The model extracts spatial features from the PM_2.5 directed flow graph. Given the PM_2.5 flow graph G, the feature representation of the central city vertex i is updated by aggregating the adjacency graph vertices through the message-passing paradigm as shown in the following equations.

(5)ξit=[Xi,Pit],∀i∈V

(6)ej→it=Ψ([ξjt,ξit,Qj→it]),∀(j,i)∈E

(7)ζit=Φ(∑j∈Ni⁡(ej→it,ei→jt)),∀i∈Vwhere V, E, Xi, P, Q are the city vertex set and the edge set of G, the PM_2.5 concentration data, meteorological features of city vertex i, and edge attribute features, respectively. Ψ and Φ are linear transformations. ξit is the feature representation of city vertex i at time t. ej→it and ei→jt are the PM_2.5 inflow impact and outflow impact of city vertex i at time t, respectively.

The deeper vertex spatial feature representation is extracted by graph attention weighting, and the graph attention layer operation is shown in the following equations.

(8)γit=[ξit,ζit],∀i∈V

(9)eij=LeakyReLU(aT[Wγit||Wγjt])

(10)αij=Softmaxj(eij)=exp(eij)∑k∈Niexp(eik)

(11)εit=δ(∑j∈NiαijWγjt)where γit is the feature representation of city vertex i at time t; W is the linear transformation matrix; || is a splicing operation. The feedforward neural network aT maps the features to real numbers and obtains the similarity degree eij of the feature representation of city vertex i and its neighbor vertex j through LeakyReLU activation. Then the Softmax function calculates the attention weight αij of city vertex i and its neighbor vertex j. Finally, the attention-weighted feature representation εit of city vertex i at time t is obtained by weighting all neighboring vertex feature representations at time t, so the spatial correlation feature extraction of city vertices in the PM_2.5 flow graph is realized in this paper.

GRU is similar to LSTM and is also able to capture the long-term dependence of urban PM_2.5 concentration data. Combined with the time-series attention mechanism focusing on the highly correlated historical time step information of the current prediction time step, a long-term prediction of PM_2.5 concentration can be achieved. For spatiotemporal features fusion, the city spatial features extracted by WGAT are used as external features and input to a single prediction unit GRU together with the corresponding historical PM_2.5 concentration data of cities and meteorological features.

First, the loop structure of the GRU outputs a multi-step prediction of the hidden state, as shown in the following equations.

(12)xit=[ξit,εit]

(13)rit=σ(Wr⋅[hit−1,xit])

(14)zit=σ(Wz⋅[hit−1,xit])

(15)h~it=tanh⁡(W⋅[rit∗hit−1,xit])

(16)hit=(1−zit)∗hit−1+zit∗h~itwhere xit is the input of the spatiotemporal fusion prediction unit of the city vertex i at the prediction time step t. Wz, Wr, and W are parameters that can be trained and learned in GRU. σ is the Sigmoid activation function. rit and zit are reset gate and update gate structures in GRU, respectively. hit−1 is the hidden state output of the prediction unit of the previous prediction time step, h~it is the candidate hidden state of the prediction unit of the prediction time step t, and hit is the hidden state output of the prediction unit of the prediction time step t.

Then the hidden state of the prediction unit output is weighted by time series attention, so that the current forecast time step focuses on the key historical time step information, as shown in the following equations.

(17)eit=tanh⁡(Whit+b)

(18)αit=exp⁡(eij)∑j=1texp⁡(eij)

(19)sit=∑j=1tαijhijwhere eit is the attention score of the hidden state hit of the city vertex i at time step t output through the fully connected network. αit is the attention weight of the city vertex i at the current time step t. sit is the spatiotemporal fusion feature hidden state.

Finally, sit output from AGRU is linearly transformed through the fully connected layer to obtain the predicted PM_2.5 concentration value X^it for city vertex i at prediction time step t.

The current PM_2.5 concentration prediction combined with the meteorological feature at the next moment predicts the output at the next moment. And at the next moment, the predicted value can be used as the PM_2.5 concentration at that time and combined with the corresponding meteorological feature to predict the PM_2.5 concentration. The multi-step prediction of PM_2.5 concentration is achieved through an iterative process.

The prediction of PM_2.5 concentration is generally regarded as the prediction of spatiotemporal series. Assume that the PM_2.5 concentration of city vertices at time t is Xt∈RN, where N represents the number of vertices in the PM_2.5 directed flow graph G, that is, the total number of cities studied in the paper. Pt∈RN×p and Qt∈RM×q. represent the vertex feature matrix and edge attribute matrix of all city vertices in PM_2.5 directed flow graph G at time t, separately, and M is the number of edges in G. The vertex feature P represents the meteorological features such as temperature and humidity of the city, and the edge attribute matrix Q is the meteorological characteristics such as wind direction and wind speed, and geographical features such as distance and altitude required to calculate the edge weight, that is, the features related to the impact of PM_2.5 concentration between cities.

The multi-step of PM_2.5 concentration prediction in the paper is realized by iteration. In order to predict PM_2.5 concentration for a period of time in the future, the vertex feature matrix [Pt+1,…,Pt+T], edge attribute matrix [Qt+1,…,Qt+T] and PM_2.5 directed flow graph structure G of the future T time step are used as the input of the prediction model. And the multi-step prediction of PM_2.5 concentration can be defined by Eq. (20).

(20)[Xt;Pt+1,…,Pt+T;Qt+1,…,Qt+T;G]→f(⋅)[X^t+1,…,X^t+T]

The iteration multi-step prediction process of the model is shown in Fig. 2. The PM_2.5 prediction model uses the predicted PM_2.5 concentration X^t+1 in the previous prediction time step as the input of the model to predict the next time step. And the PM_2.5 concentration of the city in the future T time step is predicted through continuous iterative calculation. The iterative process is shown in Eq. (21).

(21)f(⋅)=g(…g(g(⋅)))

The prediction operation f(⋅) of PM_2.5 concentration in the future T time step is achieved by iterating the T times prediction model g(⋅). The predicted value X^t+τ of PM_2.5 concentration at time step τ in the future can be expressed by Eq. (22).

(22)X^t+τ={g(X^t+τ−1,Pt+τ,Qt+τ,G,Θ),∀τ∈[1,…,T]Xt,τ=0where X^t+τ−1 is the predicted value of the PM_2.5 concentration of the city output by the prediction model at the last time. Pt+τ is the vertex feature corresponding to the city vertex of the current prediction time step. Qt+τ is the edge attribute matrix used to calculate the edge weight between city vertices in the current prediction time step. And G is the constructed PM_2.5 directed flow graph structure. Θ represents the corresponding parameters of the neural network in the prediction model. X^t+τ is the predicted value of PM_2.5 concentration in the current prediction time step output by the prediction model.

To analyze the accuracy of the proposed WGAT-AGRU, a comparison experiment is conducted with several PM_2.5 concentration prediction models on the overall data set and the heating season data set.

Table 2 (prediction time step = 24 h) demonstrates the prediction results of each model on the full data set, and the prediction step of each model was uniformly set to 24 h. In the STA-ResCNN [22] model, correlation analysis technology is used to screen the spatial information of pollution and meteorology and combine it with the time series to complete the prediction of PM_2.5 concentration in the city. WGAT-AGRU achieves a minimum RMSE of 16.61 (9.3% lower than MLP) and a minimum MAE of 13.47 (8.8% lower than MLP). And WGAT-AGRU shows the best fitting ability with a maximum R2 of 66.71%.

Fig. 3 shows the multi-step PM_2.5 concentration prediction results of the comparison models on the full data set, taking Beijing as an example. In each sub-figure, the continuous purple dash is the real value of PM_2.5 concentration in Beijing over a period of time, and the other color dashes are the multi-step prediction sequences output by the comparison model. And the multi-step prediction results are displayed every four prediction time steps. The overall fitting ability of the multi-step prediction model can be measured by observing how well the multi-step prediction sequence fits the true values. We can see that the PM_2.5 concentration in Beijing is extremely high in the mid-term. It can be seen that the prediction results of GC-LSTM and WGAT-AGRU are good and better than GRU and MLP.

Table 3 demonstrates the prediction results of each model on the heating season data set, and the experimental setup is the same as on the full data set. WGAT-AGRU achieves optimal results on the heating season dataset with a minimum RMSE of 26.71, a minimum MAE of 21.70, and a maximum R2 of 59.23%, which shows the best prediction accuracy. And WGAT-AGRU also shows the best fitting ability with the maximum CSI of 61.02%, the maximum POD of 75.11%, and the minimum FAR of 23.71%.

Fig. 4 shows more visually the multi-step PM_2.5 concentration prediction capability of the comparison models on the heating season data set, taking Xi’an as an example. The prediction result of the GC-LSTM is similar to the prediction result of the WGAT-AGRU. They both achieve better prediction results than the other two models, GRU and MLP. However, comparing the prediction results of WGAT-AGRU and GC-LSTM, it can be found that the concentration curve predicted by WGAT-AGRU fits the true value of the change better. For example, WGAT-AGRU performs better for the slightly decreasing trend of PM_2.5 concentration at the 10-time step, 30-time step, and 40-time step.

To evaluate the long-term prediction performance of the prediction models, experiments were conducted at different prediction time steps, including 24, 36, and 48 h in this paper. The long-term PM_2.5 concentration prediction performance of the models was compared by analyzing the degree of decay of the PM_2.5 concentration prediction accuracy of each model with the increase of the prediction step. The experimental results are shown in Table 2. And we can see that the experimental metrics of the proposed model are optimal at each prediction step.

The prediction results of PM_2.5 concentration for the next 72 h and the RMSE variation with the prediction step from each model with prediction steps are shown in Figs. 5 and 6. As we can see that the WGAT-AGRU model still maintains an excellent fitting ability for PM_2.5 concentration prediction in a long prediction time, and can accurately capture the trend of PM_2.5 concentration compared with other prediction models.

In order to verify the effectiveness of each module of our prediction model, ablation experiments are conducted on the full dataset. The validity of the PM_2.5 directed flow graph structure, spatial correlation feature extraction module, and spatiotemporal fusion prediction module on the model are analyzed in this paper, respectively.

The traditional PM_2.5 spatial correlation graph DG-INPUT constructs an adjacency matrix by determining whether the distance between city vertices exceeds the threshold and whether the edge weight is the reciprocal of the distance between cities. DG-INPUT and the flow graph FG-INPUT proposed in this paper are used as inputs for experiments on the full data set, respectively, with the same prediction time step set to 24 h. The results are shown in Table 4 (different graph structure input modules). We can see from the result that FG-INPUT improves the model prediction accuracy with the minimum RMSE and MAE because it simulates the impact of inter-city PM_2.5 flow transmission well.

To verify the effectiveness of the spatial correlation feature extraction module in the model, WGAT is replaced with the fully connected network (FC), and ablation weight aggregation component (GAT), ablation graph attention component (GNN), respectively. And they are combined with AGRU. The experiments on the full data set are conducted and the prediction step size is uniformly set to 24 h. The results are shown in Table 4 (different spatial correlation feature extraction modules). We can see that the combination of WGAT can effectively improve the prediction result of the model.

To analyze the effectiveness of the spatiotemporal fusion prediction module, WGAT is combined with the fully connected network (FC), ablation temporal attention component (GRU), and the spatiotemporal fusion prediction module (AGRU) of this paper, respectively. Three prediction steps of 24, 36, and 48 h are set, to analyze the effects of each component above on the multi-step prediction of PM_2.5 concentration. The experimental results are shown in Table 5. The proposed model WGAT-AGRU achieves the best results in most cases.

In this paper, the proposed model WGAT-LSTM with fused spatiotemporal features achieves the best result in most cases. It can be seen from Fig. 3, WGAT-AGRU performs better on the full data set compared with MLP. The MLP model cannot capture the time-dependent relationship, and it cannot predict such drastic changes, and is a poor fit for the peak PM_2.5 concentration. While the model proposed in this paper can cope with this situation well and predict the concentration change trend more accurately. And WGAT-AGRU can capture the directional PM_2.5 flow through the PM_2.5 directed flow graph. GC-LSTM is a graph convolutional neural network that extracts the spatial features between city vertices in the graph structure and achieves spatiotemporal prediction by combining it with LSTM. Thus the prediction result of GC-LSTM is similar to the prediction result of the WGAT-AGRU model. However, during the heating season, when more coal is burned in northern cities in China and the prevailing north wind makes the inter-city PM_2.5 concentration more affected by wind propagation, WGAT-AGRU achieves better results as shown in Figs. 4c and 4d. It can be inferred that extracts the PM_2.5 spatial flow feature between city vertices in the PM_2.5 flow map through WGAT. By analyzing the multi-step prediction ability of the model, it can be found that WGAT-AGRU has better prediction ability when the model is forecasting for a long time. FC cannot capture long-term time dependencies leading to poorer prediction results. When the predicted step size is 24 h, which is the single step length, our multi-step prediction model does not fully reflect the advantages as shown in Table 5 (prediction time step = 24 h).