Traffic flow forecasting

Traffic flow forecasting remains a fundamental task in intelligent transportation systems, aiming to predict future traffic states from historical observations. The problem is inherently complex due to nonlinear temporal dynamics, periodic variations, and intricate spatial dependencies shaped by large-scale and irregular road networks. In practice, effective forecasting models must also achieve a balance between accuracy, computational efficiency, and scalability for real-time urban operations. Early-stage approaches such as ARIMA and SVR [22,23] provided interpretable and lightweight solutions but relied on strong linear and stationary assumptions. With the emergence of deep learning, recurrent neural networks (RNNs), long short-term memory (LSTM), and gated recurrent units (GRUs) [24–26] significantly improved temporal sequence modeling. However, their inherently sequential structure restricts parallel computation and struggles to capture long-range temporal dependencies, which limits their efficiency in large-scale forecasting tasks. To overcome these limitations, convolutional architectures were subsequently introduced to enable parallel temporal modeling. Models such as DeepST [27] and ST-ResNet [28] applied temporal and spatial convolutions separately to extract local and periodic patterns, demonstrating strong short-term performance. Nonetheless, their grid-based Euclidean assumptions hindered their ability to represent the non-Euclidean topology of real traffic networks. Later designs such as STNN [29] combined convolution with attention mechanisms to enhance spatial–temporal fusion, but at the cost of higher architectural complexity. Building upon these foundations, hybrid and graph-enhanced frameworks emerged to jointly capture spatial and temporal dependencies. T-GCN [25] integrates graph convolution with GRU for dynamic spatio-temporal learning, while SRCN [30] leverages residual convolutions for end-to-end city-scale prediction. In addition, normalization-based models like ST-Norm [31] decouple spatial and temporal variations, improving stability and convergence. These efforts collectively mark the transition from purely temporal or grid-based designs to more unified spatio-temporal representations.

Collectively, these developments have paved the way for more advanced modeling paradigms that aim to better capture spatial complexity and temporal heterogeneity. In particular, recent research trends have focused on structured spatial reasoning using graph-based models and global dependency modeling via attention-based architectures. These directions will be discussed in detail in the following sections.

Graph convolution network

Graph Convolutional Networks (GCNs) have become the cornerstone of spatial modeling in non-Euclidean traffic networks, offering a principled framework for capturing interactions among irregularly connected sensors and road segments [32]. Early GCN-based approaches, such as DCRNN [18] and STGCN [17], pioneered the integration of diffusion graph convolution and temporal convolution to jointly model spatio-temporal dependencies. However, their reliance on static and predefined adjacency matrices limits adaptability to dynamic traffic conditions and context-dependent relationships. To address this limitation, subsequent models introduced adaptive or data-driven graph structures. Graph WaveNet [33] proposed a learnable adjacency matrix to decouple spatial dependencies from fixed topology, while AGCRN [34] incorporated node-specific embeddings to capture regional heterogeneity. More recently, DSTAGNN [35] extended this paradigm by dynamically updating graph connections according to evolving traffic contexts, enabling fine-grained adaptation to temporal variations. Similarly, FC-STGNN [36] explored fully connected graph structures to capture global correlations, though such approaches inevitably increase computational overhead in large-scale networks. With these advancements, dynamic and diffusion-based frameworks have further improved physical interpretability and multi-scale spatial reasoning. DGCRN [37] learns time-varying graphs conditioned on evolving traffic contexts, while MegaCRN [38] employs a meta-graph learning strategy to dynamically generate latent adjacency matrices. MTGNN [39] integrates multi-hop diffusion to model directional propagation across the network, balancing local influence and global context. STGNN-TCN [40] further couples temporal convolution with graph diffusion, demonstrating the potential of hybrid architectures for scalable and interpretable spatio-temporal learning.

Despite these developments, GCN-based frameworks continue to face challenges in achieving a balance between adaptability and computational efficiency, as well as in maintaining consistency between spatial diffusion and temporal modeling. Future research should focus on developing unified, physically interpretable spatio-temporal frameworks that integrate diffusion-based spatial reasoning with temporally coherent learning.

Transformer

Transformer-based architectures have recently emerged as a powerful paradigm for spatio-temporal modeling, owing to their ability to capture long-range dependencies and global contextual relationships. This capability has made them increasingly popular in traffic forecasting, where complex and interdependent dynamics must be modeled efficiently across both temporal and spatial domains. Early efforts sought to integrate the attention mechanism into existing spatio-temporal frameworks. For instance, ASTGCN [41] introduced spatial and temporal attention modules into graph-based networks, while GMAN [20] employs multi-head spatio-temporal attention to learn latent graphs at each time step. STAEformer [42] further introduces spatio-temporal adaptive embeddings within an encoder–decoder architecture to better capture periodic and fluctuating patterns. More recent approaches, such as HUTFormer [43], design hierarchical U-shaped Transformer structures to model multi-scale dependencies, whereas GATFormer [44] integrates graph structural priors into Transformer encoders to improve topology awareness. Despite these advances, Transformer-based models often face intrinsic challenges when applied to traffic networks. Their dense attention mechanisms compute all-to-all interactions across nodes or time steps, resulting in quadratic complexity and redundant dependency learning. STPFormer [45] introduces pivot-node-based sparse attention to mitigate the quadratic complexity of dense attention while preserving key spatio-temporal dependencies. In parallel, other domains have explored context-aware spatio-temporal reasoning under attention-driven paradigms. For instance, Chen et al. [46] proposed a multi-context aware human mobility prediction framework that integrates motif-preserving hypergraph convolution and social-gated fusion to capture higher-order mobility motifs and contextual dependencies. Although focused on individual mobility behavior rather than large-scale traffic flow, this work highlights how incorporating structured priors and multi-context attention can enhance the interpretability and generalization of spatio-temporal models.

Building on these insights, recent Transformer-based research continues to investigate structured and sparse attention mechanisms as a means to enhance scalability while maintaining essential dependencies. Sparse attention selectively emphasizes the most relevant spatial or temporal relationships, thereby reducing redundant computations and highlighting dominant propagation patterns within complex networks. However, the adaptive and topology-aware integration of such sparsity remains insufficiently explored, leaving room for more unified and physically interpretable modeling frameworks in future work.

Overall framework

The proposed DSSA-TCN model is designed to effectively capture spatio-temporal dependencies in traffic networks by integrating three key components: a temporal convolutional network for long-range temporal modeling, an adaptive sparse attention mechanism for dynamic spatial interaction, and a diffusion-based graph convolution for directional and multi-hop propagation. Given an input sequence , where T_h is the number of historical time steps, N is the number of traffic sensors, and C is the number of features per node, the goal is to predict future traffic states over the next T_f steps, denoted by , The underlying spatial structure is encoded in an adjacency matrix , which reflects the connectivity of the traffic network.

As shown in Fig 1, the model first transforms the raw inputs into a latent representation through a linear projection, and augments it with time-of-day, day-of-week, and learnable node embeddings. These embeddings help the model capture periodic traffic patterns and spatial heterogeneity. Temporal modeling is performed using stacked layers of dilated causal convolutions applied along the time axis. The causal structure ensures that predictions are made using only past information, while dilation enlarges the temporal receptive field without increasing model depth. Compared to recurrent networks, this structure enables greater parallelism and improved training efficiency. To capture spatial dependencies, each temporal block is followed by a spatial modeling module consisting of two parts. First, an adaptive attention mechanism computes relevance scores across nodes and selects a sparse subset of top-ranked neighbors for each target node. This reduces unnecessary computation and enables the model to focus on structurally meaningful connections. Then, the attended features are passed through a diffusion convolutional layer, which aggregates information from multi-hop neighbors along both forward and reverse directions on the graph. Outputs from multiple blocks are connected through skip connections to preserve multi-scale representations. After stacking all layers, a lightweight convolutional head is applied to produce the final predictions over all nodes and time steps. By alternating between temporal and spatial modules, DSSA-TCN is able to jointly learn temporally coherent and spatially structured representations, resulting in accurate and efficient traffic forecasting on complex road networks.

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Fig 1. The overall architecture of DSSA-TCN.

https://doi.org/10.1371/journal.pone.0336787.g001

Input embedding layer

To enhance the model’s ability to capture complex spatio-temporal dependencies, we construct an input embedding layer that encodes raw traffic observations into a unified latent representation. This layer integrates three complementary components: raw feature projection, temporal context embedding, and learnable node embedding. Together, they enrich the input tensor with temporal periodicity and spatial semantics, facilitating effective representation learning for traffic forecasting.

Raw feature projection. The input traffic sequence includes real-valued input features such as traffic volume or speed. A fully connected linear layer is applied to project these features into a hidden representation space of dimension d_r, producing . This transformation allows the model to extract more expressive patterns from the original input, providing a stronger foundation for downstream processing.

Temporal context embedding. Given the periodic nature of urban traffic, we introduce time-aware embeddings to capture short-term and long-term temporal patterns.

• Time-of-Day (ToD): Each timestamp is discretized into one of P intervals throughout the day (e.g., 288 intervals representing 5-minute steps), and mapped into a latent vector using a learnable embedding matrix of size ;
• Day-of-Week (DoW): Each day is encoded as one of seven categorical indices and mapped using an embedding matrix of size ;

The resulting embeddings are broadcasted across all nodes at each time step and concatenated into a joint temporal context embedding . These representations help the model to distinguish between different traffic regimes, such as weekday peak hours versus weekend off-peak periods.

Learnable node embedding. To account for spatial heterogeneity across traffic sensors, each node is assigned a trainable node-specific embedding vector. These embeddings are shared across time and replicated over the temporal dimension to obtain . By incorporating these node-specific representations, the model gains the ability to differentiate sensor behaviors and learn spatial biases adaptively, which is particularly beneficial in dynamically changing traffic environments.

Finally, all three components are concatenated along the feature axis to form the final input embedding:

(4)

where is the total embedding dimension.

Spatial-temporal encoder layer

Temporal correlation capturing.

To effectively capture temporal dependencies in traffic data, we employ a gated dilated causal convolution module as the temporal encoder in our model. Compared to recurrent architectures, this structure enables parallel computation, avoids gradient vanishing, and preserves strict time causality by constraining each output to depend solely on past inputs. Let the initial embedded input be denoted as . Before feeding into the temporal encoder, the embedding tensor is first permuted to the shape , so that a temporal 1D convolution can be applied along the time axis. A convolution is then used to project the feature dimension from to , resulting in . This transformed representation serves as the input to the first temporal block. For each layer − 1}, the input is processed by the module to produce , which will subsequently be fed into the spatial modeling unit of the same layer. The module operation is defined as:

(5)

Here, denotes a dilated causal convolution, represents element-wise multiplication, and , are the hyperbolic tangent and sigmoid activation functions. The learnable parameters include the convolutional kernels and biases , where is the kernel size.

The dilated convolution operation for a filter is given by:

(6)

where d is the dilation factor, which increases exponentially across layers (e.g., ). This design allows the model to expand the receptive field exponentially without deepening the network, making long-range temporal dependencies easier to learn.

By integrating the filter and gate branches through a gated activation mechanism, the model is able to dynamically regulate temporal information flow, retaining informative patterns while suppressing noise. The output of each layer is forwarded to the spatial modeling unit, enabling the network to alternately learn temporal and spatial structures in a unified framework.

Sparse spatial attention.

To capture dynamic and fine-grained spatial dependencies among traffic sensors, we introduce a sparse spatial attention (SSA) mechanism following each temporal convolutional layer. Unlike standard attention mechanisms methods that compute interactions between all node pairs indiscriminately, our method selectively attends to a subset of the most relevant nodes per target, thereby reducing computational overhead and improving robustness against noisy or irrelevant dependencies.

Given the output of the l temporal convolutional block , we first permute it to , and then project it into query, key, and value representations through learnable linear transformations:

(7)

where are the trainable parameters, and is the attention dimension. We compute pairwise attention scores among all nodes using scaled dot-product attention:

(8)

The Softmax operation normalizes the pairwise similarity scores into a probability distribution, ensuring that all attention weights are positive and sum to one for each target node. This normalization allows the model to interpret the relative importance of neighboring nodes and stabilizes the gradient updates during training.

To promote sparsity and interpretability, we apply an adaptive Top–k filtering strategy. Specifically, a learnable scalar α is used to dynamically determine the number of top-attended neighbors per node:

(9)

where are predefined lower and upper sparsity bounds, and is the sigmoid function. For each row of A, we retain only the Top–k values are retained while the rest are masked with , followed by a second softmax normalization to ensure valid attention weights. This process yields the sparse attention matrix .

The final sparse spatial representation is computed by aggregating the value vectors through the filtered attention weights:

(10)

This attention mechanism allows the model to automatically select the most informative spatial interactions per node at each time step, effectively filtering out irrelevant signals while preserving key structural dependencies. The resulting sparse representation is then passed to the diffusion graph convolution module for further spatial modeling.

Diffusion graph convolution.

To further capture the directional and multi-hop spatial dependencies in traffic networks, we incorporate a diffusion graph convolution (DGC) module into the model. This module performs feature propagation over a static graph topology, allowing the network to learn structured spatial representations based on physical connectivity.

Let be the predefined adjacency matrix of the traffic network, representing the connectivity among sensors. To enhance self-awareness in the diffusion process, we add self-loops and construct the one-step random walk transition matrix as:

(11)

where I is the identity matrix and D is the diagonal out-degree matrix of . Here, P represents the one-step transition probability between connected sensors, describing how traffic information from a node is distributed to its immediate neighbors. The self-loop term ensures that each node retains part of its own signal, avoiding feature dilution. The -step diffusion process is then defined as:

(12)

Intuitively, P quantifies how strongly node j influences node i after k successive propagation steps, thereby encoding multi-hop spatial dependencies across the traffic graph. Let be the input representation at layer l, produced by the preceding sparse attention module. The diffusion graph convolution aggregates spatial information from multiple hops through linear projections over the diffusion matrices:

(13)

where is a trainable weight matrix for the -th diffusion order, and is the maximum number of hops. This formulation can be viewed as a weighted combination of signals diffused through 0-hop (self), 1-hop, ... , K-hop neighbors, where each term P models a distinct receptive field and learns its importance. In essence, it allows each node to integrate both local and distant contextual information.

In traffic systems, vehicles move along directed roads, where information propagates from upstream to downstream and can also feed back upstream; employing both P and enables the model to more faithfully represent this asymmetric bidirectional interaction. The resulting spatial representation is then passed to the subsequent temporal modeling layer or the final output projection module.

Output projection layer

To generate the final traffic predictions, we design an output projection layer that aggregates the deep spatio-temporal representations from all preceding blocks and maps them into the target prediction space. This module performs two key functions: multi-scale feature fusion and step-wise forecasting. Let the output of the -th spatio-temporal block be denoted as is the number of time steps. To integrate hierarchical temporal-spatial features, we apply a skip connection mechanism by projecting each block’s output into a unified feature space of dimension f, and aggregating them as follows:

(14)

where is a learnable linear projection matrix for the l-th block, and the fused skip representation satisfies .

We then apply a lightweight prediction head consisting of two linear layers with a nonlinear activation in between to generate the final multi-step predictions:

(15)

Here, and are trainable weight matrices, is the intermediate embedding dimension, and denotes a nonlinear activation function (e.g., ReLU). This output projection layer enables the model to decode high-level spatio-temporal features into fine-grained, time-aligned predictions for each sensor over the forecast horizon.

Loss function

To train the model for accurate multi-step traffic flow forecasting, we adopt the mean absolute error (MAE) as the optimization objective. MAE is robust to outliers, numerically stable, and widely used in temporal regression tasks such as traffic prediction.

Let the model’s predicted output be , and the corresponding ground truth be . The overall loss function is defined as:

(16)

where T_f denotes the prediction horizon and N is the number of traffic sensor nodes. and represent the predicted and ground truth values for node n at future time step t. This loss function effectively quantifies the model’s forecasting accuracy across temporal and spatial dimensions, guiding parameter optimization during training to improve predictive performance.

Experimental setup

Datasets. To thoroughly evaluate the generalization and robustness of the proposed DSSA-TCN model, we conduct experiments on six widely-used benchmark datasets: METR-LA, PEMS-BAY, PEMS03, PEMS04, PEMS07, and PEMS08. These datasets differ significantly in terms of geographic coverage, spatial scale, and traffic modalities, providing a comprehensive testbed for spatio-temporal forecasting. Specifically, METR-LA and PEMS-BAY are traffic speed datasets published by Li et al. [18], containing 5-minute interval measurements collected from 207 and 325 sensors in Los Angeles and the Bay Area, respectively. Both datasets are divided into training (70%), validation (10%), and testing (20%) splits. The remaining four datasets, PEMS03, PEMS04, PEMS07, and PEMS08, are sourced from the caltrans performance measurement system (PeMS) and introduced by Zheng et al. [20]. These datasets record traffic flow at 5-minute intervals from sensors deployed in various Caltrans districts. They are split into 60% training, 20% validation, and 20% testing sets. All data are preprocessed using z-score normalization based on the statistics of the training set. A summary of dataset statistics is provided in Table 1. Notably, METR-LA and PEMS-BAY are used for traffic speed prediction, while the other four datasets are used for traffic flow forecasting.

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Table 1. Summary of benchmark datasets.

https://doi.org/10.1371/journal.pone.0336787.t001

Settings. All experiments are conducted on a Linux server equipped with an Intel i9-10900K CPU and an NVIDIA RTX 4080 GPU. The model is implemented using PyTorch 2.3.0 with CUDA 12.1 support. Optimization is performed using the Adam optimizer with an initial learning rate of 0.002 and a weight decay of 0.0003. A multi-step learning rate scheduler decays the learning rate by a factor of 0.1 at the 50th and 70th epochs. Each model is trained for 100 epochs with a batch size of 64, except for PEMS07, where a batch size of 8 is used due to its larger graph scale. The training objective is masked mean absolute error (MAE), and all results are averaged over three independent runs to reduce variance.

Each node’s input series is projected into a 12-dimensional embedding space. To capture temporal and spatial heterogeneity, three additional 12-dimensional embeddings are concatenated: time-of-day, day-of-week and a learnable node embedding. The temporal modeling backbone consists of four TCN blocks, each with two layers of gated dilated convolutions (kernel size = 2). Both the residual and dilation branches use 32 channels. Skip connections and the final projection layer use 256 and 512 channels respectively. Dropout is applied with a fixed rate of 0.3 for regularization. For spatial modeling, we employ a one-layer sparse multi-head self-attention module with 4 attention heads and a hidden feedforward dimension of 128. An adaptive top-k selection mechanism dynamically selects the most relevant neighbors, where the k is constrained within the range of 1/6 to 1/3 of the total nodes. Following the attention operation, a bidirectional diffusion graph convolution is applied using a diffusion order of 2, allowing information to propagate through both forward and reverse 2-hop neighbors based on the normalized adjacency matrices.

Metrics. To comprehensively evaluate the predictive performance of our model, we adopt three widely-used evaluation metrics: mean absolute error (MAE), mean absolute percentage error (MAPE), and root mean squared error (RMSE). These metrics are defined as follows:

• MAE measures the average absolute deviation between predicted and ground truth values:(17)
• MAPE quantifies the average relative error in percentage terms:(18)
• RMSE penalizes larger errors more heavily by computing the square root of the mean squared error:(19)

where and y_i,t denote the predicted and ground truth values for node i at time t, and is the total number of valid prediction points. All metrics are reported under three common forecasting horizons: 3-step (15 minutes), 6-step (30 minutes), and 12-step (60 minutes), enabling a comprehensive evaluation of model performance across short-term, medium-term, and long-term intervals.

Experiment result

To thoroughly evaluate the effectiveness and generalization ability of our proposed DSSA-TCN, we compare it against six representative baselines spanning a diverse set of spatio-temporal forecasting paradigms. All models are implemented or re-implemented within the BasicTS [47] framework to ensure consistency in data preprocessing, training protocols, and evaluation metrics.

• DCRNN [18]: A pioneering model that integrates recurrent neural networks with bidirectional diffusion graph convolutions. It captures temporal dynamics through an encoder-decoder architecture based on GRUs, while modeling asymmetric spatial dependencies via directed diffusion processes;
• STGCN [17]: A fully convolutional framework that alternates between temporal convolutions and spatial GCN layers. By eliminating recurrent structures, it improves training efficiency while synchronously modeling spatial and temporal patterns;
• Graph WaveNet [33]: Enhances temporal modeling with dilated causal convolutions and spatial adaptivity via a learnable adjacency matrix. This design improves robustness to noisy or incomplete graph structures and supports long-range dependency learning;
• ST-Norm [31]: Introduces spatial-temporal normalization modules that standardize node-level sequences using local statistics. It mitigates distributional heterogeneity across nodes and time, leading to improved generalization across varying traffic regimes;
• MegaCRN [38]: Employs a meta-graph convolution module to dynamically construct instance-specific graph structures during training. This allows the model to capture time-varying spatial correlations without relying on static or predefined topologies;
• STAEformer [42]: A Transformer-based architecture that introduces spatio-temporal adaptive embeddings to model periodic patterns and spatial heterogeneity.

These baselines span multiple modeling paradigms, including sequence-based RNNs, convolutional encoders, graph neural networks with adaptive structures, and Transformer-based architectures. Together, they provide a comprehensive and rigorous benchmark suite for validating the performance and generalization ability of DSSA-TCN.

We conducted a comprehensive comparison between our proposed DSSA-TCN model and six representative spatio-temporal forecasting baselines across six real-world traffic datasets: METR-LA, PEMS-BAY, PEMS03, PEMS04, PEMS07, and PEMS08. On the METR-LA and PEMS-BAY datasets, we further evaluated model performance under multiple prediction horizons, specifically 15, 30, and 60 minutes, as reported in Tables 2 and 3. Overall, DSSA-TCN achieved the best results in most experimental settings. According to Table 2, DSSA-TCN ranked first in 10 out of 18 metric-dataset combinations. The advantages were particularly evident on datasets with complex traffic dynamics such as PEMS-BAY, PEMS04, and PEMS08. For example, on the PEMS-BAY dataset, DSSA-TCN achieved MAE of 1.53, RMSE of 3.50, and MAPE of 3.42%, outperforming all baseline models. On PEMS04, it also achieved the lowest RMSE (30.51) and MAPE (12.46%), significantly surpassing competitive models like MegaCRN and ST-Norm.

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Table 2. Overall performance comparison on six benchmark traffic datasets.

https://doi.org/10.1371/journal.pone.0336787.t002

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Table 3. Performance under different prediction horizons on METR-LA and PEMS-BAY.

https://doi.org/10.1371/journal.pone.0336787.t003

Further, Table 3 presents a comparison under short-term (15-minute) and long-term (60-minute) forecasting scenarios. In 60-minute forecasts, DSSA-TCN achieved RMSE improvements of 1.4% on METR-LA and 4.3% on PEMS-BAY compared to the best-performing baselines. For the 15-minute horizon, the improvements were relatively smaller, such as a 0.9% RMSE reduction on METR-LA, while still demonstrating strong responsiveness to short-term variations. These results indicate that DSSA-TCN exhibits stronger resistance to error accumulation over longer forecasting horizons and outperforms models such as DCRNN and GWNet, whose performance degrades more significantly with increasing prediction steps. It is worth noting that the performance gap between DSSA-TCN and several baselines becomes less pronounced in short-term forecasting (e.g., 15 minutes). This observation is expected because short-term traffic dynamics are dominated by recent local trends and periodic patterns that most temporal models can effectively capture. However, DSSA-TCN demonstrates greater advantages in long-term forecasting horizons (≥60 minutes) and in datasets with higher spatial complexity, where adaptive sparse attention and bidirectional diffusion jointly enhance the modeling of long-range and directional dependencies. This finding confirms that DSSA-TCN is particularly effective for complex, multi-hop, and temporally extended traffic prediction scenarios. Among all compared methods, RNN-based models such as DCRNN exhibited relatively poor performance in long-horizon settings. This is likely due to the limited capacity of GRU encoders to capture long-range dependencies, as well as their susceptibility to gradient vanishing problems. In contrast, the convolutional architecture of DSSA-TCN provides a larger temporal receptive field and more stable optimization, resulting in better robustness and generalization across dynamic and structurally complex traffic scenarios.

Ablation study

To evaluate the contribution of each proposed module, we conduct ablation studies by systematically removing or replacing specific components from the full model. The detailed configurations are as follows:

• w/o Temporal Embedding: Removes the time-of-day and day-of-week embeddings to assess the impact of temporal context modeling.
• w/o Adaptive Embedding: Eliminates the learnable node embeddings, retaining only structural information to evaluate the effect of personalized node representations.
• w/o Top-k Attention: Replaces the sparse Top-k spatial attention mechanism with full attention to examine the benefits of sparsity in attention computation.
• w/o Diffusion GCN: Removes the diffusion graph convolution module, relying solely on attention to model spatial dependencies, thus testing the importance of explicitly modeling topological structure.
• Ours: The complete model with all proposed components, serving as the performance baseline.

To evaluate the individual contributions of each model component, we conduct a comprehensive ablation study on three benchmark datasets: METR-LA, PEMS-BAY, and PEMS03. The results, summarized in Table 4, reveal several key observations. The exclusion of the diffusion graph convolution module results in the most significant performance degradation. On METR-LA, the RMSE increases from 6.01 to 6.36, which corresponds to a relative rise of 5.8%. Similarly, on PEMS03, the RMSE increases from 26.17 to 27.51, reflecting a 5.1% decrease in accuracy. These results highlight the crucial role of explicit topological modeling in capturing spatial dependencies across traffic networks. Replacing the sparse Top-k spatial attention mechanism with fully connected attention also causes noticeable declines in accuracy. For instance, on PEMS03, the RMSE increases from 26.17 to 26.85. This indicates that the sparse attention design effectively filters irrelevant spatial interactions, helping the model focus on the most informative nodes in large-scale networks. When adaptive node embeddings are removed from the model, the performance drops slightly. On PEMS03, the RMSE rises from 26.17 to 26.24. Although the decline is relatively minor, the result still confirms that incorporating node-specific representations improves predictive precision by introducing spatial heterogeneity. Removing temporal embeddings, which encode periodic patterns such as time-of-day and day-of-week, also leads to a moderate reduction in performance. On METR-LA, the RMSE increases from 6.01 to 6.08. This suggests that temporal priors contribute auxiliary information that helps the model capture recurring fluctuations more effectively. In summary, each component of DSSA-TCN contributes meaningfully to the overall performance. Among them, diffusion graph convolution and sparse attention have the most significant impact and serve as the core of spatial modeling, while temporal and adaptive embeddings offer complementary global and node-level enhancements, jointly improving the model’s generalization and stability.

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Table 4. Ablation study of DSSA-TCN on three benchmark datasets.

https://doi.org/10.1371/journal.pone.0336787.t004

Sensitivity to sparsity level

To further evaluate the impact of the sparse spatial attention mechanism on forecasting performance, we conduct a sensitivity analysis on the sparsity-level hyperparameter. Specifically, we focus on the neighbor selection ratio range used in the Top-k sparsity strategy. This mechanism employs a learnable parameter α to dynamically determine the number of spatial neighbors retained for each node during training. Such a design enables the model to adaptively balance spatial expressiveness and computational efficiency across different scenarios. Given the varying spatial density and topological complexity of real-world traffic graphs, the appropriate sparsity level is crucial for maintaining model stability and generalization. To this end, we perform experiments on two representative traffic volume datasets, PEMS03 and PEMS04, under six different sparsity configurations: , and a fully connected setting that retains all spatial neighbors. This analysis aims to reveal how the sparsity control mechanism influences the accuracy and robustness of the DSSA-TCN framework under different traffic conditions.

The results illustrated in Fig 2 reveal a consistent U-shaped relationship between model performance and spatial sparsity level. On both PEMS03 and PEMS04 datasets, the DSSA-TCN achieves the lowest MAE under moderate sparsity configurations. The optimal sparsity ranges are for PEMS03 and for PEMS04, suggesting that a moderate number of spatial neighbors yields the best performance. At extremely high sparsity levels, where only a minimal number of spatial neighbors are preserved, the model is deprived of adequate contextual information, hindering its ability to learn meaningful spatial correlations. This deficiency often results in underfitting, as the network fails to capture essential structural dependencies. On the other hand, reducing sparsity toward a fully connected setting introduces an abundance of irrelevant or noisy spatial signals. Such excessive connectivity not only increases computational burden but also disrupts attention stability, which can deteriorate prediction performance. These observations provide empirical support for the incorporation of a learnable sparsity mechanism. By dynamically adjusting spatial connectivity during training, the model selectively attends to the most informative nodes while suppressing redundant interactions. Proper calibration of this sparsity level is therefore crucial to fully harness the representational power of DSSA-TCN in traffic networks with diverse topological characteristics.

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Fig 2. MAE performance of DSSA-TCN under varying spatial sparsity levels on the PEMS03 and PEMS04 datasets.

https://doi.org/10.1371/journal.pone.0336787.g002

Interpretability analysis of sparse attention

To assess the interpretability of DSSA-TCN, we visualize the attention maps learned in the spatial attention module. Fig 3 compares the raw attention matrix before sparsification with the sparse attention matrix after the adaptive top-k operation. The matrices are obtained from the final TCN block, and the attention values are averaged across heads to represent inter-node dependencies. The raw attention matrix appears dense, with many low but nonzero correlations across node pairs, which makes it difficult to identify meaningful spatial relationships. In contrast, the sparse attention matrix exhibits clear and structured patterns. Only a small subset of nodes retains high attention weights, forming visible vertical and horizontal concentration bands. These high-weight nodes correspond to major traffic sensors that exert strong global influence across the network. The visualization indicates that the adaptive sparse attention mechanism can capture both local and long-range dependencies between distant road segments that are not directly connected in the physical topology. This demonstrates that the attention mechanism complements the graph diffusion process by learning additional functional relationships beyond the predefined adjacency structure. Therefore, the sparse attention enhances model interpretability by revealing how remote regions interact dynamically within the traffic network.

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Fig 3. Visualization of attention matrices on the METR-LA dataset.

(a) Raw attention matrix before sparsification, showing dense and noisy correlations across nodes. (b) Sparse attention matrix after the adaptive Top-k operation, where high-weight connections capture both local and long-range dependencies beyond the physical topology.

https://doi.org/10.1371/journal.pone.0336787.g003

Efficiency study

To evaluate the practical efficiency of the proposed model, we conduct a comparative analysis on the METR-LA dataset to examine the trade-off between predictive accuracy and model complexity. Fig 4 illustrates the relationship between the total number of trainable parameters (x-axis) and the corresponding MAE (y-axis) across all competing models. Our proposed DSSA-TCN achieves the lowest overall MAE while maintaining a relatively compact architecture with only 358,060 parameters. This highlights the model’s excellent balance between predictive performance and memory efficiency. In contrast, Transformer-based models such as STAEformer tend to incur substantially higher parameter counts due to dense attention operations over time and space dimensions. While they offer competitive accuracy, their elevated memory and computation demands may hinder practical deployment in large-scale traffic networks. DSSA-TCN, on the other hand, leverages lightweight temporal convolutions and sparse spatial attention to achieve competitive convergence with shorter per-epoch runtimes and reduced memory footprint. To further quantify the computational advantages, Table 5 reports the training time, inference latency, and parameter counts of all compared models. DSSA-TCN achieves a 19.6% faster inference speed compared to MegaCRN and maintains comparable training efficiency to lightweight baselines such as STGCN and STNorm. These results demonstrate that DSSA-TCN not only delivers state-of-the-art prediction accuracy but also exhibits excellent computational efficiency, making it a practical and scalable solution for deployment in real-world intelligent transportation systems.

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Fig 4. Comparison of model efficiency on the METR-LA dataset.

https://doi.org/10.1371/journal.pone.0336787.g004

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Table 5. Computational efficiency of DSSA-TCN and baseline models on the METR-LA dataset.

https://doi.org/10.1371/journal.pone.0336787.t005