A Physics-Informed Multi-Task Soft Sensor for Uncertainty-Aware Slag Tin Content Prediction in Top-Blown Tin Smelting

Slag tin content is a key indicator of reduction efficiency and metal recovery in top-blown tin smelting, but delayed and sparse laboratory assays hinder online monitoring. This study proposes a physics-informed multi-task soft sensor (PI-MTSS) combining a shared temporal convolutional encoder, self-supervised future state reconstruction, a soft monotonicity constraint based on a simplified CO–temperature reduction-potential indicator, and Monte Carlo dropout for epistemic uncertainty estimation. The model was evaluated on a continuous 30-day dataset comprising 43,200 one-minute process observations and 240 laboratory assays, including 40 chronologically held-out test labels. Across five random seeds, PI-MTSS achieved an RMSE of 0.198, an MAE of 0.145, and an R2 of 0.889. Across three rolling-origin evaluations, it obtained a mean RMSE of 0.199 and maintained the lowest RMSE among the evaluated methods. The complete model reduced the physical violation rate to 3.5%. At 95% nominal coverage, its epistemic intervals achieved a PICP of 96.5% and an NMPIW of 0.245; the ACE across six nominal levels was 0.021, and uncertainty correlated positively with absolute error (ρs=0.691). These results support the potential of PI-MTSS for minute-scale monitoring and operator decision support under sparse labels, although validation remains limited to one month from a single furnace.