## 3 Results
### Application of Physical Error Metrics to the Models  
After the addition of a mass imbalance error term in the model tuning, all models presented a validation RMSE that was essentially unchanged (Table 2), with the exception of the CNN-LSTM model, which increased its RMSE. Nevertheless, the addition of mass conservation was able to stabilize the errors through the epochs (Figure 6), specifically for the FNO model. For the case without mass conservation information, the mass balance achieved a minimum testing RMSE and then started to increase monotonically, which does not happen in the case with mass conservation.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Model** & **CNN-LSTM** & **ConvLSTM** & **FNO** \\ \hline
**RMSE (m-s\({}^{-1}\))** & \(2.26\times 10^{-2}\) & \(7.36\times 10^{-3}\) & \(6.83\times 10^{-3}\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: RMSE obtained by all optimized model architectures after 50 epochs.  
Figure 4: Flow velocities obtained by the ConvLSTM model after tuning, for t = 0.3 s.  
Figure 5: Flow velocities obtained by the FNO model after tuning, for t = 0.3 s.  
The addition of the mass imbalance error also improved the overall mass conservation, and the impact of the weight on it is shown in Table 3. As can be seen, the presence of the error term itself already worked to diminish the mass imbalance, and an increase in the weight did not imply a further improvement, except for the CNN-LSTM model, which was less sensible to the weight and needed a higher value.  
To further illustrate the beneficial effect of the addition of a mass conservation error term in the model training, Figure 7 depicts the specific case of the mass imbalance (Equation (1)) field for the prediction of t = 1.3 s. When we refer to the mass imbalance field, we actually present the absolute calculated mass imbalance for each grid cell, which should always approximate to zero. This constraint was added to the model training. On the left plots, one can see the ground truth (CFD results), and on the right, the predicted fields are shown, without adding the mass conservation error to the models (top), and with its application to the solution (bottom). The improvement was remarkable, being the predicted field with the mass conservation constraint essentially equal to the ground truth.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & **No MC Error** & \begin{tabular}{c} **0.1 \(\times\) MC** \\ **Error** \\ \end{tabular} & \begin{tabular}{c} **1.0 \(\times\) MC** \\ **Error** \\ \end{tabular} & \
\begin{tabular}{c} **2.0 \(\times\) MC** \\ **Error** \\ \end{tabular} &
\begin{tabular}{c} **5.0 \(\times\) MC** \\ **Error** \\ \end{tabular} \\ \hline
**FNO** & \(6.56\times 10^{-3}\) & \(6.43\times 10^{-3}\) & \(6.46\times 10^{-3}\) & \(6.49\times 10^{-3}\) & \(6.48\times 10^{-3}\) \\
**CNN-LSTM** & \(1.82\times 10^{-2}\) & \(1.91\times 10^{-2}\) & \(2.57\times 10^{-2}\) & \(2.58\times 10^{-2}\) & \(8.09\times 10^{-2}\) \\
**ConvLSTM** & \(7.06\times 10^{-3}\) & \(7.03\times 10^{-3}\) & \(7.21\times 10^{-3}\) & \(7.31\times 10^{-3}\) & \(7.31\times 10^{-3}\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Impact of mass conservation weights on validation RMSE obtained by all optimized model architectures after a maximum of 300 epochs.  
Figure 6: Loss function and mass imbalance error over the training epochs of the FNO model. Testing RMSE is also presented (red line, left plot).