# Development of a CFD-Suitable Deep Neural Network Model for Laminar Burning Velocity
## 3 Comparison with Other Methods  
To test the quality of the developed DNN model and to check whether some simpler method could be sufficient, other models using popular machine learning methods were created based on the same database. All models were optimized to make a million predictions in under 2 s with a testing script. Final models were tested and compared with DNN in terms of prediction density and the most widely used metrics such as MAE, MSE and  
Figure 3: Predicted loss function behavior tested by comparing DNN models with different loss functions.  
R-squared. While density estimates are not the most optimal method of comparison, we reduced the Gaussian kernel and estimation for all methods which was carried out with the same kernel; therefore, we consider it to be sufficient for comparison purposes. For all models, the same training and testing datasets were used as for the developed DNN model (75%/25%).  
Density plots in Figure 4 explain the high spread (Table 1) of some models (RF, MARS). While DNN and k-NN both show good agreement with the density of data until LBV reaches 350 cm/s, k-NN shows worse results at higher velocities. Even more, only SVM and DNN models predict higher than 700 cm/s laminar burning velocities. On the other hand, evaluation of SVM shows that this model is most likely to make predictions too high at over 280 cm/s and otherwise, guesses too low. As stated by this study earlier, DNN was trained by minimizing MAE and MSE at the same time, which should give optimal prediction between median and mean values. It means that the model might not have the lowest MSE and MAE values; therefore, it would be wise to evaluate it based on how well it explained the data. However, the results show (see Table 1) that DNN has the lowest MEA and MSE values of all tested methods with the highest being R-Squared at approximately 0.997. The second place would go to SVM regression which shows less spread than other models (RF, k-NN, MARS). The remaining solutions performed well by showing the ability to explain most of the data; however, their main drawback is the limited ability to predict higher burning velocities.  
The good performance of the developed DNN model can be visible in Figure 5 below, which shows the comparison of experimental and predicted LBV values. The red line,  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Model** & **MSE, (cm/s2)** & **MAE, (cm/s)** & **R-Squared** \\ \hline DNN & 73.07057 & 6.094699 & 0.9972631 \\ SVM (SVR) & 843.9467 & 23.36927 & 0.9687515 \\ Random Forest & 1100.299 & 23.68055 & 0.9650517 \\ K-Nearest Neighbors (k-NN) & 1164.565 & 18.31963 & 0.9541389 \\ MARS & 1324.741 & 24.40525 & 0.9487737 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Comparison of machine learning algorithms (values calculated against testing set).  
Figure 4: Prediction densities by multiple machine learning algorithms.  
which shows linear regression of DNN predictions plotted against reference velocities, almost perfectly covers the perfect match line (light blue).  
LBV has high uncertainty in the equivalence ratio range [0.2, 0.5]. [PERSON] [18] investigated the burning behavior of hydrogen mixtures and offered an expression to calculate LBV at low equivalence ratios. A comparison of the DNN model and Malet formula predictions (Figure 6) shows that the DNN model gives predictions closer to reality with MAE equal to 0.0589 m/s, while the Malet correlation has MAE of 0.1002 m/s.  
Figure 5: DNN predictions versus experimental data.  
Figure 6: Malet and DNN comparison.