# Fast Prediction of Flow Field around Airfoils Based on Deep Convolutional Neural Network
## 3 Results and Discussion
### Comparison with POD  
For a more quantitative study on the interpretability of our CNN-DCNN model, we use the method of POD to investigate the reconstructed physical fields. POD extracts essential characteristics of discrete data by describing multidimensional stochastic processes with low-dimensional approximations. Its basic idea is to decompose the dynamic system into a set of basic functions (modes), where the principle of determining the basis function is to make the lowest order modes retain as much \"energy\" as possible from the original high-order system. Considering a given pressure or velocity fields as a two-dimensional matrix \(\mathbf{A}_{m\times n}\) (where \(m\times n\) is 208 \(\times\) 340 in this work), decomposition by POD gives:  
\[\mathbf{A}_{m\times n}=\mathbf{W}_{m\times m}\sum_{m\times n}\mathbf{Z}_{m\times n} ^{*} \tag{25}\]where the column vector of \(\mathbf{W}_{m\times n}\) is the spatial modes of \(\mathbf{A}_{m\times n}\), the row vector of \(\mathbf{Z}_{n\times n}^{*}\) represents the temporal modes and \(\sum_{m\times n}\) is the eigenvalue matrix. \(\mathbf{A}_{m\times n}\) can be further described as follows:  
\[\mathbf{A}_{m\times n}=[\mathbf{w}_{1},\ \mathbf{w}_{2},\ldots,\mathbf{w}_{n}] \begin{bmatrix}\lambda_{1}\\ &\lambda_{2}\\ &&\cdots\\ &&&\lambda_{r}\\ &&&0\end{bmatrix}\begin{bmatrix}\mathbf{z}_{1}\\ \mathbf{z}_{2}\\ \cdots\\ \mathbf{z}_{n}\end{bmatrix} =\sum_{j=1}^{r}\lambda_{j}\mathbf{w}_{j}\mathbf{z}_ {j} \tag{26}\]  
where \(\lambda_{j}\) is the non-zero eigenvalue of \(\mathbf{A}_{m\times n}\), \(\mathbf{w}_{j}\) is the column vector of \(\mathbf{W}_{m\times n}\), \(\mathbf{z}_{j}\) is the row vector of \(\mathbf{Z}_{n\times n}^{*}\) and \(\lambda_{j}\mathbf{w}_{j}\mathbf{z}_{j}\) represents the \(j\)th mode of \(\mathbf{A}_{m\times n}\). POD decomposition is \"optimal\" by its mathematical nature: the first mode (\(\lambda_{1}\mathbf{w}_{1}\mathbf{z}_{1}\)) contains more energy than the second mode (\(\lambda_{2}\mathbf{w}_{2}\mathbf{z}_{2}\)), \(\lambda_{2}\mathbf{w}_{2}\mathbf{z}_{2}\) is more energetic than \(\lambda_{3}\mathbf{w}_{3}\mathbf{z}_{3}\), which is again more energetic than \(\lambda_{4}\mathbf{w}_{4}\mathbf{z}_{4}\)... Consequently, reconstruction of the original \(\mathbf{A}_{m\times n}\) can be approximated by retaining only the leading modes in Equation (26), the number of the retained modes depend on the acceptable error threshold. After applying POD decomposition to the pressure and velocity fields from both CFD calculation and model prediction, the first thing we try to compare is the eigenvalue, as it is a direct measurement of the energy contained in the corresponding POD mode. Figure 14 gives the eigenvalues of 3 case setups of airfoil geometry, AOA and Reynolds numbers, which would be referred to as case 1, case 2 and case 3. For all three cases, the energy cascade is clearly visible from the monotonic decrease of eigenvalues, and model prediction agrees almost exactly with the original CFD calculation for the leading 20 to 30 modes. For the trailing POD modes, however, eigenvalues by the CNN-DCNN model tend to be higher than CFD by an order of magnitude. Considering the optimality characteristic of the POD method, the deviations in the high order range have litter impact on the overall accuracy of prediction, as the low order modes already extract almost the entire system energy from the original flow fields. In this sense, if we are looking at the global effect considering all the POD modes, error of eigenvalue by the CNN-DCNN model is typically lower than 1%, as given in Table 7.  
As eigenvalue provides information in terms of the energy contained by a given POD mode, the intrinsic characteristic of the fluid dynamics is only visible from the distribution of POD mode itself, or more intuitively, the reconstructed flow field by leading POD modes according to Equation (26).