Chemical processes involve complex operational mechanisms and high-dimensional variables with multiscale dependencies and temporal covariate shifts, posing significant challenges for soft sensing implementation. To address these issues, a self-constrained weighted garrote ordered memory network with adaptive distribution learning is proposed for industrial soft sensor modeling. Specifically, an improved long short-term ordered memory (LSTOM) architecture with a composite hierarchical information update mechanism is designed to capture multiscale dynamic features from industrial time-series data. Meanwhile, an adaptive distribution learning module is embedded into the LSTOM architecture to match high-order feature distributions across different temporal periods via maximally dissimilar segments. With the composite hierarchical update mechanism and embedded adaptive distribution learning, the proposed approach improves generalization in soft sensor modeling under multi-timescale dependencies and temporal covariate shifts. Finally, the shrinkage coefficients of the self-constrained weighted garrote are embedded into the LSTOM’s input weights to eliminate redundant variables and promote structural sparsity. The effectiveness of the proposed approach is validated on a public industrial-scale penicillin fermentation process and a real-world flue gas desulfurization case.

Gated recurrent units (GRUs) have been successfully applied to various industrial soft-sensor tasks. However, their linear coupling constraint in the hidden-state limits information propagation, while high-dimensional inputs and structural redundancies further challenge modeling. To address these issues, a speed-adaptive gated recurrent unit (SAGRU) network with weights dual-garrote regularization (WDG-SAGRU) is proposed for product quality prediction. Specifically, a speed-adaptive gating mechanism is integrated into the GRU update gate through a nonlinear exponential transformation. This mechanism is incorporated into the hidden-state update to break the linear coupling constraint, enhancing information flow and feature extraction. Meanwhile, the dual-garrote shrinkage coefficients are embedded into SAGRU’s input and hidden weight matrices, enabling synchronized optimization of input variable selection and structural sparsity. The proposed WDG-SAGRU is validated against state-of-the-art methods on a numerical example and an operational flue gas desulfurization system in a thermal power plant.

As modern industrial processes increase in integration and scale, there exist intricate dynamic time variability and nonlinearity within process data. Deep learning-based nonlinear dynamic models, such as Transformer, are frequently applied to soft sensor modeling for industrial time sequences due to their powerful ability in learning temporal feature representations. Nevertheless, process data gathered from industrial plants are usually sampled at irregular intervals, posing a challenge for most mainstream dynamic models to handle the resulting temporally changeable relations in process sequence data. Thus, this paper proposes a time-aware rotary Transformer (TART) for soft sensor modeling of irregular sampled time series in industrial processes, which adaptively and efficiently model the temporally changeable dynamics among series data. Specifically, a sampling interval embedding layer is devised to simultaneously encode the process variables, positional information and sampling intervals, facilitating the efficient extraction of temporal dynamic features. Accordingly, a non-increasing function-based time-aware rotary attention mechanism is proposed to deal with the temporally changeable sampling intervals via assigning proper weights to the corresponding feature representations. Theoretical analysis shows that the proposed time-aware rotary attention mechanism can adaptively assign temporal similarities according to the irregular sampling intervals. The TART-based soft sensor is applied to an actual-run industrial sugar crystallization process to predict the supersaturation and purity of mother liquor. In comparison with recent predictive modeling methods, the proposed TART achieves state-of-the-art performance, demonstrating its feasibility and efficacy for soft sensing of practical industrial processes.

Deep probabilistic learning networks have been applied in industrial soft sensors. However, they face significant challenges in latent variable inference, deep learning backend implementation, and labeled data scarcity. The first challenge arises when covariates directly infer the latent variable, potentially leading to inaccuracies. The second stems from discrepancies between the theoretical probabilistic distribution and practical instance-based deep learning backends. The third is commonly encountered in soft sensor applications, where unlabeled data often go unused, reducing accuracy. To address these challenges, this work proposes a novel semi-supervised probabilistic learning network (SS-PLN) for soft sensor modeling with partially labeled data. The first issue is addressed by formulating an optimization problem as the model’s learning objective. This optimization is efficiently solved by investigating the input of inference network through analyzing the optimal solution’s structure. For the second issue, mean and covariance equations are used to represent probabilistic distributions, ensuring effective deep learning backends. The third issue is addressed by integrating supervised and unsupervised probabilistic learning networks to form the SS-PLN model, thereby maximizing the use of unlabeled data and enhancing the soft sensing performance. The feasibility and effectiveness of the proposed SS-PLN are validated through comparisons with recent semi-supervised learning methods, using data from two industrial processes.

For soft sensor modeling of multimode chemical processes, a common method is to build an individual model corresponding to each mode. However, certain individual mode models may not be reliable due to the imbalanced data across different modes. The soft sensor built for one mode with sufficient data exhibits suboptimal performance for other modes with insufficient data. To address this issue, a deep transfer learning method is introduced for soft sensor and a deep transfer spatial–temporal slow feature regression framework (STSFE) is proposed. In the framework, a Siamese network is employed for slow feature extraction. In addition, the encoder–decoder structure is embedded for input reconstruction verifying the effectiveness of slow features. However, the Siamese network fails to consider correlation of variables in quality prediction. To address this limitation, the Siamese network is designed with an embedded spatial–temporal attention mechanism to construct the STSFE model, and the spatial–temporal slow features are augmented with the historical quality variable for current quality prediction. The STSFE model is initially trained for the mode with sufficient data (source domain), and then transfer learning is employed to facilitate knowledge transfer from the source domain to the target domain (the mode with insufficient data) by reusing the lower level extraction part of STSFE. The effectiveness of the proposed method is validated through a benchmark sewage treatment case and a real chemical process.

Developing a fault detection model for the wastewater treatment process that combines satisfactory accuracy with comparatively low time overhead remains an exceedingly formidable endeavor. Fortunately, the broad slow feature neural network (BSFNN) perfectly embodies the dual advantages mentioned above. The BSFNN utilizes both slow feature windows and enhancement windows to extract significant and slowly varying information characterized by different velocities, which facilitates the learning of nonlinear and dynamic features related to superior monitoring accuracy. Another benefit of the BSFNN model is that it continues to retain the efficiency of the broad learning system with regard to time overhead, which is considerably decreased through employing the pseudoinverse strategy to determine network parameters. The operational environment often undergoes nonstationary dynamic changes in actual wastewater treatment processes. Especially in scenarios where higher monitoring accuracy is demanded or the network structure needs online adjustments to real-time update, and yet network adjustments can be time-consuming, the number of node parameters within incremental windows can be flexibly determined by dynamically adding enhancement nodes, which better obviates the necessity of retraining the entire BSFNN system from scratch, thereby allowing for online real-time adjustments to the structure and fault detection accuracy to achieve the desired performance. A case study using benchmark wastewater treatment platforms demonstrates that the suggested method outperforms advanced fault detection methods.

In industrial processes, quality variables are typically sampled at a considerably lower frequency than system inputs due to technical or cost constraints. Dynamic soft sensors utilize temporal prediction to bridge these sampling gaps, thus enabling real-time closed-loop control. However, existing approaches primarily focus on one-step prediction accuracy, potentially leading to significant deviations in long-term predictions. In addition, these methods are incapable of evaluating the reliability of prediction results, subsequently increasing the potential risk of closed-loop systems. To tackle these challenges, this study presents a novel regression modeling approach based on the conditional variational autoencoder (CVAE) framework. In contrast to traditional regression approaches, this method focuses on modeling the transition probability distribution of the system, allowing the model to produce a range of credible quality variable predictions via Monte Carlo (MC) sampling. Based on the CVAEs, the sequential MC method is further employed to simulate diverse potential system state trajectories, thereby achieving multistep soft measurement prediction. Compared with traditional soft measurement techniques, the proposed method demonstrates lower prediction biases and the capacity to assess the credibility of prediction results from a probabilistic standpoint. When online quality variables are assessed by the laboratory, this method can update predictions utilizing the resampling scheme. Two case studies are offered to validate the effectiveness of the proposed scheme.

Chemical processes involve complex physical and chemical mechanisms that exhibit slow and fast-varying features and nonstationary characteristics, making it difficult for single model-based methods to satisfactorily extract both slow and nonstationary fast-varying features for soft sensing. To address this issue, we propose a two-stream slow and nonstationary fast feature (TS-SNFF) model. This model includes a slow feature stream (SF-stream) and a nonstationary fast feature stream (NFF-streama). In the SF-stream, an encoder-decoder based Siamese network and a linear mapping layer are used for slow feature extraction. It employs long-short term memory (LSTM) networks as encoder and decoder units. Meanwhile, the NFF-stream utilizes the LSTM, differential LSTM (D-LSTM), and linear mapping layers for nonstationary fast feature extraction. The D-LSTM unit is established by embedding differential operations into the LSTM cell to obtain the nonstationary information. Then, the obtained features are fused using the merging layer, followed by a multilayer perceptron as the regressor. The proposed TS-SNFF model is utilized to address the slow and fast-varying dynamics in nonstationary conditions and nonlinearity problem in chemical processes. The effectiveness and superiority of the proposed method are demonstrated in a numerical example and an industrial process case