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.

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.

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.

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.

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.

In this article, a novel data-driven fault diagnosis method by combining deep canonical variate analysis and Fisher discriminant analysis (DCVA-FDA) is proposed for complex industrial processes. Inspired by the recently developed deep canonical correlation analysis, a new nonlinear canonical variate analysis (CVA) called DCVA is first developed by incorporating deep neural networks into CVA. Based on DCVA, a residual generator is designed for the fault diagnosis process. FDA is applied in the feature space spanned by residual vectors. Then, a Bayesian inference classifier is performed in the reduced dimensional space of FDA to label the class of process data. A continuous stirred-tank reactor and an industrial benchmark of the Tennessee Eastman process are carried out to test the performance of DCVA-FDA fault diagnosis. The experimental results demonstrate that the proposed DCVA-FDA fault diagnosis is able to significantly improve the fault diagnosis performance when compared to other methods also examined in this article.