Physics-Guided Multi-Task Learning for Small-Sample Soft Sensing: Simultaneous Prediction of Kappa Number and Viscosity in Continuous Kraft Pulping

In continuous kraft pulping, key quality indicators such as Kappa number and pulp viscosity are usually measured offline at low frequency, which limits real-time quality monitoring and control. Although data-driven soft sensors have shown potential for quality prediction, their performance is often restricted by limited labeled data and weak physical consistency. In addition, existing studies have focused mainly on single-target prediction, while the coupled prediction of Kappa number and pulp viscosity remains insufficiently explored despite their common dependence on cooking conditions and degradation kinetics. To address these issues, this study proposes a physics-guided multi-task learning framework (PG-MTL) for simultaneous prediction of Kappa number and pulp viscosity. The model combines a hard-parameter-sharing multi-task architecture with a physics-guided monotonicity constraint that enforces the expected non-increasing Kappa trend with increasing H-factor. Homoscedastic uncertainty weighting is also used to balance the two regression tasks during optimization. Experiments on industrial operating data show that PG-MTL achieved R2 = 0.920 for the Kappa number and R2 = 0.910 for pulp viscosity. Compared with the strongest benchmark model, RMSE was reduced by 23.2% and 29.5% for the Kappa number and pulp viscosity, respectively. These results demonstrate that PG-MTL provides an effective and physically consistent solution for pulp-quality soft sensing under small-sample industrial conditions.