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Updated: May 5, 2026

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Physics-Guided Multi-Task Learning for Small-Sample Soft Sensing: Simultaneous Prediction of Kappa Number and

Bing Zhang1, Liuxin Shi1, Xiao Zhang1

  • 1College of Light Industry and Food Engineering, Nanjing Forestry University, Nanjing 210037, China.

Sensors (Basel, Switzerland)
|May 4, 2026
PubMed
Summary

This study introduces a physics-guided multi-task learning framework (PG-MTL) for real-time prediction of Kappa number and pulp viscosity in kraft pulping. PG-MTL significantly improves prediction accuracy and physical consistency, addressing limitations of current methods.

Keywords:
Kappa numbercontinuous kraft pulpingmulti-task learningphysics-guided learningpulp viscositysmall-sample modelingsoft sensor

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Area of Science:

  • Chemical Engineering
  • Process Control
  • Machine Learning

Background:

  • Offline measurement of Kappa number and pulp viscosity limits real-time quality control in continuous kraft pulping.
  • Existing data-driven soft sensors face challenges with limited data and lack of physical consistency.
  • Simultaneous prediction of coupled quality indicators like Kappa number and pulp viscosity is underexplored.

Purpose of the Study:

  • To develop a physics-guided multi-task learning framework (PG-MTL) for simultaneous prediction of Kappa number and pulp viscosity.
  • To enhance the physical consistency and accuracy of soft sensors for pulp quality.
  • To address the limitations of small-sample industrial data in soft sensor development.

Main Methods:

  • Implemented a hard-parameter-sharing multi-task architecture.
  • Incorporated a physics-guided monotonicity constraint for Kappa number prediction (non-increasing trend with H-factor).
  • Utilized homoscedastic uncertainty weighting to balance regression tasks.

Main Results:

  • PG-MTL achieved high prediction accuracy: R2 = 0.920 for Kappa number and R2 = 0.910 for pulp viscosity.
  • Reduced Root Mean Square Error (RMSE) by 23.2% for Kappa number and 29.5% for pulp viscosity compared to benchmark models.
  • Demonstrated effective and physically consistent soft sensing performance on industrial data.

Conclusions:

  • PG-MTL offers an effective solution for simultaneous pulp quality prediction.
  • The physics-guided approach enhances model reliability and physical consistency.
  • This framework is particularly suitable for small-sample industrial conditions in pulp and paper manufacturing.