reduce the uncertainty in audio-tomotion mapping. Despite these advancements, existing methods share common limitations. Most approaches rely on deterministic models, which generate identical 3D facial motions for the same input conditions. This contradicts realworld behavior, where non-verbal facial cues are inherently nondeterministic [Ng et al., 2022]. Even when the same individual repeats the same sentence, subtle variations in facial motion naturally occur, reflecting the complexity of human communication. By failing to capture this variability, current methods often suffer from modality inconsistencies, particularly misalignment between audio and mesh. These issues lead to reduced motion diversity and compromised lip-sync accuracy, ultimately undermining the realism and expressiveness of generated animations.

To tackle the issues, we propose GLDiTalker, a Graph Latent Diffusion Transformer for audio-driven talking head generation. GLDiTalker first encodes facial motion into a spatio-temporal quantized latent space, effectively addressing the audio-mesh modality misalignment to ensure accurate lip synchronization. It then diffuses the signal within this latent quantized spatio-temporal space, introducing a controlled degree of stochasticity to enhance motion diversity. Existing methods excel in either lip-sync accuracy or motion diversity but struggle with both due to audio-mesh modality misalignment. In contrast, GLDiTalker resolves this issue and achieves superior performance in both aspects, as shown in Fig. 1.

To achieve this goal, GLDiTalker adopts a carefully designed quantized space-time diffusion training pipeline, which consists of two complementary stages: the Graph Enhanced Quantized Space Learning Stage and the SpaceTime Powered Latent Diffusion Stage. In the first stage, GLDiTalker focuses on encoding facial motion into a spatiotemporal quantized latent space to ensure accurate lip synchronization. Unlike existing approaches that primarily emphasize temporal facial motion, our method simultaneously captures temporal correlations and spatial connectivities between vertices. To this end, we propose a novel Spatial Pyramidal SpiralConv Encoder, which extracts multi-scale features using spiral convolution across multiple levels. This design enables GLDiTalker to effectively integrate diverse levels of semantic information, laying a solid foundation for accurate and realistic lip synchronization. In the second stage, GLDiTalker employs a diffusion model to iteratively add and remove noise to the latent quantized facial motion features obtained from the first stage. This process introduces a controlled degree of stochasticity, enhancing motion diversity while maintaining the integrity of the learned features. Together, these two stages ensure that GLDiTalker achieves a balance between precise lip-sync and rich motion diversity, overcoming the limitations of previous methods.

Extensive experiments on existing datasets demonstrate that our method significantly outperforms previous state-ofthe-art approaches in both lip-sync accuracy and motion diversity, effectively addressing the limitations of current techniques. Our contribution can be summarized as:

• We introduce GLDiTalker, a novel framework that addresses the conflict between achieving accurate lip synchronization and diverse facial motion caused by audiomesh modality misalignment.
• GLDiTalker features a two-stage design: a Graph Enhanced Quantized Space Learning Stage for encoding spatio-temporal facial motion and a Space-Time Powered Latent Diffusion Stage for enhancing motion diversity.
• Experiments on VOCASET and BIWI datasets demon-

strate that GLDiTalker outperforms state-of-the-art methods in both lip-sync accuracy and motion diversity.

2 RelatedWork

2.1 Speech-Driven 3D Facial Animation

Speech-driven talking face generation has attracted significant attention and achieved remarkable progress in recent years [Karras et al., 2017; Richard et al., 2021; Fan et al., 2022; Xing et al., 2023; Peng et al., 2023a; Wu et al., 2023; Stan et al., 2023; Shen et al., 2025b]. Early methods, such as VOCA [Cudeiro et al., 2019], introduce CNN-based frameworks for cross-identity face animation, while later works like Mimic [Fu et al., 2024] and EmoTalk [Peng et al., 2023b] focus on decoupling speech content, style, and emotion. Transformer-based approaches such as FaceFormer [Fan et al., 2022] improve audio-visual mapping through attention mechanisms, and CodeTalker [Xing et al., 2023] reduces cross-modal uncertainty using discrete motion priors. Recently, FaceDiffuser [Stan et al., 2023] applies diffusion mechanisms to enhance the diversity of facial motion generation. Although these methods address various challenges, they still struggle to simultaneously achieve high accuracy and diversity due to limitations in modality alignment and inherent model constraints. To overcome these issues, we propose GLDiTalker, which introduces a quantized space-time diffusion training pipeline to better balance precision and diversity in speech-driven talking face generation.

2.2 Diffusion in Talking Head Animation

Inspired by the diffusion processes in non-equilibrium thermodynamics, [Sohl-Dickstein et al., 2015] proposes diffusion, which differs from generative models like GAN [Goodfellow et al., 2020], VAE [Kingma and Welling, 2014], and Flow [Kingma and Dhariwal, 2018]. Diffusion is a Markov process capable of generating high-quality images from Gaussian noise, exhibiting a certain degree of stochasticity [Shen and Tang, 2024; Shen et al., 2025a]. Recently, many 2D talking head generation methods based on diffusion models have achieved highly realistic results [Park et al., 2022; Shen et al., 2022; Zhang et al., 2023; Shen et al., 2023; Xu et al., 2024]. For example, using audio as a condition, DREAM-Talk [Zhang et al., 2023] designs a twostage diffusion-based framework that generates diverse expressions while maintaining accurate audio-lip synchronization. DiffTalk [Shen et al., 2023] further incorporates reference images and facial landmarks, enabling personalityaware synthesis and generalization across identities without fine-tuning. Similarly, VASA-1 [Xu et al., 2024] employs diffusion in a latent space to model overall facial dynamics and head movements, effectively capturing subtle facial expressions and head motions. In the context of 3D facial animation, FaceDiffuser [Stan et al., 2023] is the first to explicitly apply diffusion for speech-driven 3D facial motion generation. While diffusion methods inherently enhance diversity due to their stochastic nature, they often struggle to achieve high accuracy, particularly in terms of lip synchronization and motion precision. This trade-off between diversity and accuracy remains a significant challenge. To address this, we propose GLDiTalker, which leverages a quantized space-time diffusion training pipeline to balance diversity and accuracy.

3 Method

3.1 Overview

Problem Definition

The facial motion M1:T = (m1,m2, ...,mT ) is a sequence of vertex displacements mi ∈ RV ×3 applied to a neutral template face mesh f ∈ RV ×3 , which consists of V vertices. The audio A1:T = (a1, a2, ..., aT ) is a sequence of speech snippets, where each ai ∈ RC contains C samples aligned with the corresponding motion mi. Our objective is to predict a mesh sequence ˆM1:T based on the audio A1:T and the talking style represented by one-hot vectors S = (s1, s2, ..., sI), where I denotes the number of speaking styles. The final face animations are obtained by summing the neutral template and the predicted motion, expressed as ˆ F1:T = ˆm1+f , ..., ˆmT+f . The output sequence ˆ F1:T is expected to achieve accurate lip synchronization with the input audio while exhibiting diverse and expressive facial motions that reflect different speaking styles.

Quantized Space-Time Diffusion Training Pipeline

To solve the problem of failing to simultaneously achieve high lip-sync accuracy and motion diversity caused by modality misalignment, GLDiTalker follows a two-stage pipeline that consists of a Graph Enhanced Quantized Space Learning Stage and a Space-Time Powered Latent Diffusion Stage, as illustrated in Fig. 2. In the first stage, GLDiTalker focuses on quantization to enhance lip-sync accuracy. Specifically, it employs a spatio-temporal VQ-VAE [Van Den Oord et al., 2017] based on graph convolution and transformer to model facial motion into a quantized facial motion prior, ensuring precise alignment between audio and facial motion. In the second stage, the focus shifts to diffusion for generating diverse motions. A transformer-based denoising network is adopted for the reverse diffusion process, which iteratively transforms a standard Gaussian distribution into the facial motion prior. This process is conditioned on audio, speaker identity, and the diffusion step, enabling the generation of realistic and expressive facial animations. The animation results can be directly obtained from the denoising outputs by using the decoder from the first stage.

Next, we introduce the Graph Enhanced Quantized Space Learning Stage and the Space-Time Powered Latent Diffusion Stage in detail, respectively.

3.2 Stage 1: Graph Enhanced Quantized Space Learning for Lip-sync Accuracy Improvement

Existing methods have limited results on lip-sync, mostly limited by the ability to model the explicit space or the accumulation of errors due to autoregression. To improve the robustness to cross-modal uncertainty, GLDiTalker proposes a Graph Enhanced Quantized Space Learning Stage inspired by VQ-VAE to model the facial motion into a quantized motion prior, as illustrated in the first stage in Fig. 2. A graph based encoder is trained with both the temporal information dependency and the spatial connectivity of mesh vertices taken into account. Specifically, spatial and temporal information are serially processed to get latent feature ˆ Z ∈ RT×H×C with a novel designed Spatial Pyramidal SpiralConv Encoder EV following by a temporal transformer encoder ET :

where H and C are the number of face components and channels, respectively.

The Spatial Pyramidal SpiralConv Encoder consists of SpiralConv layers, downsampling layers and a fusion layer, as shown in Fig. 3. SpiralConv [Gong et al., 2019], a graphbased convolution operator is introduced to process mesh data, which determines the convolution centre and produces a sequence of enumerated centre vertices based on adjacency, followed by the 1-ring vertices, the 2-ring vertices, and so on, until all vertices containing k rings are included. SpiralConv determines the adjacency in the following way:

where NH(V ) selects all vertices in the neighborhood of any vertex in set V . SpiralConv generates spiral sequences only once and explicitly encodes local information. It adopts a fully-connected layer for feature fusion:

where W and b are learnable weights and bias.

Then the SpiralConv layer and the downsampling layer are novelly composed into a Spiral Block, which reduces the number of vertices and increases the feature channels during feature extraction. After stacking multiple Spiral Blocks, the resulting graphs with a small scale are flattened and fed to the fusion layer (an MLP layer).

The Spatial Pyramidal SpiralConv Encoder is capable of extracting multi-scale features from the input graph by performing SpiralConv at multiple scales. This module enables the network to integrate semantic-level information at different levels. Lower-level pyramids can capture more detailed information, while higher-level pyramids focus more on abstract and semantic information. Pyramid network can effectively reduce the computational cost and improve the efficiency of the model in comparison to a single-scale network.

After getting the latent feature ˆ Z ∈ RT×C , the discrete facial motion latent sequence Zq ∈ RT×C is obtained by an element-wise quantization function Q(·), which calculates by a nearest neighbour look-up in codebook Z = {zm ∈ RC }Mm=1 :

Finally, temporal transformer decoder DT and spatial MLP decoder DV are adopted for self-reconstruction:

Constraints

In this stage, we train with the following loss functions:

where λrec1 = λquant = 1.

Motion Reconstruction Loss calculates the L1 loss between the predicted animation sequence ˆM1:T and the reference facial motion M1:T :

Quantization Loss contains two intermediate code-level losses that update the codebook items by reducing the MSE distance between the codebook Z and latent features ˆ Z:

where sg(·) stands for the stop-gradient operator, which is defined as identity in the forward computation with zero partial derivatives; β denotes a weighted hyperparameter, which is 0.25 in all our experiments.

3.3 Stage 2: Space-Time Powered Latent Diffusion for Motion Diversity Enhancement

To enhance the motion diversity that neglected by the deterministic models, GLDiTalker designs a Space-Time Powered Latent Diffusion Stage to introduce the quantized facial motion prior obtained from the first stage and subsequently train a speech-conditioned diffusion-based model on top of it. During the forward process, a Markov chain q(xn|xn-1) for n ∈ {1, ..., N} gradually introduces Gaussian noise to the clean data sample Zq (denoted as x0), resulting in a standard normal distribution q(xN|x0):

where N is the diffusion step.

On the contrary, the distribution p(xn-1|xn) is employed to reconstruct the clean data sample x0 from the noise xN during the backward process. Conditioning on the audio A1:T , one-hot speaker style embedding si and diffusion step n, we apply a Markov chain p(xn-1|xn,A1:T , si, n) to sequentially transform the distribution p(xN) into p(x0):

where p(xN) = N(0, I). In practice, we use a denoising network G that directly predicts the clean sample ˆx0:

As the second stage shown in Fig. 2, the architecture of the denoising network G consists of five modules: (1) a noise encoder Enoise; (2) a style encoder Es; (3) an audio encoder Ea; (4) a denoising step encoder Ed and (5) a diffusion facial decoder D.

Noise Encoder Enoise reduces the dimension of the latent features, thereby retaining the most informative features while reducing the computational complexity of model training and inference, thus improving the efficiency and speed of the model.

Style Encoder Es encodes the one-hot speaker style embedding si ∈ RI into a latent code.

Audio Encoder Ea uses the released hubert-base-ls960 version of the HuBERT architecture pre-trained on 960 hours of 16kHz sampled speech audio. The feature extractor, feature projection layer and the initial two layers of the encoder are frozen, while the remaining parameters are set to be trainable, which enables HuBERT to better explore the motion information from audio. Since the facial motions might be captured with a frequency different from that of the speech tokens, we adjust input representation similar to [Stan et al., 2023].

Denoising Step Encoder Ed encodes the denoising step n into a latent code.

Diffusion Facial Decoder D is a transformer decoder that produces the final animation latent feature sample. The decoder process can be abstracted as follows:

where CA(q, kv), SA(·) and f(·) mean Biased CrossModal Multi-Head Attention, Biased Causal Multi-Head Self-Attention and feed forward network, respectively; q denotes query input features and kv denotes key and value input features. The alignment mask [Fan et al., 2022] in CA makes the motion features only attend to the speech features at the same position.

Constraints

The loss function of this stage consists of two items:

where λrec2 = λvel = 1.

Latent Feature Reconstruction Loss uses a Huber loss to supervise the recovered ˆx0:

Besides, Velocity Loss also employs a Huber loss to supervise inter-frame variations for smooth animation:

4 Experiments

4.1 Datasets and Implementations

We conduct abundant experiments on two public 3D facial datasets, BIWI [Fanelli et al., 2010] and VOCASET [Cudeiro et al., 2019], both of which have 4D face scans along with audio recordings.

BIWI dataset. The BIWI dataset contains 40 sentences spoken by 14 subjects. The recordings are captured at 25 fps with 23370 vertices per mesh. We follow the data splits of the previous work [Fan et al., 2022] and only use the emotional data for fair comparisons. Specifically, the training set (BIWITrain) contains 192 sentences, the validation set (BIWI-Val) contains 24 sentences, and the testing set are divided into two subsets, in which BIWI-Test-A contains 24 sentences spoken by 6 seen subjects during training and BIWI-Test-B contains 32 sentences spoken by 8 unseen subjects during training. BIWI-Test-A is used for both quantitative and qualitative evaluation and BIWI-Test-B is used for qualitative testing.

VOCASET dataset. The VOCASET dataset consists of 480 audio-visual pairs from 12 subjects. The facial motion sequences are captured at 60 fps with about 4 seconds in length. The 3D face meshes in VOCASET are registered to the FLAME topology [Li et al., 2017], with 5023 vertices per mesh. Similar to [Fan et al., 2022], we adopt the same training (VOCA-Train), validation (VOCA-Val) and testing (VOCA-Test) splits for qualitative testing.

4.2 Quantitative Evaluation

We adopt the following metrics for quantitative evaluation:

• Lip vertex error (LVE) measures the deviation of the lip vertices of the generated sequences relative to the ground truth by calculating the maximal L2 loss for each frame and averaging over all frames.
• Facial Dynamics Deviation (FDD) measures the deviation of the upper face motion variation of the generated sequences relative to the ground truth. The standard deviation of the elements-wise L2 norm along the temporal axis at the upper face vertices of the generated sequence and the ground truth are initially calculated. Subsequently, their differences are solved and averaged.
• Diversity measures the diversity of the generated facial motions from the same audio input. We calculate this metric across all samples in the BIWI-Test-A following [Ren et al., 2023].

Since all subjects in BIWI-Test-A have all been seen in the training dataset, the model can learn the motion styles of the subjects in the testing set, which exhibits a relatively fixed output under speech-driven conditions. In contrast, the subjects in BIWI-Test-B and VOCASET-Test have not been included in the training dataset. Consequently, the model is only able to predict the animation based on the styles of the GroundTruth FaceFormer CodeTalker FaceDiffuser GLDiTalker subjects in the training set. The results may differ significantly from the ground truth, but it may be reasonable to consider without the styles. Therefore, we only quantitatively compare the performance of our GLDiTalker with state-ofthe-art methods on BIWI-Test-A, as in previous work [Xing et al., 2023] and [Stan et al., 2023].

Figure 5: Standard deviation motion heatmaps of facial motion within a random sampled sequence, where dark blue means less motion variations and bright red means more motion variations.

Quantitative evaluations are shown in Tab. 1. Similar to the superior performance of CodeTalker compared to FaceFormer, GLDiTalker outperforms FaceDiffuser in terms of lip movement and upper facial expression motion prediction (0.1383×10-4mm and 0.0751×10-5mm decrease in LVE and FDD, respectively) by leveraging VQ-VAE to introduce a motion prior that enhances the robustness of the crossmodal mapping from audio to 3D vertex displacements. This strongly proves the effectiveness of our quantized space-time diffusion pipeline. Although TalkingStyle stands out with the lowest LVE and FDD, our approach ranks closely behind, showcasing only a minor difference. Furthermore, the highest Diversity (8.2176×10-4mm) demonstrates the superiority of our model in capturing many-to-many mappings between audio and facial motion. The latent diffusion model that starts with random noise and generates latent features through multiple iterations of denoising extremely enhances motion diversity compared to deterministic methods and explicit diffusion model method.

4.3 Qualitative Evaluation

We visually compare our GLDiTalker with FaceFormer, CodeTalker and FaceDiffuser and randomly assign the same speaking style as the conditional input to make a fair comparison. Fig. 4 shows the animation results of the audio sequences from VOCASET-Test and BIWI-Test-B. We can observe that the lip movements produced by GLDiTalker are more accuracy. For vowel articulation, GLDiTalker exhibits a larger mouth opening than FaceFormer and FaceDiffuser, which is more consistent with GroundTruth. This can be observed in the mouth shape of the three examples in VOCASET-Test as well as 'I' and 'on' in BIWI-Test-B. For consonant articulation, GLDiTalker can fully close the mouth but CodeTalker and FaceDiffuser are less effective, which can be reflected by 'table' in BIWI-Test-B. Readers are recommended to watch the the Supplemental Video for more detail. In addition, we also visualize the standard deviation of facial movements of the four methods. Fig. 5 shows that our GLDiTalker outperforms others in the range of facial motion variations, illustrating the generation diversity. In addition, the right side of Fig. 1 demonstrates the highest motion diversity of GLDiTalker than the other methods. More comparisons are shown in Supplementary Video.

4.4 User Study

In order to evaluate our model more comprehensively, we conduct a user study to evaluate realism and lip synchronization. Specifically, a questionnaire is designed for the purpose of comparing the effectiveness of FaceFormer, CodeTalker, FaceDiffuser and GLDiTalker with ground truth on 20 randomly selected samples from BIWI-Test-B and VOCASETTest, respectively. Twenty participants are shown the questionnaire and asked to rate on a scale of 1-5. We count the Mean Opinion Score (MOS) of all methods. Tab. 2 shows that GLDiTalker gets the highest score in both two metrics, demonstrating its superiority on semantic comprehension and natural realism over other methods.

4.5 Ablation Study

In this section, we conduct an ablation study to the impact of our proposed quantized space-time diffusion training pipeline on the quality of generated 3D talking faces. All the experiments are conducted on BIWI dataset and Tab. 3 and Tab. 4 are the results from BIWI-Test-A.

Impact of Graph Enhanced Quantized Space Learning Stage

Tab. 3 illustrates that our model achieves obvious superiority in lip-sync accuracy due to the Graph Enhanced Quantized Space Learning Stage with Spatial Pyramidal SpiralConv Encoder. The lowest LVE and FDD (4.6440×10-4mm and 3.8474×10-5mm in LVE and FDD, respectively) indicate the considerable impact on the overall visual quality of the generated faces and the comprehensibility of lip movements. With regard to the local connectivity of the graph, Spatial Pyramidal SpiralConv Encoder updates the representation of each mesh vertex by fusing the features with those of the neighbouring vertices. This process preserves the topology information of the vertices, thereby encouraging the model to effectively learn the representation of each vertex and the whole graph structure. In contrast, MLP is typically oriented towards regular input data with overly dense connections that do not take into account the structure of the graph.

A series of ablation experiments on kernel size, dilation coefficient, and layer numbers of Spatial Pyramidal SpiralConv Encoder are provided, as shown in Tab. 4. With respect to kernel size, the data in the first and third rows indicates that a large kernel size confers a significant advantage in LVE and FDD (0.2159×10-4mm and 0.1852×10-5mm decrease in LVE and FDD, respectively). This may be attributed to the fact that larger convolution kernels have larger receptive field, thus enabling the integration of a greater quantity of information. Furthermore, in terms of the dilation coefficients, the third and fifth rows show that larger dilation coefficients lead to better lip synchronisation, although this is associated with a reduction in facial quality performance (0.1305×10-4mm decrease in LVE and 0.1074×10-5mm increase in FDD). We believe that the effect of lip synchronisation is more significant, so we retain the option of using a large dilation factor. Finally, for the number of layers, two sets of experiments are conducted to demonstrate the importance of the pyramid layer number. With a dilation factor of 1 (no dilation convolution operation), changing the number of layers from 4 to 2 leads to a decline in performance across all metrics (0.1944×10-4mm and 0.3487×10-5mm increase in LVE and FDD, respectively), as illustrated in the second and third rows; in the case of dilation factor of 2, modifying the number of layers from 4 to 3 also leads to worse performance (0.5208×10-4mm and 0.3493×10-5mm increase in LVE and FDD, respectively), as shown in the fourth and fifth rows. This suggests that an increase in the layer number can enhance feature extraction capability to comprehensively capture the intrinsic structure and regularity of the sparse vertex data effectively.

Impact of Space-Time Powered Latent Diffusion Stage

The last column of the Tab. 3 shows that using diffusion in the quantized space of Graph Enhanced Quantized Space Learning Stage leads to significant improvements in terms of motion diversity (8.2176 × 10-4mm). With the same input, the deterministic model can only produce a fixed output, which does not reflect the many-to-many correspondence between speech and facial motions in reality. Instead, our approach uses diffusion in the quantized space, which even has an advantage of several orders of magnitude over FaceDiffuser, which adopts diffusion in the explicit space. We also conducted experiments to remove the Noise Encoder Enoise, which resulted in a performance degradation, as shown in Tab. 5, indicating the need of Noise Encoder.

5 Conclusion and Discussion

In conclusion, we propose GLDiTalker, a novel speech-driven 3D facial animation model that addresses the challenges of modality misalignment, lip-sync accuracy, and motion diversity in talking head generation. By introducing a twostage pipeline, GLDiTalker ensures precise lip synchronization through the Graph-Enhanced Quantized Space Learning Stage and enhances motion diversity with the SpaceTime Powered Latent Diffusion Stage. Extensive evaluations demonstrate that GLDiTalker achieves state-of-the-art performance, generating realistic, temporally stable 3D facial animations with both high lip-sync accuracy and diverse motion, making it a robust solution for speech-driven animation tasks.

Ethical Statement

Face data can be used for generating content that may jeopardize privacy. We must act responsibly by considering the aspects related to privacy and ethics.

Acknowledgements

This work was supported by the National Natural Science Foundation of China under Grant No. 62441617. It was also supported by the Beijing Natural Science Foundation under Grant No. 4254100, the Fundamental Research Funds for the Central Universities under Grant No. KG16336301, the China Postdoctoral Science Foundation under Grant No. 2024M764093, and by Beijing Advanced Innovation Center for Future Blockchain and Privacy Computing.

Contribution Statement

Xiandong Li and Wenxiong Kang are the corresponding authors. They led the planning and execution of the research and provided financial support for the project leading to this publication. Yihong Lin and Zhaoxin Fan contributed equally to this work as co-first authors. Yihong Lin and Xiandong Li proposed the ideas. In addition, Yihong Lin designed the model and the implementation of the computer code, performed the experiments and wrote the initial draft. Zhaoxin Fan analyzed the study data and made critical review, commentary and revision including pre and post publication stages. Xianjia Wu and Lingyu Xiong assisted with figure preparation. Liang Peng, Songju Lei and Huang Xu proofread the manuscript. All authors have read and approved the final manuscript.