An Online Updating Robust Soft Sensor for Nonstationary Industrial Processes Based on Bidirectional Long Short-Term Memory and Integrated Gradients

In modern process industries, data-driven soft sensors have become indispensable for monitoring critical process variables that are inaccessible to direct measurement. Nevertheless, the accurate modeling of industrial processes remains challenging due to their intrinsic complexities, such as time-series behaviors, measurement outliers, redundant variables, and potential concept drift. Existing approaches can address subsets of these challenges but generally lack a unified mechanism that integrates robust offline modeling, variable-importance analysis, and efficient online adaptation. To address these issues, this study proposes an online-updating robust soft sensor framework based on bidirectional long short-term memory (BiLSTM) with integrated gradients (IG) and smoothed quantile loss (SQLoss). During offline modeling, a soft-sensing model is constructed using a BiLSTM, and the proposed SQLoss is introduced to reduce the influence of outliers; the IG method is then employed to evaluate the importance of input variables, enabling input variable selection. During online operation, model parameters associated with significant variables are selectively updated based on IG-derived variable importance, thereby addressing concept drift. Finally, experimental results on an industrial desulfurization process demonstrate that, compared with the best-performing competing basic learner, the proposed SQLoss-BiLSTM-IG reduces the average root mean squared error (RMSE) and mean absolute percentage error by 5.26% and 1.87%, respectively, while increasing the average correlation coefficient by 1.94%; in the online evaluation, the proposed updating strategy achieves a mean RMSE of 2.441, demonstrating its effectiveness in handling concept drift. Moreover, the analysis of key variable importance is consistent with field experience, offering valuable insights for optimizing the desulfurization control system.