{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,17]],"date-time":"2026-08-17T11:27:08Z","timestamp":1786966028327,"version":"build-2736575974"},"reference-count":35,"publisher":"MDPI AG","issue":"16","license":[{"start":{"date-parts":[[2026,8,14]],"date-time":"2026-08-14T00:00:00Z","timestamp":1786665600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["62273357"],"award-info":[{"award-number":["62273357"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"award":["62273357"],"award-info":[{"award-number":["62273357"]}],"id":[{"id":"https:\/\/ror.org\/01h0zpd94","id-type":"ROR","asserted-by":"publisher"}]},{"name":"Yunnan Fundamental Research Projects","award":["202501BC070019-009"],"award-info":[{"award-number":["202501BC070019-009"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>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\u2013temperature 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 (\u03c1s=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.<\/jats:p>","DOI":"10.3390\/s26165157","type":"journal-article","created":{"date-parts":[[2026,8,14]],"date-time":"2026-08-14T13:12:56Z","timestamp":1786713176000},"page":"5157","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["A Physics-Informed Multi-Task Soft Sensor for Uncertainty-Aware Slag Tin Content Prediction in Top-Blown Tin Smelting"],"prefix":"10.3390","volume":"26","author":[{"given":"Zhaojun","family":"Ma","sequence":"first","affiliation":[{"name":"School of Resources and Environment, Yunnan University, Kunming 650500, China"},{"name":"Yunnan Tin Group (Holding) Co., Ltd., No. 471 Minhang Road, Guandu District, Kunming 650200, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jubo","family":"Peng","sequence":"additional","affiliation":[{"name":"Yunnan Tin Group (Holding) Co., Ltd., No. 471 Minhang Road, Guandu District, Kunming 650200, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Xiaojun","family":"Zhou","sequence":"additional","affiliation":[{"name":"School of Automation, Central South University, Changsha 410083, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Kui","family":"Xie","sequence":"additional","affiliation":[{"name":"School of Automation, Central South University, Changsha 410083, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Zhang","family":"Zhang","sequence":"additional","affiliation":[{"name":"Yunnan Tin Group (Holding) Co., Ltd., No. 471 Minhang Road, Guandu District, Kunming 650200, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2026,8,14]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Kandalam, A., Reuter, M.A., Stelter, M., Reinm\u00f6ller, M., Gr\u00e4bner, M., Richter, A., and Charitos, A. 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