Inferring lag times between blast furnace operations and hot metal silicon using an interpretable machine learning augmented stochastic differential equation

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. The 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 trajectory fits with R² > 0.90 in most months. The inferred lag structure exhibits a clear separation of time scales: blast and pulverized coal injection actions respond rapidly, whereas burden- and stockline-related variables show slower responses that depend on the charging regime. Local and sample-level SHAP analyses further show that the operational effects on Si are context-dependent rather than fixed, and that high-Si and low-Si states typically arise from balanced combinations of multiple lag-aligned contributors rather than from any single dominant variable. Overall, the proposed approach elevates lag-time inference from statistical alignment to interpretable identification of physically meaningful process parameters, supporting improved multi-timescale Si control and furnace-heat diagnosis.