## 3 Results  
To evaluate the model, several key points must be addressed. In Figure 7, the true pressure values of an experiment are shown, revealing significant fluctuations in the pressure trend Figure 7a. To fairly assess the prediction error, we first calculated a moving average (MA) and standard deviation (STD) for the experiments, as described by the following equations:  
\[\text{MA}(t)=\frac{1}{n}\sum_{i=t-n+1}^{t}x_{i}, \tag{5}\]  
\[\text{STD}=\sqrt{\frac{1}{n}\sum_{i=t-n+1}^{t}(x_{i}-\text{MA}(t))^{2}}, \tag{6}\]  
Figure 6: Error metrics MSE, MAE, and \(R^{2}\) for training and validation data during training of pressure and flow models. Figure (**a,b**) represent FFNN and RNN models for pressure prediction, respectively, while (**c,d**) depict FFNN and RNN models for flow prediction.  
where \(n\) is the size of the averaging window, \(x_{i}\) represents the data points, and \(n\) is the number of data points.  
Next, a 90% confidence interval (CI90%) is defined using the calculated STD:  
\[\text{CI90}=\text{MA}\pm z\cdot\text{STD}, \tag{7}\]  
where \(z=1.645\) is the \(z\)-score corresponding to the 90% percentile. This interval is used to assess how many predictions fall within the CI90%.  
By applying these equations to the experiments, an averaged line is obtained with upper and lower bounds, which represent the fluctuations in pressure and flow rate, as shown for the pressure in Figure 7b. This was applied for the flow rate in the same manner.  
In addition to MSE and RMSE, the relative \(L^{2}\)-norm (RL2N) is also used to estimate the percentage error relative to the total magnitude of the experiments. It is defined as  
\[\text{RMSE} =\sqrt{\frac{1}{n}\sum_{i=1}^{n}(y_{i}-\hat{y}_{i})^{2}}, \tag{8}\] \[\text{RL2N} =\frac{\sqrt{\sum_{i=1}^{n}(y_{i}-\hat{y}_{i})^{2}}}{\sqrt{\sum_{ i=1}^{n}y_{i}^{2}}}\cdot 100, \tag{9}\]  
where \(y_{i}\) represents the true values and \(\hat{y}_{i}\) represents the predicted values. In order to obtain an idea of the deviation of points in regards to the CI90% bounds, the \(L^{2}\)-norm bound (RL2N-B) is introduced and is defined as  
\[\text{RL2N-B} =\sqrt{\frac{\sum_{i=1}^{n}e^{2}(x_{i})}{\sum_{i=1}^{n}f^{2}(x_{i })}}\cdot 100, \tag{10}\] \[e^{2}(x_{i}) =\begin{cases}0,&\text{if }\hat{f}(x_{i})\in[l(x_{i}),u(x_{i})],\\ \min\big{(}(\hat{f}(x_{i})-l(x_{i}))^{2},(\hat{f}(x_{i})-u(x_{i}))^{2}\big{)},& \text{if }\hat{f}(x_{i})\
otin[l(x_{i}),u(x_{i})].\end{cases} \tag{11}\]  
where \(e^{2}(x_{i})\) represents the squared error for the value \(x_{i}\). It is zero if the predicted value \(\hat{f}(x_{i})\) lies within the confidence band \([l(x_{i}),u(x_{i})]\) defined by a lower and upper bound. Otherwise, \(e^{2}(x_{i})\) is computed as the square of the minimum distance between \(\hat{f}(x_{i})\) and the nearest bound of the confidence interval. This formulation ensures that only deviations outside the tolerated range contribute to the total error. The RL2N-B metric thus quantifies how far and how often the prediction violates the expected confidence bounds, rather than simply measuring deviation from a reference value. This is particularly useful in scenarios where predictions are allowed to vary within a known uncertainty band and should only be penalized when they exceed those limits. The denominator \(\sum_{i=1}^{n}f^{2}(x_{i})\) serves as a normalization factor, representing the total squared magnitude of the reference values \(f(x_{i})\). This allows the error to be interpreted as a relative percentage of the baseline signal, enabling fair comparisons across datasets or scales while maintaining a focus on confidence-aware model performance.  
Last but not least, the percentage of points inside the CI90% bounds (PIB) is also used for evaluation.  
In this section, two experimental datasets are considered: Table 2, which contains experiments used for training and validation (split as 80%/20%), and Table 3, which contains completely unknown experiments, including one with an unknown concentration to the model.