## 2 Mathematical Methods
### Reconstruction Algorithm Based on POD  
Temperature data from the objective plane are used to present the two-dimensional temperature field. CFD simulations can be used to obtain temperature data under different boundary conditions; temperature data under the \(i\)th boundary condition are denoted \(\mathbf{t}_{\mathrm{i}}\), and \(\mathbf{t}_{\mathrm{i}}\) is an n-dimensional vector, showing the temperature values that are sampled in a fixed order from \(n\) different CFD grid nodes on the objective plane. After \(m\) times CFD calculation, a temperature database of \(\mathbf{T}_{\mathrm{nxm}}[\mathbf{t}_{\mathrm{I}},\mathbf{t}_{\mathrm{2}}, \mathbf{t}_{\mathrm{3}},\cdots,\mathbf{t}_{\mathrm{i}},\cdots,\mathbf{t}_{ \mathrm{m}}]\) is set up. Mathematically, \(\mathbf{T}\) is a high-dimensional matrix, and directly processing this matrix turns out to be an impossible task. In the POD framework, this task is simplified by extracting a dimension reduction model \(\mathbf{X}_{\mathrm{lxm}}[\mathbf{x}_{\mathrm{1}},\mathbf{x}_{\mathrm{2}},\mathbf{x}_{\mathrm{3 }},\cdots,\mathbf{x}_{\mathrm{m}},\mathrm{1<n}]\) from \(\mathbf{T}\), while retaining the most important information. This process can be described as follows:  
\[\mathbf{X}=\mathbf{W}^{\mathrm{T}}\mathbf{T}, \tag{1}\]  
where \(\mathbf{W}_{\mathrm{nxd}}\) is constituted by \(\mathbf{w}_{\mathrm{1}},\mathbf{w}_{\mathrm{2}},\mathbf{w}_{\mathrm{3}},\cdots,\mathbf{w}_{ \mathrm{1}}\), and \(\mathbf{w}_{\mathrm{1}},\mathbf{w}_{\mathrm{2}},\mathbf{w}_{\mathrm{3}},\cdots,\mathbf{w}_{ \mathrm{1}}\) are eigenvectors of the \(\mathbf{T}\) covariance matrix, which are associated with the first \(l\) largest eigenvalues. Moreover, the importance of the POD basis vector \(\mathbf{w}_{\mathrm{i}}\) is related to the magnitude of the eigenvalue \(\lambda_{\mathrm{i}}\). It is commonly defined as follows:  
\[\mathbf{r}=\frac{\lambda_{\mathrm{i}}}{\sum_{\mathrm{1}}^{\mathrm{n}}\lambda_{ \mathrm{i}}}, \tag{2}\]  
where the ratio \(\mathbf{r}\) shows the \"energy\" content of \(\mathbf{w}_{\mathrm{i}}\)[34]. In addition, when the dimension of matrix \(\mathbf{T}\) is too high to calculate the covariance matrix, using a singular value decomposition (SVD) can also obtain POD basis vectors [12]. Equation (1) can be written as \((\mathbf{x}_{\mathrm{1}},\mathbf{x}_{\mathrm{2}},\cdots,\mathbf{x}_{\mathrm{i}},\cdots,\bm {x}_{\mathrm{m}})=\mathbf{W}^{\mathrm{T}}(\mathbf{t}_{\mathrm{1}},\mathbf{t}_{ \mathrm{2}},\cdots,\mathbf{t}_{\mathrm{i}},\cdots,\mathbf{t}_{\mathrm{m}})\), and combining it with \(\mathbf{W}^{\mathrm{T}}\mathbf{W}=\mathbf{I}\), for \(\forall\mathbf{t}_{\mathrm{i}}(\mathbf{t}_{\mathrm{i}}\in\mathbf{T})\) yields Equation (3):  
\[\mathbf{f}_{\mathrm{i}}=\mathbf{W}\mathbf{x}_{\mathrm{i}}, \tag{3}\]  
where \(\mathbf{x}_{\mathrm{i}}\) is used as a coefficient vector; thus, temperature information can be represented as a linear combination of the POD basis vectors and a coefficient vector. In practical application, the POD basis vectors and limited temperature data are obtained before the reconstruction process, and only the coefficient vector is unknown. Thus, the temperature field reconstruction task is transformed into a coefficient optimization problem. Some temperature data can be measured from the undetermined temperature field; this measurement process can be presented as follows:  
\[\mathbf{f}_{\mathrm{M}}=\mathbf{M}\mathbf{t}, \tag{4}\]  
where \(\mathbf{f}_{\mathrm{M}}\) is a k-dimensional vector, and represents the measurement data, \(\mathbf{t}\) is an n-dimensional vector that represents the undetermined temperature field, \(\mathbf{M}_{\mathrm{kxm}}\) is a measurement matrix that represents the measurement sites, and k shows the sensor number that is used in reconstruction.