# Development of a CFD-Suitable Deep Neural Network Model for Laminar Burning Velocity
## 2 Developed Deep Neural Network Model for Laminar Burning Velocity  
The presented model was based on a previously developed (further referred to as the original) large DNN [17], which was too slow for CFD application. The further model development aimed to improve the calculation speed without a significant accuracy loss. The original DNN model was trained on the dataset of experimental hydrogen-air mixtures LBV at various temperatures, pressures and hydrogen concentrations, collected from the open literature [17]. For the training of the new DNN model, the original database was expanded to 2871 data points [22; 23; 24; 25; 26; 27; 28; 29; 30; 31; 32; 33; 34; 35; 36; 37; 38; 39; 40; 41; 42; 43; 44; 45; 46; 47]. Around 33% of all of them are experimental values; however, the number of pure experimental values was insufficient for the reliable training of DNN. Therefore, additional values were interpolated from the experimental values, by adding points from the correlation curve produced from the experimental points of the given experiment, close to the actual experimental points. Since the results of the newest studies are most likely to have a higher accuracy due to technological development, they were prioritized for the interpolation, in this way giving them more 'weight' in the final database, since more data points have been added based on the newest observations and fewer points in the area near old observations. As can be noticed from Figure 1, most of the data are around 1 bar pressure or 298 K temperature, because most experimental works start by taking normal conditions and changing only a single parameter. However, the expansion of the database was performed to make the model more reliable while predicting LBVs at higher temperatures and higher pressures, since most of the new data are under non-normal conditions. These changes should have made the model more reliable in an entire applicability range. The used experimental data model is applicable in the range of 295-600 K temperature and 0.1-5 bar pressure. Data were split randomly into training and testing sets with around 75% of data used to train the model and the remaining data used for validation. Further modifications such as outlier removal were focused on the training set, avoiding modification of the testing set, this way trying to obtain as reliable data as possible for validation. Four hidden layers were left in the new model (same as in the original); however, to ensure faster calculations, the number of neurons was changed to (7,10,7,5) (Figure 2) from (40,50,30,10) in previous work [17]. The number of neurons was reduced by trying various combinations and balancing accuracy against prediction speed. In general, reducing the number of neurons increased the risk that the model will not be as robust to data as the previous model. Due to this, we had to re-train the model multiple times as changes in weights could affect the predictions. Overall, this meant that  
Figure 1: Gathered data statistics [22; 23; 24; 25; 26; 27; 28; 29; 30; 31; 32; 33; 34; 35; 36; 37; 38; 39; 40; 41; 42; 43; 44; 45; 46; 47]: (a) count against pressure and (b) count against temperature. Figure 2: Architecture of the improved model. the smaller neural network required more human monitoring during the training to less accuracy loss. In addition, we increased the risk that fewer neurons will lack data to learn in some areas; to counter this, we had to increase our database. In the end, the main drawback related to the reduced number of neurons was that we had to put in substantial additional effort to obtain the model with results comparable to those of larger DNN. Several different sets of activation functions were also tested and the best predictions were obtained when using _tanh_ activation function for the first (dense) hidden layer (Equation (1)) and _ReLU_ for the rest (Equation (2)):  
\[tanh(z)\frac{e^{z}-e^{-z}}{e^{z}+e^{-z}} \tag{1}\]  
\[ReLU(output)=\left\{ \begin{aligned} & output,ifoutput>0;\\ & 0,else\end{aligned} \right.