# A Study on Aeroengine Conceptual Design Considering Multi-Mission Performance Reliability
## 3 Basic Theory
### Generation and Pre-Processing for Training Scenarios  
In this study, training scenarios are generated by Latin Hypercube Sampling (LHS) method, which is capable to achieve a relatively small but a representative number of samples with different combinations of input parameters. In order to extract m samples in n-dimensional vector space, the implementation of LHS is interpreted as follows [28]:  
1. uniformly divide the cumulative distribution of each variable dimension into m non-overlapping intervals;
2. randomly select a value from each interval;
3. randomly combine m values of each variable are with that of other variable.  
To eliminate the negative influence from the different measurement units of each input parameter, sample pre-processing technique is adopted to normalize each variable dimension to the same numerical range. Based on the Z-score method, the normalized value of samples \(\{x^{(i)}\), \(i=1...\), N\(\}\) in jth dimension is  
Figure 1: Artificial neural network.  
\[\hat{x}_{j}^{(i)}=\frac{x_{j}^{(i)}-\mu_{j}}{\sigma_{j}} \tag{2}\]  
where, \(\mu\) and \(\sigma\) respectively are the mean and standard deviation of samples \(\mathbf{x}^{(i)}\) in jth dimension. Thus, each element of \(\mathbf{x}^{(i)}\) such that columns of \(\mathbf{x}^{(i)}\) are centered to have mean 0 and scaled to have standard deviation 1.  
The probability statistical analysis is undertaken to determine the performance reliability of each training sample. Based on the MC simulation of thermodynamic-based aeroengine model, the performance reliability output of \(k\) state is presented as  
\[R_{k}^{(i)}=\Phi\Bigg{(}\frac{f_{Man}\Big{(}y_{k}\big{(}\mathbf{x}^{(i)},\mathbf{r} \big{)}-y_{k}^{*}\Big{)}}{f_{Std}\Big{(}y_{k}\big{(}\mathbf{x}^{(i)},\mathbf{r}\big{)}- y_{k}^{*}\Big{)}}\Bigg{)} \tag{3}\]  
where, \(\Phi(\cdot)\) is the standard normal distribution function; \(f_{Man}(\cdot)\) and \(f_{Std}(\cdot)\) are the functions for calculating mean and standard deviation, respectively; \(\mathbf{r}\) is the vector of uncertainty parameters characterized by a predefined probability distribution; \(y(\cdot)\) is the performance output response of \(k\) state; \(y^{*}\) is the required performance output. By the way, the obtained samples could be easily transformed by adjusting the value of \(y^{*}\).  
Above all, the pre-processed training samples, which contain normalized input parameters and corresponding reliability outputs in all states, are  
\[\Big{\{}\Big{(}\hat{\mathbf{x}}^{(i)},\mathbf{R}^{(i)}\Big{)},i=1,\ldots,N\Big{\}} \tag{4}\]  
In general, the ANN is trained by Levenberg-Marquardt Algorithm, which started from a set of random initial network parameters according to a given prior distribution. Especially, these different initial networks parameters can converge to different local optima and may exhibit different convergence properties and network performance. Differing from a random initial network, a well pre-trained initial network is sited closer to designated optimum, which improves the convergence property and shortens the follow-up training time [29]. Thus, pre-training technique is adopted to acquire better-conditioned initial network.  
To address this issue, a new hybrid algorithm of APSO and LM (HAPSOLM) is presented to integrate the implementation of pre-training and training process.