## 2 Materials and Methods
### Load Identification Method  
For conceptual introduction, we first assume a simple linear elastic problem. Non-uniform distributed lift loads acting on the eight wing segments of the wing model, a load vector \(\mathbf{L}\), induce strains at individual FBGs, strain vector \(\varepsilon\), which is expressed as  
\[\varepsilon=\mathbf{SL}, \tag{1}\]  
where \(\mathbf{S}\) is a transfer matrix. This is a direct problem where loads are input and strains are output. When applied loads are to be identified from measured strains, it is inversely solved as  
\[\mathbf{L}=\mathbf{S}^{+}\varepsilon, \tag{2}\]  
where \(\mathbf{S}^{+}\) is a generalized inverse matrix of \(\mathbf{S}\). This solution tends to be unstable. In the neural network approach, a neural network is used to replace \(\mathbf{S}^{+}\) as  
\[\mathbf{L}=NN(\varepsilon), \tag{3}\]  
where \(NN\)(x) expresses the output of a neural network when x was the input. In detail, an output of \(i\)th neuron in \(l\)th layer, \(a_{i}^{(l)}\), is expressed by a weighted sum from the previous layer and a bias as  
\[a_{i}^{(l)}=\sigma^{(l)}\left(\sum_{j}w_{ji}^{(l-1)}a_{j}^{(l-1)}+b_{i}^{(l)} \right), \tag{4}\]  
where \(b_{i}^{(l)}\) is the bias of the \(i\)th neuron in the \(l\)th layer and \(w_{ji}^{(l-1)}\) is the weighted coefficient. The weight's subscript \(ji\) represents the link from the \(j\)th neuron in the \((l-1)\)th layer to the \(i\)th neuron  
Figure 6: Examples of flap angles for “random” test cases. \begin{table}
\begin{tabular}{c c} \hline \hline
**Flap No.** & **Flap Angle (deg)** \\ \hline
1 & 15.0 \\
2 & 15.0 \\
3 & 13.0 \\
4 & 0.8 \\
5 & \(-12.1\) \\
6 & \(-15.0\) \\
7 & \(-15.0\) \\
8 & \(-15.0\) \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Flap angle combination for “controlled” test cases. in \(l\)th layer. The weighted sum with the bias is activated through the activation function \(\sigma\). When a simple three-layer neural network is used, the \(i\)th element of \(\mathbf{L}\), \(L_{i}\), is calculated from the strain input as  
\[L_{i}=\left[\sigma^{(3)}\left(\sum_{j}w_{ji}^{(2)}\left[\sigma^{(2)}\left(\sum _{k}w_{ij}^{(1)}\epsilon_{k}+b_{j}^{(2)}\right)\right]_{i}+b_{i}^{(3)}\right) \right]_{i}. \tag{5}\]  
In this manner, the strain-to-load transfer can be represented by the neural network whose parameters are optimized through regression. This contributes to the stability of solutions [16]. In addition, the activation functions enable non-linear transformation, which enhances the applicability to a non-linear problem. We created a neural network to calculate the lift load distribution. The architecture of the neural network, which was a feed-forward neural network with one hidden layer, is depicted in Figure 7. The input layer had 68 neurons, the hidden layer 12 neurons, and the output layer eight neurons, and they were fully-connected. The number of neurons and layers were determined experimentally. For the input layer, we used data from 60 FBGs bonded along the main and aft spars, as well as the data of the eight flap angles. It was evident that the strain had a mechanical cause-and-effect (or rather effect-and-cause) relationship and, therefore, the FBG signals should be used for load identification. Theoretically, the input of the strain information should be sufficient for the load identification as seen in Equations (3) and (5); however, it is anticipated that the strain variations would be less responsive with respect to local load variations specifically at the wing root due to the high stiffness. In order to enhance the responsiveness between the input and output of the neural network, the flap angle information was added as an input in this study. Therefore, the load identification process is redefined as  
\[\mathbf{L}=NN_{load}(\boldsymbol{\varepsilon},\boldsymbol{\delta}), \tag{6}\]  
where \(\boldsymbol{\delta}\) is the eight flap angles.