## 4 Deep Belief Network for Peak Wavelength Prediction  
A DBN is a generative graphical model consisting of multiple hidden unit layers. Each pair of connected layers forms a restricted Boltzmann machine (RBM) [22; 23; 24; 25; 26], which serves as the core building block of a DBN. An RBM is a two-layer network capable of learning to represent complex data distributions [22; 25]. It comprises a visible layer (v) for input data and a hidden layer (h) for learned features. The nodes in these layers are interconnected by undirected edges, with no connections within a layer, simplifying the learning process and rendering the RBM efficient for training. The energy of a configuration, including visible and hidden states in an RBM, is determined by [25]:  
\[E(\mathrm{v},\mathrm{h})=-\sum_{i}a_{i}v_{i}-\sum_{j}b_{j}h_{j}-\sum_{i,j}v_{i }w_{i,j}h_{j} \tag{1}\]  
where \(v_{i}\) is visible units, \(h_{j}\) is hidden units, \(a_{i}\) is bias for visible units, \(b_{j}\) is bias for hidden units, and \(w_{i,j}\) is the weight matrix between visible and hidden units. Figure 4: At each strain step, the wavelength information of the two FBG wavelengths and the wavelength information of the interferometric spectrum which is assumed to be used for sensing. Figure 3: (**a**) The strain on the FBG that senses the strain is 0 \(\upmu\) for the three different interference scenarios. (**b**) The strain on the FBG that senses the strain is 420 \(\upmu\) for the three different interference scenarios. (**c**) The strain on the FBG that senses the strain is 720 \(\upmu\) for the three different interference scenarios. The architecture of a DBN is constructed by stacking several RBMs on top of each other, as shown in Figure 5. The visible layer of the first RBM serves as the input layer of the DBN. The hidden layer of the first RBM becomes the visible layer for the next RBM in the stack, and this process continues for each subsequent layer. This hierarchical structure allows the DBN to learn increasingly abstract representations of the input data as it moves up the layers. The training of a DBN typically involves a two-step process: pre-training and fine-tuning. During the pre-training phase, each RBM is trained independently in a greedy, layer-by-layer manner, starting from the bottom-most RBM. This unsupervised pre-training step is crucial in effectively initializing the network's parameters, like the network's weights, in a way that captures the underlying structure of the data. The pre-training acts as a form of feature extraction, allowing the network to learn useful representations of the data [23]. Once all the layers are pre-trained, the entire DBN can be fine-tuned using supervised learning techniques if labeled data are available. This involves adding a final output layer and using backpropagation to adjust the weights across the entire network to minimize the error in predictions [24]. The fine-tuning process ensures that the learned representations are optimized for the specific task at hand, such as classification or regression. The combination of unsupervised pre-training and supervised fine-tuning makes DBNs powerful tools for learning from complex, high-dimensional data, providing a robust framework for capturing intricate patterns and structures in the data. Their ability to learn hierarchical representations of data makes them powerful for handling complex datasets. The hierarchical nature of DBNs means that higher layers capture more abstract features, enabling more accurate and efficient learning and prediction [23; 24]. The process of utilizing a DBN for peak wavelength detection in FBGs and interferences involves several well-defined steps, each crucial for building a robust and accurate model. There were 3 different interferometric modes and 13 different FBG strained settings in the experiment, and the total number of interferometric spectral modes and FBG spectral modes of union was 39 different spectra.