## 2 Methods  
### Improved N-BEATS Architecture  
The N-BEATS architecture in this paper differs from the existing architecture. Its fundamental block employs a fully connected nonlinear regressor (Figure 2a), using nonlinear multivariate regression to predict the next step's solution [21]. Moreover, N-BEATS uses the principle of residual stacking to stack multiple layers together (Figure 2b), and its superiority has been fully demonstrated through extensive experimentation. By leveraging this principle, the model can effectively stack layers according to different tasks, thus fully learning the underlying patterns of the time series and showing excellent performance in subsequent predictions. At this stage, the basic module predicts future outputs while backtracking the input sequence to decompose its contribution to future predictions. The sum of the contributions from multiple basic blocks forms the output of the corresponding stack (Figure 2c). Conceptually, each fully connected layer can be regarded as a multivariate linear regression block, followed by a Rectified Linear Unit (ReLU) nonlinear layer. Therefore, improving the structure of the N-BEATS basic block can be viewed as a repeated multivariate regression with nonlinearity. Mathematically, each block of the improved N-BEATS comprises a sequence of fully connected layers, ending with a forecast/backcast split. This architecture performs residual recursion within each stack and sums the outputs of all stacks to make the final prediction. Assuming that the model has \(M\) stacks and \(R\) residual blocks, with each residual block containing \(L\) hidden layers, the architecture takes \(x\in\mathbb{R}^{w}\) as the input. The model predicts a window of length \(H\) and backcasts over a window of length \(\omega\), using superscripts (\(r,l\)) for the residual blocks and layers, and denotes the Fully Connected Layers (FC) with weights \(\mathbb{W}^{r,l}\) and \(\mathbb{b}^{r,l}\) as follows:  
\[\text{FC}_{r,l}\Big{(}\mathbb{h}^{r,l-1}\Big{)}\equiv\text{ReLU}\Big{(} \mathbb{W}^{r,l}\mathbb{h}^{r,l-1}+\mathbb{b}^{r,l}\Big{)} \tag{1}\]  
Figure 1: Framework of blade stiffness degradation prediction. Assume that \(\xi^{0}\equiv 0,x^{0}\equiv x\), where the projection matrix has dimensions \(\text{B}^{\prime}\in\text{R}^{u\times d_{h}},\text{F}^{\prime}\in\text{R}^{H \times d_{h}}\), The operations for improving the N-BEATS model can be described as follows:  
\[ \begin{array}{c}x^{\prime}=\text{ReLu}\left[x^{\prime-1}-\hat{x}^{\prime-1} \right]\\ h^{\prime,1}=\text{FC}_{\tau,1}(x^{\prime}),\ldots,h^{\prime,L}=\text{FC}_{ \tau,L}\left(h^{\prime,L-1}\right)\\ \hat{x}^{\prime}=\text{B}^{\prime}h^{\prime,L},\hat{y}^{\prime}=\text{F}h^{ \prime,L}\end{array}  \tag{2}\]  
In the formula, \(x^{\prime}\) represents the input of the \(r\)th block and \(\hat{x}^{\prime}\) represents the approximate backcast output of the \(r\)th block. \(\hat{y}^{\prime}\) represents the forecast output of the \(r\)th block. \(\text{B}^{\prime}\) and \(\text{F}^{\prime}\) represent the backcast matrix and forecast matrix, respectively. \(h^{\prime,L}\) represents the output of the \(L\)th layer in the \(r\)th block. \(d_{h}\) represents the dimension of the hidden layer. The first residual block serves as the initial input of the model. For the other block, the backward residual branch can be considered as a continuous analysis of the input signal, where the preceding block removes the signal components that it can approximate well \(\hat{x}\), thereby making the prediction task for downstream blocks easier. This structure also facilitates smoother gradient backpropagation. More importantly, the partial predictions output by each block are first aggregated at the stack level and then at the entire network level, providing a hierarchical decomposition. The final prediction is the sum of all partial predictions. The model's hierarchical aggregation can be formulated as follows:  
\[ \begin{array}{c}y_{m}=\sum\limits_{r=1}^{R}\hat{y}^{\prime}\\ y=\sum\limits_{m=1}^{M}y^{m}\end{array}