## 6 Experiments
### Experimental Setup
#### 6.2.2 Physical Informed Loss Function  
In our endeavor to incorporate the physical dynamics of the mill into the predictive model, as we saw in Equation (8), we introduce a loss function, \(\mathcal{L}_{f}(\hat{u},\hat{y})\), where \(\hat{u}_{k}(\mathbf{x}_{k})\) is a function with each \(\mathbf{x}_{k}\) representing the pairs \((\mathbf{x}^{(i)},\mathbf{x}^{(j)})\) of selected features for each rule \(k\). The indices \(i\) and \(j\) correspond to specific variables relevant to each rule. These variables are selected based on the physical conditions that the rule is designed to evaluate. Each \(\hat{u}_{k}(\mathbf{x}_{k})\) function is structured to evaluate the following conditions: \begin{table}
\begin{tabular}{c c c} \hline
**Variable** & **Criteria** & **Unit** \\ \hline Fine Granulometry & \textgreater{}0 & \% \\ Coarse Granulometry & \textgreater{}0 & \% \\ Intermediate Granulometry & \textgreater{}25 & bar \\ Mill Power & \textgreater{}14,000 & kW \\ Mill Pressure & \textgreater{}6000 \(\wedge\) \textgreater{}7250 & kPa \\ Solid Percentage & \textgreater{}68 \(\wedge\) \textgreater{}77 & \% \\ Throughput & \textless{}8500 & ton per hour \\ Mill Speed & \textgreater{}7.2 & rpm \\ \hline \end{tabular} Note: Percentages described in the table regarding the throughput.  
\end{table} 
Table 1: Criteria for data filtering based on expert judgment. 1. The increase in _pressure (pre)_ concurrent with a decrease in _power (pow)_, symbolizing a overload condition on the mill [39; 40]: \[\hat{u}_{1}(\mathbf{x}_{1})=\sigma\Big{(}\
abla_{t}\mathbf{x}_{1}^{(pre)}\Big{)} \sigma\Big{(}\
abla_{t}\mathbf{x}_{1}^{(pow)}\Big{)}.\] (12)
2. Also, an increase in _speed_ alongside a decrease in _power_, represents an overload scenario [41]: \[\hat{u}_{2}(\mathbf{x}_{2})=\sigma\Big{(}\
abla_{t}\mathbf{x}_{2}^{(spe)} \Big{)}\sigma\Big{(}\
abla_{t}\mathbf{x}_{2}^{(pow)}\Big{)}.\] (13)
3. Sufficient granulometry to ensure the flow of material within the mill. An excess of _fine (fin)_ over _coarse (coa)_ particles leading to high pulp density, which makes material movement difficult and can cause the mill to jam or overload [42]: \[\hat{u}_{3}(\mathbf{x}_{3})=\sigma\Big{(}-\
abla_{t}\mathbf{x}_{3}^{(fin)} \Big{)}\sigma\Big{(}-\
abla_{t}\mathbf{x}_{3}^{(coa)}\Big{)}.\] (14)
4. The relationship between _power consumed_ and _time elapsed since the last maintenance (tlm)_ of the mill is an indicator that there is a greater probability of overload [43; 44]: \[\hat{u}_{4}(\mathbf{x}_{4})=\sigma\Big{(}-\
abla_{t}\mathbf{x}_{4}^{(pow)} \Big{)}\sigma\Big{(}-\
abla_{t}\mathbf{x}_{4}^{(tlm)}\Big{)}.\] (15)
5. The _fine_ and _intermediate granulometry (int)_ ratio acting as an indicator of increasing overload probability in SAG mills [42]: \[\hat{u}_{5}(\mathbf{x}_{5})=\sigma\Big{(}-\
abla_{t}\mathbf{x}_{5}^{(fin)} \Big{)}\sigma\Big{(}\
abla_{t}\mathbf{x}_{5}^{(int)}\Big{)}.\] (16)
6. The ratio between _coarse_ and _intermediate particle size_ acting as an overload probability increase indicator in SAG mills [42]: \[\hat{u}_{6}(\mathbf{x}_{6})=\sigma\Big{(}\
abla_{t}\mathbf{x}_{6}^{(coa)} \Big{)}\sigma\Big{(}\
abla_{t}\mathbf{x}_{5}^{(int)}\Big{)}.\] (17)
7. The relationship between _pressure_ and the _time elapsed since the last maintenance_ of the mill acting as a high probability of overload indicator [43]: \[\hat{u}_{7}(\mathbf{x}_{7})=\sigma\Big{(}\
abla_{t}\mathbf{x}_{7}^{(pre)} \Big{)}\sigma\Big{(}-\
abla_{t}\mathbf{x}_{7}^{(tlm)}\Big{)}.\] (18)
8. The relationship between _intermediate granulometry_ and _pressure_ is an indicator of a higher probability of overload [45]: \[\hat{u}_{8}(\mathbf{x}_{8})=\sigma\Big{(}\
abla_{t}\mathbf{x}_{8}^{(int)} \Big{)}\sigma\Big{(}-\
abla_{t}\mathbf{x}_{8}^{(pre)}\Big{)}.\] (19)  
Finally, we obtain the value of penalization by multiplying the value of Equation (8) with the model's predicted value to obtain the respective value of \(\mathcal{L}\). Each condition was mathematically formulated as the difference of derivatives (slopes) of the curves representing the values in the measured windows interval.