2.1. Spatiotemporal Characteristics of the BF Ironmaking Process

A BF is a large-scale countercurrent reactor with intensive heat and mass transfer. Its internal conditions evolve through coupled physicochemical reactions occurring in multiple zones, including the lump, cohesive, dripping, raceway, and hearths. Among these zones, the hearth plays a critical role because its thermal state directly determines the residual lining thickness and campaign safety ¹⁹. Therefore, understanding the spatiotemporal evolution of the hearth lining temperature is essential for establishing an effective prediction model.

From a modeling perspective, two fundamental characteristics make the prediction problem extremely challenging:

1. (1)

   The operational parameters affect the lining temperature with distinct lags. Changes in the gas flow and thermal load appear after 2–4 hours, while cooling adjustments respond within 1–2 hours. These uneven delays require models that can represent variable temporal dependencies.

2. (2)

   BF variables interact rather than acting independently. Local changes in the blast temperature, pressure, or permeability propagate and couple with other factors, generating complex spatial correlations and structures in multivariate time series that extend beyond point-wise temporal models.

Conventional data-driven models often fail to simultaneously capture these temporal and spatial complexities, resulting in inaccurate predictions and limited operational utility. BFs are complex reactors involving a series of chemical and physical reactions. Inside a BF, iron ore and other iron-containing compounds undergo redox reactions with coke at high temperatures and pressures to produce molten iron and BF gas ²⁰. Fig. 1 illustrates the BF ironmaking process.

2.2. Integrated Spatiotemporal Modeling Framework

To address the aforementioned challenges, we propose an integrated spatiotemporal modeling framework designed for accurately forecasting the BF hearth lining temperature. The core idea is to model multi-scale temporal dependencies and inter-variable spatial correlations using a specialized neural network.

The overall structure of the proposed model is shown in Fig. 2. It comprises two synergistic components:

1. (1)

   Temporal feature extraction using the GRU. GRU networks were employed to capture long-term temporal dependencies and cumulative process effects while avoiding vanishing gradients. They provide a compact representation of historical evolution by generating hidden states for multivariate inputs.

2. (2)

   Feature-dimensional attention for non-uniform delay modeling. A feature-dimensional attention mechanism is introduced to dynamically evaluate the relevance of each variable at each time step. This component enables the model to automatically learn variable-specific influence intensities and effectively approximate heterogeneous delay vectors.

3. (3)

   CNN-based local spatial feature perception. To capture the spatial correlations among variables within a short time span, a CNN module was applied to the GRU-generated hidden state matrix. This operation recognizes localized structural patterns, such as rising trends, coordinated fluctuations, and segmental correlations, which reflect the underlying metallurgical interactions.

Together, these modules form an integrated spatiotemporal architecture that transforms raw BF operational data into a physically meaningful and highly predictive representation of the hearth lining temperature.

3.1. Problem Formulation and Modeling Rationale

Accurate prediction of the hearth lining temperature in the BF requires the formalization of the complex spatiotemporal relationships between process variables and the target quality variable ²¹. The problem is defined as follows: given \(N\) process variables, each forming a time series of length \(M\), the objective is to predict the future value of quality variable \(y_t\) (hearth lining temperature) at time \(t\).

Conventional approaches often rely on simplified temporal assumptions. The most basic form is the synchronous model that assumes all variables exert an instantaneous influence at time \(t\).

A common refinement is the uniform-lag model that introduces a single time delay \(\tau\) for all variables

Although these formulations provide tractable baselines, they fundamentally misrepresent the BF ironmaking process. The synchronous model ignores temporal causality, whereas the uniform-lag model fails to capture the empirically observed variable-specific delay times that range from one to four hours for different parameters.

A more realistic approach involves modeling a distinct time delay \(\tau_{i}\) for each variable \(i\), defined by a delay vector

This formulation accounts for non-uniform delays. However, it remains limited by focusing on a single discrete historical time step for each variable, thereby neglecting the cumulative effects of the entire recent history of each variable and the dynamic interactions between variables. To overcome these limitations, a comprehensive spatiotemporal feature representation is proposed. This representation integrates the cumulative influence of all variables over a historical window of length \(L\), weighted by their dynamically computed importance

3.2. Temporal Feature Extraction Using GRU and Feature Attention

The extended smelting cycle of a BF necessitates modeling the continuous cumulative effects of the process variables over time. Conventional approaches struggle to handle the non-uniform time delays inherent in the process, in which different parameters affect the target quality variable over varying lag times ²². To address this issue, a multi-scale temporal feature extraction network is presented, integrating GRU with a feature-dimension attention mechanism.

The GRU architecture serves as the foundational temporal modeling component, selected for its efficacy in capturing long-range dependencies while mitigating the vanishing gradient problem ²³. Compared with the LSTM network, the GRU adopts a more streamlined gating mechanism that often results in faster convergence and comparable performance on industrial-scale datasets. This capability to learn complex temporal patterns is further enhanced by a feature-dimension attention mechanism that explicitly addresses the challenge of non-uniform variable influence.

As illustrated in Fig. 3, the attention component dynamically recalibrates the importance of each process variable at every time step by performing soft-feature weighting based on its instantaneous relevance to the prediction target. Together, this integrated design enables the model to model the evolution of process states and adaptively focus on the most pertinent variables from the high-dimensional input space, directly addressing the core challenges of dynamic coupling and variable-specific time delays.

The overall architecture of the proposed temporal feature extraction model is shown in Fig. 4. The input is a multivariate time series \(X = [x_{1}, x_{2}, \dots, x_{T}]\), where \(x_{t} = (x_{t}^{1}, x_{t}^{2}, \dots, x_{t}^{n}) \in \mathbb{R}^{n}\) denotes the values of \(n\) process variables at time \(t\). The model processes this input through three coordinated stages to generate a dynamically weighted temporal representation.

The process begins with the input sequence being encoded by a GRU network that captures the fundamental temporal dependencies and produces a sequence of hidden states \(H = [h_{1}, h_{2}, \dots, h_{T}]\). Each hidden state \(h_t\) is updated based on current input \(x_t\) and previous hidden state \(h_{t-1}\), following the gating mechanisms defined in Eqs. 5–8.

Here \(r_t\) and \(z_t\) denote the reset and update gate, respectively; \(h'_{t}\) denotes the candidate hidden state; \(W_r\), \(W_z\), and \(W_h\) denote learnable weight matrices; \(\odot\) denotes the element-wise multiplication.

A feature-dimension attention layer is applied at each time step to dynamically quantify the influence of each variable ²⁴. This layer evaluates the relevance of current input \(x_t\) to previous context \(h_{t-1}\) by computing an unnormalized attention score

The original input vector \(x_t\) is modulated by these weights to form a refined input vector

3.3. CNN-Based Local Spatial Feature Perception

Although the temporal model, as described in Section 3.2, effectively captures dynamic dependencies and non-uniform delays, it primarily operates on pointwise interactions between current input \(\tilde{x}_t\) and historical context \(h_{t-1}\). This approach computes feature weights based on the similarity between discrete points and may overlook richer local spatial patterns (for example, sequential shapes and structural trends) that exist among multiple variables over short time windows. Ignoring these inter-variable correlations can lead to feature mismatches during training, thereby limiting the prediction accuracy and model interpretability.

3.3.1. Motivation for Local Spatial Feature Optimization

Process variables temporally affect the quality variable and exhibit local spatial characteristics that reflect mechanistic interactions across different furnace zones ²⁵. These local patterns are critical to accurately represent the underlying process dynamics and ensure robust feature learning.

In this study, the term “spatial” refers to the feature space rather than the physical geometric zones of the BF. Specifically, it denotes the structural correlations and proportional relationships between the different process variables in the input matrix. Although conventional correlation coefficients capture global linear relationships, they often fail to recognize local structural patterns (such as synchronized fluctuations across multiple sensors) within short time windows. The CNN module addresses this by leveraging its sliding window mechanism to capture the local “shapes” in the feature matrix.

As shown in Fig. 5(a), we consider three feature points (a, b, and c) within a multivariate time window that exhibit similar proportional relationships. A point-wise similarity method (as described in Section 3.1) evaluates the matches between (a, b), (a, c), and (b, c) with equal probabilistic considerations. However, as illustrated in Fig. 5(b), points b and c share higher similarity in terms of local trend structure and contextual causality, making them the more meaningful match from a process perspective. Therefore, relying solely on point-wise attention can introduce mismatches that accumulate during training, leading to distorted feature representations and reduced prediction accuracy.

To address this issue, CNNs are introduced for local spatial feature perception. CNNs are well-suited for this task because of their sliding kernel mechanism that provides inherent translation invariance and enables robust pattern recognition within local receptive fields. By capturing inter-variable structures over short sequences, CNNs complement the temporal modeling capabilities of GRU and attention mechanisms, resulting in a more holistic spatiotemporal feature representation.

3.3.2. CNN-Based Local Spatial Feature Perception Model

The CNN module is applied to hidden state matrix \(H = [h_{1}, h_{2}, \dots, h_{T}]\) that is output by the GRU network, where each \(h_{t} \in \mathbb{R}^{d}\). To preserve the sequence boundaries, the matrix is padded appropriately, forming an expanded matrix \(Y \in \mathbb{R}^{d \times (T+l-1)}\), where \(l\) denotes the convolutional kernel size.

Let \(W^{k} \in \mathbb{R}^{m \times d \times l}\) and \(W^{q} \in \mathbb{R}^{m \times d \times l}\) denote the learnable kernel parameters for the key and query mappings, respectively, where \(m\) denotes the number of filters. The convolutional operation for the key matrix at position \(i\) is computed as follows:

\begin{equation} o_{i}^{k} = \begin{bmatrix} W_{1}^{k} \cdot Y[i:i+l-1] \\ W_{2}^{k} \cdot Y[i:i+l-1] \\ \vdots \\ W_{m}^{k} \cdot Y[i:i+l-1] \end{bmatrix}. \label{eq:eq12} \end{equation}

Subsequently, key vector \(K_i\) is obtained by

\begin{equation} K_{i} = \tanh \bigl(o_{i}^{k} + b^{k}\bigr), \label{eq:eq13} \end{equation}

where \(b^k\) represents a learnable bias term. The query \(o^q\) matrix is derived from the most recent \(l\) steps of the hidden state matrix \(Y\).

\begin{equation} o^{q} = \begin{bmatrix} W_{1}^{q} \cdot Y[T-l+1:T] \\ W_{2}^{q} \cdot Y[T-l+1:T] \\ \vdots \\ W_{m}^{q} \cdot Y[T-l+1:T] \end{bmatrix} \label{eq:eq14} \end{equation}

Then, query vector \(Q \in \mathbb{R}^m\) is then given by

\begin{equation} Q = f \bigl(o^{q} + b^{q}\bigr), \label{eq:eq15} \end{equation}

where \(f(\cdot)\) represents a nonlinear projection function. The final output of the quality variable is computed as follows:

\begin{equation} y = \sigma \left(\operatorname{softmax} \left(\frac{Q^{T}K}{\sqrt{m}}\right) H^{T} W + b\right), \label{eq:eq16} \end{equation}

where \(\sigma(\cdot)\) denotes the sigmoid activation, and \(W\) and \(b\) denote the learnable parameters of the output layer.

The complete architecture that integrates the proposed CNN-based spatial perception module with a temporal feature extraction network is shown in Fig. 6. This integrated design enhances feature alignment, reduces mismatches, and improves the overall robustness and accuracy of hearth lining temperature prediction.

4.1. Data Preprocessing and Input Variable Selection

A BF is an ultra-large nonlinear integrated reactor, and its hearth lining temperature is influenced by numerous parameters. Based on the process mechanism analysis, as described in Section 2.2, and expert experience, the following parameters affecting the lining temperature were identified across the ironmaking timeline: raw material input properties, operational parameters during smelting, and output hot metal characteristics.

The experimental dataset was collected from a large-scale industrial BF over two days (May 15–16, 2023), yielding 12,000 valid samples. Although the time span was relatively short, the sampling frequency was high, and the furnace was operated under stable and representative working conditions, ensuring that the dataset effectively captured the thermal and operational dynamics relevant to the hearth lining temperature prediction. Similar short-range high-resolution datasets have been widely used in BF-related temperature forecasting studies.

Considering the data quality and integrity, 28 parameters across the four categories were initially selected as characteristic inputs, as summarized in Table 1. Given the strong nonlinearity and coupling in BF processes, the maximal information coefficient (MIC) was employed instead of the Pearson correlation coefficient to evaluate both the linear and nonlinear relationships between the candidate variables and the target temperature. The MIC is calculated as follows:

The preprocessed dataset comprised 12,000 samples collected from a steel plant between May 15 and 16, 2023. It was split into training (9,000 samples) and test sets (3,000 samples) at a ratio of \(3:1\), with an additional validation set (1,500 samples) extracted from the training set to monitor the model training and prevent overfitting. To ensure data quality, the raw dataset was rigorously preprocessed. Outliers caused by sensor malfunctions were identified and removed using the \(3\sigma\) rule, and missing values were filled via linear interpolation. For variable selection, the MIC threshold was set to 0.3; variables with MIC values below this threshold were considered weakly correlated and excluded from the input set.

Although the dataset spanned a relatively short period (two days), the high sampling frequency yielded 12,000 samples that was sufficient for training the deep learning model to convergence. The selected period represents a stable operation, making it suitable for validating the proposed methodology, although long-term seasonal trends remain a subject for future studies.

4.2. Verification of the Attention Mechanism

The standard LSTM algorithm was selected as the baseline to validate the effectiveness of the proposed attention mechanism. Comparative experiments were conducted between the standard LSTM and an enhanced variant, namely feature attention-LSTM (FLSTM) that integrated the feature-dimension attention mechanism proposed in this study.

Figure 7 illustrates the features of the weight distribution enhanced by the attention mechanism. Notably, the contribution weights of different features at the same time step are adaptively adjusted during the training process. When combined with temporal attention, the model further dynamically modified the influence weights of different time steps for the same feature, thereby effectively achieving nonlinear time-delay adaptation within the sequence window. Notably, this learned distribution aligns well with the metallurgical time lags discussed in the Introduction section. As observed in Fig. 7, variables related to gas flow (for example, Sensor id: 4) exhibited higher attention weights at earlier time steps in the history window (reflecting a lag of \(>\) 2 h), whereas cooling-related variables (for example, Sensor id: 6) showed a peak influence at more recent time steps. This correspondence confirmed that the model successfully captured the underlying physical characteristics of the BF process.

The prediction results on the test set are presented in Fig. 8. The cyan curve represents the actual hearth lining temperature, blue solid line denotes the prediction results of the standard LSTM, and red solid line indicates the prediction results of the FLSTM model. The model performance was evaluated using two metrics, hit rate (HR) and root mean square error (RMSE), that are defined as follows:

Thresholds \(\delta\) were selected based on the industrial operational standards. A threshold of 0.1 corresponds to a 10% relative error that is acceptable for general trend monitoring, whereas 0.01 represents a stricter 1% precision required for high-risk safety alerts.

As summarized in Table 2, the FLSTM model significantly outperformed the standard LSTM, with RMSE reduced from 0.1758 to 0.0976 and the HR(0.1) increased from 29.93% to 68.32%. This demonstrated that the attention mechanism markedly enhanced the ability of the model to track the hearth lining temperature and improved the prediction accuracy.

4.3. Functional Verification of the Integrated Spatiotemporal Prediction Model

An experimental approach to hearth lining temperature prediction was conducted based on the full spatiotemporal feature fusion method described in this subsection. Consistent with the previous experiment, the dataset was split into training (9,000 samples) and test sets (3,000 samples), with a validation set (1,500 samples) extracted from the training set to optimize the hyperparameters.

During model training, the input sequence length and convolutional kernel size were identified as the key hyperparameters. Considering the model depth, potential overfitting risks, and relatively short time lag of lining temperature responses to internal reactions, the input sequence length was set to 15 based on the maximum process lag (approximately 4 h) observed in the domain knowledge. A sensitivity analysis via grid search indicated that a kernel size of 3 offered the best trade-off between feature extraction capability and computational efficiency. Regarding the computational cost, the average inference time of the proposed feature-time attention (FTA) model was approximately 15 ms per sample on a standard workstation. This speed is significantly faster than the data acquisition interval of the BF SCADA system, confirming the model’s feasibility for real-time online deployment.

The integrated model (denoted as FTA) dynamically predicts hearth lining temperature through a sequential multi-step process: it first uses GRU units to extract hidden state information from feature sequences, then applies a feature-dimension attention mechanism to weight variables according to their similarity to historical states, subsequently employs a CNN module to capture deep local spatial patterns from the hidden state matrix, and finally utilizes a self-attention mechanism to fuse the extracted spatiotemporal features before inputting them into the output layer to compute the final prediction of the hearth lining temperature.

Traditional shallow machine learning models, such as SVR or ELM, were not included in this comparison. Because the hearth lining temperature is governed by cumulative physicochemical reactions with distinct time lags, deep recurrent architectures (such as LSTM and GRU) are theoretically superior to static models in capturing these temporal dependencies.

For performance validation, the proposed FTA model was compared with standard LSTM and GRU models. The prediction curves and actual values are shown in Fig. 9, and the performance metrics are compared in Table 3. The FTA model achieved a superior RMSE of 0.0523 and an HR(0.1) of 94.26%, significantly outperforming the LSTM (RMSE: 0.1758, HR(0.1): 25.93%) and GRU (RMSE: 0.0973, HR(0.1): 67.06%) benchmarks.

The prediction of the BF hearth lining temperature was constrained by the diversity of the operational conditions and the complexity of the state parameters, exhibiting strong nonlinear and time-varying characteristics. This attention mechanism effectively addressed variable-specific time delays by continuously updating and quantifying the influence of each variable at different time steps. Furthermore, the CNN module enhanced the perception of local spatial patterns among variables, thereby reducing feature mismatches. These results confirmed that the constructed hearth lining temperature prediction model was effective and that the proposed spatiotemporal feature extraction framework achieves significantly higher accuracy than conventional models within the specified error range.