{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,23]],"date-time":"2026-10-23T21:14:29Z","timestamp":1792790069960,"version":"4.5.0"},"reference-count":31,"publisher":"MDPI AG","issue":"19","license":[{"start":{"date-parts":[[2026,9,29]],"date-time":"2026-09-29T00:00:00Z","timestamp":1790640000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"King Saud University","award":["ORF-2026-2179"],"award-info":[{"award-number":["ORF-2026-2179"]}],"id":[{"id":"https:\/\/ror.org\/02f81g417","id-type":"ROR","asserted-by":"crossref"}]},{"award":["ORF-2026-2179"],"award-info":[{"award-number":["ORF-2026-2179"]}],"id":[{"id":"https:\/\/ror.org\/02f81g417","id-type":"ROR","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>Accurate sulfur dioxide (SO2) estimation is important for sulfur recovery unit (SRU) monitoring when reliable one-step information is required between analyzer observations. This study proposes the Gated Residual Autoregressive\u2013Exogenous GRU Network (GRAG-Net), which separates multivariate process dynamics from strictly causal SO2 history, predicts branch-specific residual corrections relative to the latest measured target, and combines them through a sample-dependent gate. The model was evaluated on 10,081 chronologically ordered industrial observations using three expanding future test periods and five final random seeds. Comparators included persistence, linear extrapolation, absolute and residual classical regressors, process-only and autoregressive GRUs, a joint GRU, LSTM\u2013ARX, TCN\u2013ARX, fixed fusion, and global gating. Across 15 descriptive fold\u2013seed evaluations, GRAG-Net achieved MAE 0.00315\u00b10.00044 mole%, RMSE 0.00790\u00b10.00064 mole%, and R2=0.97753\u00b10.00331. Pooled seed-ensemble MAE and RMSE were 0.00291 and 0.00766 mole%, reductions of 32.5% and 18.0% relative to AR\u2013GRU. Independently trained ablations identified residual reconstruction and sample-dependent fusion as the main architectural contributors, while dependence-aware bootstrap analysis, causal gate diagnostics, lag analysis, and cross-fold\/cross-seed SHAP stability supported robust and interpretable one-step soft sensing.<\/jats:p>","DOI":"10.3390\/s26196168","type":"journal-article","created":{"date-parts":[[2026,9,29]],"date-time":"2026-09-29T12:09:25Z","timestamp":1790683765000},"page":"6168","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["GRAG-Net: A Gated Residual Autoregressive\u2013Exogenous GRU Soft Sensor for Dynamic SO2 Estimation in Sulfur Recovery Units"],"prefix":"10.3390","volume":"26","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6731-7614","authenticated-orcid":false,"given":"Maha Mesfer","family":"Alghamdi","sequence":"first","affiliation":[{"name":"Department of Computer Science and Engineering, College of Applied Studies, King Saud University, Riyadh 11451, Saudi Arabia"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2026,9,29]]},"reference":[{"key":"ref_1","unstructured":"Seinfeld, J.H., and Pandis, S.N. 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