Causality in an industrial process provides insights into how various process variables interact and affect each other within the system. It reveals the underlying mechanisms of industrial processes, which ensures predictive reliability and facilitates physical interpretability. However, existing causality-based techniques have limitations, as they neglect the temporal factor in causal description, introduce spurious causal associations in causal discovery, and fail to consider the spatial-temporal synchronicity in causal utilization. To address these issues, this article proposes a spatial-temporal causality modeling approach. A novel spatial-temporal causal digraph (STCG) is proposed to describe causal dependencies among process variables, which considers both spatial and temporal factors encompassing causal relationships and time delays. The STCG identification procedure is formulated as a Markov decision process, and knowledge-data guided reinforcement learning is developed to acquire the optimal identification policy and avoid spurious causal associations. With the identified STCG, a graph attention gate recurrent unit (GAGRU) is constructed for spatial-temporal process modeling, which is able to capture the synchronous evolution of industrial data in spatial-temporal dimensions. Finally, the effectiveness of the proposed modeling approach is verified by applying to two real industrial cases, including soft sensing for a sulfur recovery unit and anomaly detection for an argon distillation system. The experimental results demonstrate that the STCG-based industrial process modeling outperforms classical and state-of-the-art comparison methods in terms of reliability and interpretability.

Sensor faults are non-negligible issues for soft sensor modeling. However, existing deep learning-based soft sensors are fragile and sensitive when considering sensor faults. To improve the robustness against sensor faults, this article proposes a deep subdomain learning adaptation network (DSLAN) to develop a sensor fault-tolerant soft sensor, which is capable of handling both sensor degradation and sensor failure simultaneously. Primarily, domain adaptation works for process data with sensor degradation in industrial processes. Being founded on the basic structure of deep domain adaptation, a novel subdomain learner is added to automatically learn the subdomain division, enabling DSLAN adaptable to multimode industrial processes. Notably, the subdomain structure of each sample follows a categorical distribution parameterized by output of the subdomain learner. Based on the designed subdomain learner, a new probabilistic local maximum mean discrepancy (PLMMD) is presented to measure the difference in distribution between source and target features. In addition, a generator for failure data imputation is integrated in the framework, making DSLAN handle sensor failure simultaneously. Finally, the Tennessee Eastman (TE) benchmark process and two real industrial processes are used to verify the effectiveness of the proposed method. With the fault tolerance ability, soft sensing technology will take a step toward practical applications.