## 3 The Proposed Method  
### Cable Forces Optimization  
In the beginning of this section, the method for determining the optimum cable forces, i.e., Formula (2), is discussed, which builds a foundation for establishing the training database of the force predictor. It should be noted that a rational distribution of cable forces in cable-stayed bridges is generally smoothly distributed without sudden changes. Therefore, B-spline curve presented by French engineer [PERSON], is handy for fitting the cable force distribution in cable-stayed bridges [20; 21; 22]. Compared to the precise optimization of each original cable force, introducing B-spline greatly condenses design variables into a few control points, while causing only a limited impact on the overall structural behavior.  
Suppose a \(p\)-th degree B-spline curve is introduced, the original dense cable prestressing forces \(\mathbf{Q}=\left\{q_{1},q_{2},q_{3},\ldots\ldots q_{n_{q}}\right\}\) can be replaced by its control points \(\mathbf{P}=\left\{p_{1},p_{2},p_{3},\ldots\ldots p_{n_{p}}\right\}\), where \(n_{p}\) is far less than \(n_{q}\). The relation between \(\mathbf{Q}\) and \(\mathbf{P}\) can be described as  
\[q_{k}=\sum_{i=0}^{n_{p}}N_{i,p}(u_{k})p_{i}\text{ for all }k\in\left\{1,2,3, \ldots\ldots,n_{q}\right\} \tag{4}\]  
where \(u_{k}\) is the relative position of the \(k\)-th cable (\(0\leqslant u_{k}\leqslant 1\)), and \(N_{i,p}\) is the shape function. More detail can be found in [22].  
In this work, assisted by the B-spline technique, the following design variables are considered: four fitting points (\(d_{1}\)-\(d_{4}\)) in the side span, four fitting points (\(d_{5}\)-\(d_{8}\)) in the main span, the force of the No. 1 cable (\(d_{9}\)), the range of the counterweight (\(d_{10}\)), and the load magnitude (\(d_{11}\)), as shown in Figure 3. The counterweight and its distribution range are set as variables as a complement to deal with the unbalanced self-weight between the main span and the side.  
Figure 3: Cable forces optimization with B-spline interpolation curve.  
Since Formula (2) is a typical nonlinear optimization problem and the number of design variables is condensed with B-spline curve, it can be efficiently solved with several optimization methods, such as the differential evolution and surrogate-model-assisted differential evolution [17]. The sequential quadratic programming (SQP) solver [23] is adopted in this research due to its high efficiency (note that the initial point should be carefully selected to avoid local optimum solutions).  
During the sampling to construct the predictor, Formula (2) is solved in collaboration with MATLAB R2018b and ANSYS 2022 R1 APDL. For each sample, modeling parameters are derived in the beginning. An initial FEM is built with these modeling parameters in ANSYS. When the cable forces and counterweight are updated during the cable forces optimization in MATLAB, the initial strains of the cables are modified for the FEM and the counterweight is imposed in ANSYS. Then, the static analysis is launched to obtain the bending energy. If convergence is checked, the result of the sample is output to a text file. The sample process is completed when it reaches the target sample count.