as shift identifiers, temperature zones, and laboratory-assay latency indicators with the objective function (5). L(θ) = Prediction Error + Regularization weight L(θ)=∑I ∥Δyi − fθ(xi)∥ 2 + λ Reg(θ) (5) Where: L(θ) = Loss function Δyi = True residual for sample i fθ(xi) = Model-predicted residual λ = Regularization weight (also called regularization coefficient or penalty weight) Reg(θ) = Regularization term (e.g., L2-norm) Reg(θ) (5) controls model complexity, mitigating overfitting while promoting interpretability. Model validation consists of feature-importance diagnostics, multicollinearity assessment, and ensuring residual consistency across diverse operating regimes, including high-iron intervals and varying head-grade conditions. The residual model thus forms a data-driven representation of unmodeled first principle, equipment characteristics, and operational non-idealities. 3.3.4 Hybrid Model – First principle -ML fusion With Physical Feasibility: In the hybrid model formulation, the machine-learning component does not replace the first principle -based solver; rather, it augments it by providing a first principle -informed correction. The learned residual ΔyML is combined with the first principle -based prediction yphys (6) through the composite mapping: ℎ = ( ℎ , ) (6) Where: ℎ = Hybrid model output ℎ = First-principles (physics-based) model prediction = ML-predicted residual (correction term) f (⋅) = Fusion function combining physics and ML (typically additive) The coupling function f(⋅) can represent additive, multiplicative, nonlinear, or operator-aware correction strategies depending on system requirements. After fusion, a feasibility-projection operator (. )ensures that the output adheres to process-level first principle and chemical constraints in eq [7-9]. Non-negativity of species concentrations for all the parameters in the input data and the system should satisfy the operational chemistry bounds (e.g., acidity, alkalinity, ionic balance): ≥0, ∀ (7) Where: = Decision variables (non-negative) ∀ = Constraint applies to all indexed variables pHmin ≤ pH(y) ≤ pHmax (8)
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