404 with equilibrium isotherms, rate‑based kinetic descriptions, and detailed mass transfer formulations would further strengthen its predictive Capability and applicability to more complex circuits [1,4]. 3.3. Hybrid Approach — Training Workflow (Hybrid Twin Calibration) 3.3.1 Inputs and baseline first principle simulation. Plant inputs (PLS Composition, O/A by stage, flow rates, temperatures, mixer volumes, and extractant concentration etc.) are provided to the first principle -based SX model. Stagewise predictions— raffinate Cu/Fe, organic loading, pH/H₂SO₄ profile, and strip outputs—are computed using Equations (1)– (4) (chelation equilibria, distribution ratios, single-stage extraction fraction, and approach-to-equilibrium corrections). 3.3.2 Residual Model – Learning Systematic Mismatch: The residual model àA(() designed to learn the structured mismatch Δy between plant measurements and first principle -based model outputs. The model is provided with a comprehensive feature vector X, which includes plant operating variables, selected outputs from the first principle solver (predicted pH, flows, PLS compositions), and contextual metadata such 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: N ℎ'K =àÖN&ℎ'E,êNMNÜ (6) Where: N ℎ'K = Hybrid model output N&ℎ'E = First-principles (physics-based) model prediction êNMN = 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.
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