# Energy Optimisation of Industrial Limestone Grinding Using ANN
### Comparison of Black-Box, Physics-Informed, and Hybrid Models in Energy Systems  
The mineral-processing literature increasingly compares purely data-driven black-box predictors, usually deep neural networks, with classical physics-based simulators and newer hybrid architectures that fuse the two. Smith & Jones systematically benchmarked an encoder-decoder LSTM against a population-balance-model (PBM) simulator for closed-circuit ball-milling. Although the LSTM cut the mean absolute error in product P\({}_{80}\) by 35%, it exhibited 5-fold higher variance when feed hardness drifted outside the training envelope [30]. A similar conclusion emerged from [PERSON] & [PERSON]'s flotation soft-sensor study: their CNN predictor reached an \(R^{2}\) of 0.96 on within-range data but collapsed to 0.62 under unseen reagent regimes, whereas a first-principle mass balance retained \(R^{2}>0.9\) throughout [31].  
Hybrid models attempt to blend the strengths of both worlds. [PERSON] et al. embedded breakage-selection kernels as differentiable layers inside a physics-informed neural network (PINN) for semi-autogenous grinding; the resulting model matched the LSTM's accuracy while preserving physical consistency and requiring 40% fewer training samples [32]. [PERSON] et al. extended this idea to a plant-wide digital twin, coupling a convolutional auto-encoder with finite-volume transport equations for energy balance. When piloting at a copper concentrator, the hybrid twin enabled a model-predictive control (MPC) layer that cut specific energy by 11%, doubling the gain achieved with either component alone [33].  
Interpretability and maintainability are additional decision criteria. [PERSON] et al. quantified explanation quality using SHAP scores and found that operators trusted a grey-box (hybrid) mill power model 1.7 times more than a pure CNN, leading to measurably faster alarm responses in the control room [34]. [PERSON] & [PERSON] introduced a \"dual-twin\" architecture in which a recurrent neural net constantly recalibrates a mechanistic crusher model, their Bayesian updating scheme delivered calibrated prediction intervals that satisfied ISO-95 % coverage on three months of production data [35].  
Economic longevity also matters. In a five-year life-of-mine analysis, [PERSON] et al. showed that maintaining a physics-based HPGR model costs roughly one-quarter as much as retraining a large transformer every quarter, yet it yields a comparable net present value when the two are wrapped in an MPC framework [36]. However, [PERSON] et al. demonstrated that coupling discrete element method (DEM) breakage simulations with a lightweight graph neural network can achieve near-real-time crushing optimisation on commodity GPUs, narrowing the cost gap with first-principle codes while retaining DEM's extrapolation fidelity [37].  
Finally, two meta-analyses provide cross-industry context. [PERSON] & [PERSON] reviewed 78 studies across mining, cement, and steel, concluding that hybrids outperform black-box or physics-only models in 68% of energy-efficiency metrics, albeit at higher development complexity [38]. [PERSON] et al. focused on uncertainty quantification, finding that only 12% of black-box papers report calibrated prediction intervals versus 47% for hybrids and 71% for physics-based models, underscoring the maturity gap in risk-aware deployment [39].  
Collectively, these comparative studies suggest a nuanced trade-off: black-box networks excel in narrowly defined regimes with abundant data, first-principle simulators offer robustness and lower life-cycle cost, while hybrid structures increasingly dominate when both accuracy and generalisability are mission-critical. Future research should therefore target automated hybridisation frameworks and standardised uncertainty metrics to accelerate technology transfer from pilot to full-scale operations.