Statement autoformalization, a crucial first step in formal verification, aims to transform informal descriptions of math problems into machine-verifiable formal representations but remains a significant challenge.The core difficulty lies in the fact that existing language models hallucinate formal dependencies, including missing or incorrect definitions, lemmas, and theorems.Current dependency retrieval approaches exhibit poor precision and recall, and lack the scalability to leverage ever-growing public datasets.To bridge this gap, we propose a novel retrieval-augmented framework based on Direct Dependency Retrieval (DDR).DDR directly generates candidate formal dependencies from natural-language mathematical descriptions and verifies their existence in the formal library via an efficient Suffix Array Check (SAC).Built on a SAC-constructed dependency retrieval dataset of over 500,000 samples, a high-precision DDR model is fine-tuned and shown to significantly outperform state-ofthe-art methods in both retrieval precision and recall, leading to superior advantage in the autoformalization tasks.SAC also contributes in assessing formalization difficulty and enabling explicit quantification of the hallucination in In-Context Learning (ICL).

The Industrial Internet of Things (IIoT) has become a crucial infrastructure in the process industry, particularly in the era of Industry 4.0. Ensuring operational safety in industrial processes necessitates fault detection techniques, which play a pivotal role in IIoT systems. These systems continuously collect high-dimensional process data, which often exhibit dynamic behavior due to the inherent complexity of industrial operations. Consequently, the dynamic characteristics of such data pose significant challenges for fault detection. As a powerful dimensionality reduction technique, Dynamical Component Analysis (DyCA) decomposes multivariate measurements of a dynamical system into a deterministic component which can be described by a system of differential equations and independent noise components. DyCA incorporates the covariance matrices of both the signals, and their derivative, as well as their cross-correlation. By doing so, it identifies a low-dimensional subspace that minimizes the error in the underlying ordinary differential equations. The DyCA components are estimated to capture low-dimensional trajectories that characterize the process dynamics. This study proposes a novel data-driven fault detection method based on dynamical component analysis for dynamic processes. Leveraging these DyCA components that represent the low-dimensional trajectories to describe the process dynamics, Hotelling’sT2and Square Prediction Error (SPE) statistics are utilized as monitoring metrics for fault detection. Case studies on the widely utilized Tennessee Eastman process benchmark and a real-world blast furnace ironmaking process are conducted to demonstrate the effectiveness and capability of the proposed DyCA based fault detection method, comparing its performance with other relevant methods.

The blast furnace ironmaking process (BFIP) serves as the core unit within the iron and steel industry. However, its harsh operating environment often leads to process abnormalities, resulting in unplanned shutdowns and product quality degradation. Data driven fault detection is crucial for ensuring operational safety and maintaining product quality in this context. Nevertheless, BFIP data are inherently characterized by strong dynamic and highly correlated properties, which present significant challenges for effective fault detection. To address these challenges, this work proposes a novel data driven fault detection method based on Dynamic and Canonical Variate Dissimilarity Analysis (DyCVDA). Within the developed DyCVDA framework, high-dimensional process data are first decomposed into deterministic components and noise components by optimally satisfying a set of coupled ordinary differential equations, where deterministic components consist of time-dependent amplitudes and corresponding multivariate modes. Thereby, the process’s dynamic characteristic can be captured. Subsequently, low-dimensional subspaces of deterministic components are extracted to represent the underlying dynamic behavior. Canonical subspaces are further derived by maximizing the correlation between past and future observations of these low-dimensional projections to exploit the correlation structure within the process data. Finally, a canonical variate dissimilarity index is employed as the monitoring statistic for fault detection. Experimental results on two case studies, including the popular Tennessee Eastman Process industrial benchmark and a real world blast furnace ironmaking process, demonstrate the effectiveness of the proposed DyCVDA method, with favorable performance comparisons against other relevant approaches.

Anomaly detection is a cornerstone of industrial safety, enabling real-time monitoring of process operations by identifying deviations from normal conditions through statistical analysis. In real-world industrial scenarios, the nonstationary properties of multivariate time-series data present a common and substantial challenge. Existing methods for extracting stationary sources $(\mathcal {SS}s)$ mainly rely on weak stationarity (i.e., mean and variance), but their performance is limited by the long-tailed distributions common in industrial datasets. Higher-order moments, in contrast, provide a more comprehensive statistical description, capturing complex data characteristics that the mean and variance overlook. To bridge this significant gap, we propose a continuous stationary moment analysis (Co-SMA) anomaly detection framework. Its core innovation is the SMA algorithm, which introduces a novel objective function to minimize cumulative sum of the differences in multiorder moments between each epoch and the overall data, effectively fulfilling the $\mathcal {SS}$ estimation task. Furthermore, to overcome the inefficiencies of traditional model updating methods, we develop an event-triggered model updating framework based on the model bias index and first-order perturbation theory. Within this framework, we introduce a convex hull coverage metric, which enables the model to be adjusted efficiently according to the data distribution drift. The framework also incorporates iterative refinement of detection statistics and thresholds, establishing a dynamic adjustment mechanism that ensures optimal performance across diverse operating conditions. The theoretical basis of Co-SMA's properties is rigorously established. Experimental evaluations on numerical simulations and real-world datasets from the ironmaking process demonstrate Co-SMA's superior capabilities in $\mathcal {SS}$ estimation and anomaly detection.

Sintering, as a preliminary step in the blast furnace, has a profound influence on the ultimate quality of the iron product. Accurate forecasting of the chemical composition during sintering operations has become crucial to facilitate the production of higher quality inputs for downstream processes. However, modeling sintering is complex due to its long timescales, multistage transfers, and intricate redox reactions. To address this, a novel regression neural network based on orthogonal basis decomposition and reconstruction with implicit subspace identification is proposed. First, a recursive Fourier-transform-like enoding block is implemented to extract feature capturing long-term memory via orthogonal basis decomposition. Subsequently, an stochastic-gradient-based identification algorithm is used to approximate the ground truth system and model the output. The feasibility and utility of the approach are demonstrated using simulated and real-world sintering plant data. Considering encoding and identification separately offers deeper insights into sintering processes, resulting in enhanced explicability of model behaviors and a significant improvement of 22.15% loss reduction in forecasting performance.

Data-driven soft sensor modeling has received much attention in industrial processes. Most of the existing soft sensor approaches have not considered the complex dynamic spatial coupling characteristics between process variables. Recently, graph-based soft sensor modeling methods have started to show powerful expressive ability in capturing relational dependencies. However, existing graph-based soft sensor models still confront several limitations: 1) these models usually depend on predefined graph structures or local dynamic graph; 2) they fail to study dynamic message passing mechanism; 3) they have not considered the importance of extracted features from the entire graph. To handle these problems, in this study, we develop a dynamic adaptive message passing neural network (DAMPNN) for industrial soft sensor. The main novelty lies in an integration of our designed three modules into DAMPNN. First, we propose an adaptive graph learning module to automatically capture mutual relationships between process variables instead of a predefined adjacency matrix. Then, we design a dynamic message passing module to aggregate neighborhood information and update graph representation. In addition, a dual self-attention module is embedded into the top layer to concurrently emphasize informative features and time points for fine-grained soft sensor modeling. Finally, comprehensive comparison results on two real-world industrial cases demonstrate that DAMPNN outperforms the existing graph-based soft sensor methods.

The last decade has witnessed the growing prevalence of deep models on soft sensing in industrial processes. However, most of the existing soft sensing models are developed to learn from regular data in the Euclidean space, ignoring the complex coupling relations among process variables. On the other hand, graph networks are gaining attraction in handling non-Euclidean relations in industrial data. However, the existing graph networks on soft sensing models still suffer from two major issues: 1) how to capture the intervariable structural relations and intravariable temporal dependencies from dynamic and strongly coupled industrial data and 2) how to learn from nodes with distinctive importance for the soft sensing task. To address these problems, we propose a self-learning evolutionary and node-aware graph network (SENGraph) for industrial soft sensing. We first develop a self-learning graph generation (SLG) module to combine the coarse- and fine-grained graphs to capture the global trend and local dynamics from process data. Then, we build a self-evolutionary graph module (EGM) to obtain diversified node features from the entire graph using mutation and crossover strategies. Finally, we design a node-aware module (NAM) to highlight the informative nodes and suppress the less significant ones to further improve the discriminative ability of the downstream soft sensing. Extensive experimental results and analysis on four real-world industrial datasets demonstrate that our proposed SENGraph model outperforms the existing state-of-the-art (SOTA) soft sensing methods.

Burn-through point (BTP) is a very key factor in maintaining the normal operation of the sintering process, which guarantees the yield and quality of sinter ore. Due to the characteristics of time-varying and multivariable coupling in the actual sintering process, it is difficult for traditional soft-sensor models to extract spatial-temporal features and reduce multistep prediction error accumulation. To address these issues, in this study, we propose a probabilistic spatial-temporal aware network, called BTPNet, which is used to extract spatial-temporal feature for accurate BTP multistep prediction. The BTPNet model consists of two parts: an encoder network and a decoder network. In the encoder network, the multichannel temporal convolutional network (MTCN) is employed to extract the temporal features. Meanwhile, we also propose a novel architectural unit called variables interaction-aware module (VIAM) to extract the spatial features. In the decoder network, to reduce the accumulated errors of the last step prediction, a probabilistic estimation (PE) method is proposed to improve the performance of multistep prediction. Finally, the experimental results on a real sintering process demonstrate the proposed BTPNet model outperforms state-of-the-art multistep prediction models.

Broad learning system (BLS), a tri-layer feedforward neural network, has gained widespread recognition for its exceptional scalability and computational efficiency. However, BLS and its derivatives encounter several challenges: 1) overlooking the uncertainty introduced by numerous nonlinear random mappings; 2) failing to cope with the misalignment of model inputs with the output sampling rate; 3) lack of attention to nonstationary scenarios; and 4) absence of theoretical optimization for incremental learning. To overcome these obstacles, we propose a regression modeling and anomaly detection scheme rooted in a temporal kernel stationary BLS (TKS-BLS). We first create a nonlinear kernel broad representation (NKBR) extraction strategy, providing a robust nonlinear foundation for random feature mapping via kernel technology. Following this, we probe the mechanism of temporal matching between model inputs and outputs through a temporal alignment parameter, interpretable under a latent variable relationship. In the integration phase, we establish a Kullback-Leibler divergence objective function to facilitate the capture of stationary relationships within time-series data, in conjunction with the regression error. Subsequently, a double-loop parameter optimization algorithm and an independent incremental learning mechanism are put forth, both backed by comprehensive theoretical analyses. Our method’s superiority is thoroughly confirmed by experimental outcomes from extensive case studies across seven real ironmaking process datasets.

Semantic segmentation is one of the tasks concerned in the field of computer vision. However, the cost of capturing large numbers of pixel-level annotations is expensive. Semi-supervised learning can utilize labeled and unlabeled data, providing new ideas for solving the problem of insufficient labeled data. In this work, we propose a data-reliability weighted multi-phase learning method for semi-supervised segmentation (RWMS). Under the framework of self-training, we train two different teacher models to evaluate the reliability of pseudo labels. By selecting reliable data at the image level and reweighting pseudo labels at the pixel level, multi-phase training is guided to focus on more reliable knowledge. Besides, we also inject strong data augmentations on unlabeled images while training. Through extensive experiments, we demonstrate that our method performs remarkably well compared to baseline methods and substantially outperforms them, more than 3% on VOC and Cityscapes.

Neural networks (NN) are extensively studied in cutting-edge soft sensor models due to their feature extraction and function approximation capabilities. Current research into network-based methods primarily focuses on models' offline accuracy. Notably, in industrial soft sensor context, online optimizing stability and interpretability are prioritized, followed by accuracy. This requires a clearer understanding of network's training process. To bridge this gap, we propose a novel NN named the Approximated Orthogonal Projection Unit (AOPU) which has solid mathematical basis and presents superior training stability. AOPU truncates the gradient backpropagation at dual parameters, optimizes the trackable parameters updates, and enhances the robustness of training. We further prove that AOPU attains minimum variance estimation in NN, wherein the truncated gradient approximates the natural gradient. Empirical results on two chemical process datasets clearly show that AOPU outperforms other models in achieving stable convergence, marking a significant advancement in soft sensor field.

Blast furnace iron-making process (BFIP) is one of the most critical procedures in the iron and steel industry where timely detection and accurate classification of faults have always been of core focus. However, the coupling effects of system’s nonlinear and nonstationary characteristics often cause process consistent underlying information to be buried, allowing accurate extraction to be a significant challenge. This also complicates the development of BFIP fault diagnosis model. Therefore, we propose a novel data-driven joint fault diagnosis strategy that employs regularized mutual kernel analytic stationary subspace analysis (RMK-ASSA) and deep broad stationary kernel network (DBSKNet) to eliminate this interference. To develop this method, we first construct an RMK-ASSA approach to address the poor modeling accuracy caused by standard analytic stationary subspace analysis (ASSA)’s inability to handle complex process nonlinearity. Global and local kernels are utilized to account for multiple nonlinearities in BFIP data. The weight of different nonlinear data is calculated by regularized principal component analysis, and the main information is imported into ASSA to obtain more robust and accurate modeling results by eliminating the interference of redundant noise. Subsequently, we design a DBSKNet-based classifier to implement the fault diagnosis task. This network further considers the nonlinearity by boosting kernel structure in depth and width while distinguishing the respective contributions of different kernels to fault diagnosis results. Finally, a double-layer loop parameter optimization algorithm is used for optimizing. Simulated cases and practical BFIP tests validate that RMK-ASSA eliminates the negative impact caused by nonstationary data and that the proposed joint fault diagnosis strategy outperforms other methods.Note to Practitioners—BFIP’s nonlinear and nonstationary coupling properties pose unique challenges in eliminating distractions, constructing fault classifiers and accurately detecting process anomalies. To tackle these challenges, this paper proposes a joint fault diagnosis strategy based on RMK-ASSA and DBSKNet. RMK-ASSA effectively estimates nonlinear consistent features, while DBSKNet mines rich deep nonlinear information, accurately distinguishing variations in BFIP data under different working conditions. Experimental results demonstrate that this data-driven strategy can perform high-quality fault diagnosis, enabling field engineers to execute operations efficiently.

Deep learning-based soft sensor modeling methods have been extensively studied and applied to industrial processes in the last decade. However, existing soft sensor models mainly focus on the current step prediction in real time and ignore the multistep prediction in advance. In actual industrial applications, compared to the current step prediction, it is more useful for on-site workers to predict some key performance indicators in advance. Nowadays, multistep prediction task still suffers from two key issues: 1) complex coupling relationships between process variables and 2) long-term dependency learning. To ravel out these two problems, in this article, we propose a graph-based time-frequency two-stream network to achieve multistep prediction. Specifically, a multigraph attention layer is proposed to model the dynamical coupling relationships between process variables from the graph perspective. Then, in the time-frequency two-stream network, multi-GAT is used to extract time-domain features and frequency-domain features for long-term dependency, respectively. Furthermore, we propose a feature fusion module to combine these two kinds of features based on the minimum redundancy and maximum correlation learning paradigm. Finally, extensive experiments on two real-world industrial datasets show that the proposed multistep prediction model outperforms the state-of-the-art models. In particular, compared to the existing SOTA method, the proposed method has achieved 12.40%, 22.49%, and 21.98% improvement in RMSE, MAE, and MAPE on the three-step prediction task using waste incineration dataset.

In this article, a novel time-constrained global and local nonlinear analytic stationary subspace analysis (Tc-GLNASSA) is proposed to enhance blast furnace ironmaking process (BFIP) monitoring. Although the existing analytic stationary subspace analysis method has been available for deriving process consistent relationships. However, the presence of complex nonlinear, periodic nonstationary, and time-varying smelting conditions renders the satisfactory estimation of stationary projections unattainable. To this end, we leverage multiple kernel functions and manifold learning methods to establish a global and local nonlinear structure with time constraints, which will identify the unique nonlinearities excited by periodic nonstationarity. Meanwhile, a singular value decomposition-based modeling efficiency promotion strategy is constructed to reduce the proposed Tc-GLNASSA's computational complexity significantly. The orthogonality of model update scheme is analyzed theoretically, and an overall BFIP monitoring framework is given. Ultimately, practical BFIP case studies fully demonstrate the effectiveness of our proposal.

As the infrastructure industry continues to evolve toward digitalization, ongoing development of spatial and temporal data-based intelligent sensing guarantees safe operation. However, blast furnace ironmaking processes (BFIP) encounter a tricky dilemma in this revolution. Data-driven multivariate statistical analysis always fails for expected diagnosis performance due to complex dynamic, nonlinear, and nonstationary characteristics. To address this issue, we propose a novel method named modified mixed kernel-aided canonical stationary variate analysis (M2KCSVA). To start with, the past and future matrices and multiview nonlinear mapping of mixed kernel are properly considered to explore canonical stationary variables (CSVs) for both temporal correlation and weak stationarity. Especially, efficiency improvement procedures based on singular value decomposition and iterative modeling flow are deployed to reduce the computational cost and estimate accurate CSVs. In addition, we retain the smooth information without autocorrelation in the residuals for further analysis using stationary subspace analysis to generate static stationary variables. The corresponding two statistics and exponential difference contributions are computed for simultaneous fault detection and identification with an intuitive interpretation of dynamic and static stationary information. Experiments through an actual BFIP demonstrate that M2KCSVA surpasses comparison methods in terms of efficiency and accuracy.

Key performance indicators (KPIs), such as product quality variables or critical parameters in major units, play a crucial role in ensuring the desired performances in industrial processes. Nonetheless, focusing solely on monitoring process variables may result in the generation of nuisance alarms in response to disturbances that do not have a significant or meaningful impact on KPI variables. In this article, a novel KPI-related process monitoring method based on slow feature extraction and elastic net orthonormal subspace analysis (SFENOSA) is proposed. Traditional orthonormal subspace analysis (OSA) divides process data and KPI data subspaces into three orthonormal subspaces using least squares. To deal with the overfitting problem in high-dimensional space and enhance the robustness caused by correlated variables, the elastic net orthonormal subspace analysis (ENOSA) is developed by employing elastic net regularization in the OSA. Furthermore, to address the dynamic characteristics inherent in industrial processes, the slow feature analysis is naturally integrated into the framework of ENOSA for KPI-related process monitoring. Specifically, using the slow features extracted from process variables as the input and the KPI variables as the output, an ENOSA model is built. Based on the developed SFENOSA model, several monitoring statistics are established for KPI-related process monitoring. Experimental results on a numerical example, the well-known Tennessee Eastman process, and a real blast furnace ironmaking process demonstrate the superior performance of the proposed SFENOSA compared to the related methods.

The blast furnace is a highly energy-intensive, highly polluting, and extremely complex reactor in the ironmaking process. Soft sensors are a key technology for predicting molten iron quality indices reflecting blast furnace energy consumption and operation stability, and play an important role in saving energy, reducing emissions, improving product quality, and producing economic benefits. With the advancement of the Internet of Things, big data, and artificial intelligence, data-driven soft sensors in blast furnace ironmaking processes have attracted increasing attention from researchers, but there has been no systematic review of the data-driven soft sensors in the blast furnace ironmaking process. This review covers the state-of-the-art studies of data-driven soft sensors technologies in the blast furnace ironmaking process. Specifically, we first conduct a comprehensive overview of various data-driven soft sensor modeling methods (multiscale methods, adaptive methods, deep learning, etc.) used in blast furnace ironmaking. Second, the important applications of data-driven soft sensors in blast furnace ironmaking (silicon content, molten iron temperature, gas utilization rate, etc.) are classified. Finally, the potential challenges and future development trends of data-driven soft sensors in blast furnace ironmaking applications are discussed, including digital twin, multi-source data fusion, and carbon peaking and carbon neutrality.

In order to guarantee the smooth operation of the blast furnace ironmaking process, it is essential to consider the constraints for the optimization of this process. Due to the complexity of the process, it is challenging to obtain the description of the constraint functions and quantify the feasibility of solutions, making traditional optimization methods helpless. To address this problem, a transfer optimization framework is proposed that consists of a two-stage generation mapping algorithm, and an improved grey wolf optimizer (GWO) algorithm. The method achieves the transformation from constrained optimization to unconstrained optimization by establishing a proper mapping with distribution- and boundary-sensitive two-stage generation algorithm. Meanwhile, the density-amended GWO algorithm with adaptive search steps depended on the solution density distribution is applied to locate the optimal solutions. The intelligent transfer optimization method is validated by both numerical tests and practical data, and the results demonstrate the effectiveness of the proposed algorithm.

For the actual blast furnace ironmaking process (BFIP), sophisticated dynamic, nonlinear, and nonstationary characteristics make it hard to be modeled accurately with conventional monitoring methods. In this article, local dynamic broad kernel stationary subspace analysis (local-DBKSSA) is developed to improve the monitoring performance. Faced with complex dynamic nonlinear characteristics, a single model is considered to be unable for accurate representation. Thus, dynamic broad nonlinear features established by time shift and multikernel projection are adopted from more perspectives. Subsequently, the above features are integrated into stationary subspace analysis (SSA) to estimate stationary projections from time-varying data. In order to reduce the impact of large fluctuations and improve fault detection capability, a local statistic is further proposed. The effects of nonstationary characteristic on monitoring capability and the excellent performance of the local statistic are also theoretically analyzed. Finally, a case study based on actual BFIP data presents that the proposed method can discriminate between normal and sample faults more accurately and timely, and has better robustness to nonstationary perturbations under normal conditions by providing fewer false alarms.

Nowadays, data-driven soft sensors have become mainstream for the key performance indicators prediction, which guarantees the safety and stability of the industrial process. The typical autoencoder (AE) has been widely used to extract potential features through unsupervised pretraining and supervised fine-tuning. However, most existing studies fail to consider both the time-varying features of the process and the differences in the contributions of the hidden features to the target variable. Therefore, in this article, a stacked spatial–temporal autoencoder (S2TAE) is proposed to enhance the representation learning capability for soft sensor modeling by taking the spatial–temporal correlations into consideration. Specifically, to effectively model the temporal dependence from nearby times, a temporal autoencoder is proposed, in which a memory module is devised and integrated to learn valuable historical information. Moreover, a “feature recalibration” block is developed and embedded into the spatial–temporal autoencoder (STAE) to selectively capture more informative features and suppress the less useful ones in a supervised way. Then, multiple STAEs are stacked to construct the S2TAE network to extract more robust high-level features. Finally, the experimental results on two real-world datasets of a sorbent decontamination system (SDS) desulfurization process and a high–low transformer demonstrate that the S2TAE-based soft sensor is effective and feasible.

As a key thermal-state indicator of the iron ore sintering process, the content of ferrous oxide (FeO) in the finished sinter is directly related to product quality. Based on the massive data of sintering process, the data-driven soft sensor model provides a good choice for real-time FeO content detection. However, the complex characteristics of the data, including dynamics, nonlinearity, and multisource heterogeneity, are still the main obstacles to improving the modeling accuracy. To solve this problem, in this article, a multisource information fusion autoformer (MIF-Autoformer) model is introduced. First, feature-level information fusion and data-level information fusion are implemented based on the MIF strategy. Then, the comprehensive information of the sintering process is fed to the downstream autoformer model in a serial manner, which not only improves the information capacity but also provides additional prior information about the FeO content grade. This is helpful for autoformer to capture the complex temporal distributions in the sintering process. Finally, the proposed model is applied to a real sintering plant. Experimental results show that the hybrid image features provided by the MIF strategy have a general optimization effect on different competitive models, and MIF-Autoformer exhibits the lowest prediction error on the test set.

Blast furnace ironmaking is one of the most complicated industrial processes. As an essential reference index for blast furnace operation, the prediction of silicon content in hot metal is important. Most previous works focus on the dynamics and nonlinearity of the process without comprehensively considering the correlation between the process variables and the silicon content. Besides, the timing mismatch of input and output variables caused by the inertia of the process still cannot be handled effectively. To solve these problems, in this article, the proposed model puts extra attention on input variables to strengthen the information of key variables, and introduces causal convolution-based self-attention to incorporate local context into attention mechanism in the temporal dimension, which realizes local awareness enhancement and variables soft alignment. With a series of theoretical and practical verification, the hybrid model shows significant improvement at hit rate and mean-square error.

In complex industrial processes, it is difficult to measure the key quality variables online. It takes a long time to obtain quality variables through offline testing, which makes it difficult to get timely information to guide the production process. Therefore, this article proposes a novel dynamic time feature expanding and extracting framework for the sinter quality prediction. First, the original data are differentiated, compensated for time delay, expanded, and serialized by using time characteristics, and the input time series is reconstructed. Second, the integrated time features extractor is used to obtain the process information. Then, the recurrent neural network regression is applied to obtain the prediction of key quality variables. Finally, the effectiveness of the proposed method is verified by a numerical example, the actual data of sintering process and various comparative experiments, and the prediction effect of FeO content in sintering process is improved.

The papermaking industry supplies the carrier for literature and packaging via chemi-mechanical pulping, which is the most commonly used approach for converting wood chips into cellulose fibers. In 2016, Chinese papermaking enterprises consumed 41.05 million tons of standard coal. Thus, it is important to reduce the electrical energy consumed in mechanical pulp production by optimizing the operating conditions of primary stage refiners. This article reports a two-stage optimization algorithm to achieve this objective. First, mixed data sampling regression combined with a new weighting scheme derived from mixed flow conditions is used to encode the characteristics of high consistency refiner (HCR) in the manufacturing system. Then, the genetic algorithm with self-adaptive population searches for the optimal operating parameters that can minimize the power cost of the papermaking process. The proposed mixed framework and its separate components are validated through numerical functions and practical datasets, all of which have competitive performances compared to other algorithms. After being applied to a papermaking plant for two months, the intelligent optimization algorithm is found to have significant economic achievements in reducing the HCR energy consumption by 1.67 kWh/adt on an average scale.