## 4 BP NEURAL Network Modeling  
As discussed in Section 3, each combination of shear stiffnesses of the asphaltic bearings of U-shaped flumes (\(K_{flume}\)), the dynamic elastic modulus of arch trusses (\(E_{d\_arch}\)) and the dynamic elastic modulus of bent frames and moment frame supports (\(E_{d\_frame}\)) will yield a set of primary modal frequencies of the aqueduct in three directions. So, according to the Universal Approximation Theorem of a feed-forward neural network [27], \(K_{flume}\), \(E_{d\_arch}\) and \(E_{d\_frame}\) of the aqueduct can be approximated with a BP neural network using the primary modal frequencies of the structure obtained from the in situ dynamic tests. To fulfill the BP neural network modeling, a large number of modal analyses of the 3D aqueduct model need to be conducted to obtain the training data. Using Equation (2), for a rubber bearing with \(A\) and \(t\) being 250 \(\times\) 400 mm\({}^{2}\) and 2.0 mm, respectively, the shear modulus \(G\) can be taken as 1.2 MPa per the Chinese code \"Laminated Bearing for Highway Bridge (JT/T 4-2019)\", and then the bearing shear stiffness would be calculated as 6 \(\times\) 10\({}^{4}\) kN/m. However, the asphaltic bearing material is not rubber, and its mechanical property is difficult to determine. Taking the hardened asphalt as a reference, the elastic modulus typically falls into a range from 1000 MPa to 9000 MPa, so the bearing shear stiffness could be several thousand times the rubber bearing. Considering other unknown factors, \(K_{flume}\) array is then formed using a geometric progression in a wide range from 4.8 \(\times\) 10\({}^{5}\) kN/m to 1.2288 \(\times\) 10\({}^{8}\) kN/m with a constant multiplier of 2, which contains 9 elements. \(E_{d\_arch}\) and \(E_{d\_frame}\) are taken from two arithmetic progressions from 35 GPa to 60 GPa and from 30 GPa to 55 GPa with a constant increment of 5, respectively, and 20 pairs are randomly selected. Each pair of \(E_{d\_arch}\) and \(E_{d\_frame}\) is combined with each element of the \(K_{flume}\) array, which leads to a total of 180 combinations. These 180 combinations of \(K_{flume}\), \(E_{d\_arch}\) and \(E_{d\_frame}\) are put back into the 3D model individually to perform modal analyses, while the \"m\" value of foundation soil and the dynamic elastic modulus of pile caps and piles are taken as constant values (300 MN/m\({}^{4}\) and 35 GPa, respectively). Each combination will generate one set of primary modal frequencies of the aqueduct (i.e., the first, second and third lateral modes \(f_{T1}\), \(f_{T2}\) and \(f_{T3}\), the first and second longitudinal modes \(f_{L1}\) and \(f_{L2}\), and the first vertical mode \(f_{V1}\),), which finally creates a dataset with 180 lines, and each line contains 9 elements, namely, the six inputs/\(f_{T1}\),/\(f_{T2}\),/\(f_{T3}\),/\(f_{L1}\),/\(f_{L2}\),/\(f_{V1}\) and three outputs \(K_{flume}\), \(E_{d\_arch}\), \(E_{d\_frame}\). In BP modeling, this total dataset is randomly divided into training, validation and testing sets with a division ratio of 8:1:1, so the sizes for the three sets are 144, 18 and 18, respectively. The validation set is employed to optimize the hyperparameters of the BP neural network (i.e., the number of hidden layers, the size of each hidden layer and the transfer function of each layer). Since the size of the training and validation dataset (180) is relatively small, a \(K\)-fold cross-validation process is utilized with \(K=9\). So, the BP neural network will be trained 9 times, and the average MSE (mean square error) and the average MAE (mean absolute relative error) of the 9 validations will be used to determine the optimal architecture of the network. Since the ratio of the minimum input \(K_{flume}\) to its maximum value is only 1/256, it is found in the BP neural network pretraining that approximations are not satisfactory.