## 4 Deformation Force Monitoring Based on Adaptive Adjustment of Clamping Force
### Deformation Force Prediction Based on Deep Autoregressive Neural Network  
To reduce the risk of supports failure after the machining and prevent excessive clamping deformation due to the application of excessive pre-support force, a probabilistic model is proposed to predict the machining deformation force, specifically targeting a part like aero-engine casing. Based on the estimated distribution parameters, a confidence level \(\alpha\) can be selected so that \(P\left(\hat{F}_{b}\geq F_{b}\right)=\alpha\). The proposed model employs the deep autoregressive neural network [42], which is a probabilistic model based on LSTM [43], which can predict a probability interval.  
Let \(\mathbf{z}_{i,t}\) denote the value of a time series \(i\) at time \(t\); the objective of a probabilistic model is to establish a conditional distribution as in (15):  
\[P\left(\mathbf{z}_{i,t_{0:T}}\left|\mathbf{z}_{i,1:t_{0-1}},\mathbf{x}_{i,1:T}\right.\right) \tag{15}\]  
where \(\mathbf{x}_{i,t}\) is the covariate, which is known at any time. This distribution can predict the current value of system \(\mathbf{z}_{i,T}\) based on the historical data \(\mathbf{z}_{i,1:t_{0}-1}:=\left[z_{i,1},z_{i,1}\dots z_{i,t_{0}-2},z_{i,t_{0}-1}\right]\) and covariate \(\mathbf{x}_{i,t}\). The model is based on an autoregressive neural network [44; 45] (the autoregressive recurrent network architecture). The model distribution \(Q_{\Theta}\left(\mathbf{z}_{i,t_{0:T}}\mid\mathbf{z}_{i,1:t_{0}-1},\mathbf{x}_{i,1:T}\right)\) consists of multiple likelihood multipliers, as in (16):  
\[Q_{\Theta}\left(\mathbf{z}_{i,t_{0:T}}\mid\mathbf{z}_{i,1:t_{0}-1},\mathbf{x}_{i,1:T}\right) =\prod_{t=t_{0}}^{T}Q_{\Theta}(\mathbf{z}_{i,t}\mid\mathbf{z}_{i,1:t-1},\mathbf{x}_{i,1:T })=\prod_{t=t_{0}}^{T}\ell(\mathbf{z}_{i,t}|\theta(\mathbf{h}_{i,t},\Theta)) \tag{16}\]  
where \(\mathbf{h}_{i,t}=h(\mathbf{h}_{i,t-1},z_{i,t-1},\mathbf{x}_{i,t},\Theta)\) and \(h\) is a recurrent neural network composed of LSTM units. The likelihood function \(\ell(z_{i,t}\mid\theta(\mathbf{h}_{i,t},\Theta))\) is a fixed distribution whose parameters are output from the neural network. This distribution can be selected according to the learning objective, e.g., the Gaussian distribution or Binomial distribution. Based on our past experiences, in this work the Gaussian distribution is selected, which is determined by two parameters \(\theta=(\mu,\sigma)\), where \(\mu\) and \(\sigma\) are respectively the mean and variance of Gaussian distribution, and the likelihood function can be calculated as in (17):  
\[\ell_{\text{G}}(\mathbf{z}\mid\mu,\sigma)=\left(2\pi\sigma^{2}\right)^{-\frac{1 }{2}}\exp\left(-(\mathbf{z}-\mu)^{2}/\left(2\sigma^{2}\right)\right) \tag{17}\]  
The mean and variance of Gaussian distribution are calculated according to (18) and (19):  
\[\mu(\mathbf{h}_{i,t})=\mathbf{w}_{\mu}^{T}\mathbf{h}_{i,t}+b_{\mu} \tag{18}\]  
\[\sigma(\mathbf{h}_{i,t})=\log\left(1+\exp\left(\mathbf{w}_{\sigma}^{T} \mathbf{h}_{i,t}+b_{\sigma}\right)\right) \tag{19}\]  
Finally, the model is trained by maximizing the likelihood function, as in (20):  
\[\mathcal{L}=\sum_{i=1}^{N}\sum_{t=l_{0}}^{T}\log\ell(\mathbf{z}_{i,t}|\Theta( \mathbf{h}_{i,t})) \tag{20}\]  
The structure of our deformation force prediction model based on the deep autoregressive neural network is shown in Figure 3. It takes the historical deformation force data \(F_{b0:t-1}\) as input and outputs the subsequent machining step deformation force.