# A Deep Learning-Based Approach for the Identification of a Multi-Parameter BWBN Model
## 2 The BWBN Model
### Sensitivity Identification  
The sensitivity of every parameter of BWBN model is obviously different for RC columns under different failure modes. The importance of parameters can be sorted out by sensitivity identification, and some insensitive parameters can be fixed in the calculation. This improves the efficiency of parameter identification, which can retain those parameters that are more sensitive to the structural response and lock the other parameters to reduce the calculation amount and improve the computational efficiency. The commonly used sensitivity identification method is global sensitivity analysis. The global sensitivity analysis method can consider the influence of the interaction between multiple parameters on the output response of the model, and the representative method is the Sobol method [67; 68]. The Sobol method decomposes the model into a single parameter and a function consisting of the combination of parameters. By analyzing the influence of the variance of the single parameter and the influence of the combination of the parameters on the total variance of the model, the importance of the parameters and the degree of the interaction between the parameters are evaluated. The Sobol method decomposes the model \(f(X)\) into orthogonal polynomials of all subterms. \[f(X)=f_{0}+\sum_{i=1}^{n}f_{i}(x_{i})+\sum_{i=1}^{n}\sum_{j=i+1}^{n}f_{i,j}(x_{i },x_{j})+\ldots+f_{1,2,\ldots,n}(x_{1},\ldots,x_{n}) \tag{16}\]  
where \(f_{0}\) is the integral of all parameters, \(f_{i}(x_{i})\) is the integral of all parameters except \(x_{i}\), and \(f_{i,j}(x_{i},x_{j})\) is the integral of all parameters except \(x_{i}\) and \(x_{j}\). \[ \begin{split}& f_{0}=\int f(X)dX=E(Y)\\ & f_{i}(x_{i})=\int_{0}^{1}\ldots\int_{0}^{1}f(X)dX_{-i}-f_{0}=E( Y|x_{i})-E(Y)\\ & f_{i,j}(x_{i},x_{j})=\int_{0}^{1}\ldots\int_{0}^{1}f(X)dX_{-(i, j)}-f_{0}-f_{i}(x_{i})-f_{j}(x_{j})=E\big{(}Y|x_{i},x_{j}\big{)}-E(Y)-f_{i}-f_{j} \end{split}  \tag{17}\]  
In order to describe the influence degree of all parameters on the output response of the model, the total variance is derived by taking the variance on both sides of Equation (15) by using the orthogonal condition:  
\[ \begin{split}& V=\int f^{2}(X)dX-f_{0}{}^{2}\\ &=E\big{(}V(Y|x_{i})\big{)}+V(E(Y|x_{i})\big{)}\end{split}  \tag{18}\]  
The partial square error of a single parameter on the output response of the model can be given by:  
\[V_{i}=\int f_{i}^{2}(x_{i})dx_{i} \tag{19}\]  
which can describe the influence degree of multiple parameters on the output response of the model. The partial square difference of multiple model parameters can be expressed by:  
\[V_{i_{1},\ldots,i_{k}}=\int_{0}^{1}\ldots\int_{0}^{1}f^{2}{}_{i_{1},\ldots,i_ {k}}(x_{1},\ldots,x_{s})dx_{i_{1}}\ldots dx_{i_{k}} \tag{20}\]  
The differences between conditional variance \(V(Y|x_{i})\) and total variance \(V(Y)\) express the influence of model parameters \(x_{i}\) for the model \(Y\). The Sobol main effect sensitivity index is defined by:  
\[S_{X_{i}}=\frac{V(E(Y|x_{i}))}{V(Y)} \tag{21}\]  
where \(S_{X_{i}}\) represents the influence of variables on the total variance of model. It is the main effect sensitivity index of \(X_{i}\). In order to consider the interaction between the main effect index of the model parameter \(x_{i}\) with other remaining parameters, the total effect-sensitivity coefficient of the model parameter can be obtained by:  
\[S_{X_{i}}^{T}=\frac{V(Y)-V(E(Y|x_{-i}))}{V(Y)} \tag{22}\]  
where \(S_{X_{i}}^{T}\) includes the main effect of parameter \(x_{i}\) and the interaction influence of parameter \(x_{i}\) with other parameters. The multi-parameter BWBN model can be used to analyze various structural hysteresis phenomena, such as mechanical engineering, civil engineering and geotechnical engineering. Although the BWBN model has strong universality, the model parameters do not have specific physical meaning. Not only the influence of the change of single parameter value, but also the combination of multiple parameters should be considered.