Inferring operation–response lag times for hot-metal silicon in blast furnaces using an interpretable machine learning–augmented stochastic differential equation framework

Reliable blast-furnace control is challenged by multi-source, time-varying process lags between operational changes and state responses. Accurate inference of these lag times is essential for precise control and mechanistic interpretation; however, conventional regression approaches rarely make them explicit and often struggle under stochastic fluctuations. In this study, hot-metal silicon content (Si) in a 1750 m³ blast furnace is treated as the state variable, and an interpretable machine learning–augmented Itô stochastic differential equation framework is developed. Machine learning is employed to learn the governing drift and diffusion terms of the stochastic differential equation from data, capturing both deterministic evolution and random disturbances. Operation–Si response lag times are formulated as key parameters to be identified and are inferred via Bayesian optimization under constraints of feasible lag windows and cluster-based (mechanism-guided) grouping, which stabilizes lag estimates among related variables. The resulting model is further interpreted using SHapley Additive exPlanations (SHAP) to quantify the magnitude and timing of each operation’s contribution within its inferred lag window, while also revealing regime-dependent local influence rules, interaction-conditioned effects, and representative sample-level contribution paths. Monthly models built from hourly data for July–December 2023 achieve good predictive performance, with the coefficient of determination (R²) exceeding 0.90 in most months. The inferred lag structure shows a clear separation of time scales: blast and pulverized coal injection variables respond rapidly, whereas burden- and stockline-related variables exhibit slower, regime-dependent responses. SHAP-based analyses further indicate that the influence of operations on Si is strongly context dependent rather than fixed, and representative-case explanations resolve the contributions of lagged inputs to the current model update with the preceding predicted state held fixed. Overall, the proposed approach shifts lag-time inference from statistical alignment toward interpretable identification of physically meaningful process parameters, supporting improved multi-timescale Si control and furnace heat diagnosis.