PSI - Issue 84

Elisa Tomassini et al. / Procedia Structural Integrity 84 (2026) 288–295

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training dataset of 1024 samples. For each sampled configuration, a modal analysis was performed using the calibrated FEM, yielding resonant frequencies and mode shapes. The mode shapes were projected onto 30 modal displacements, matching the positions of the uniaxial accelerometers installed beneath the deck along the five spans. The resulting training dataset captures the functional relationship between the non-dimensional damage-sensitive input parameters and the modal properties of the bridge, providing a physically consistent and damage-sensitive basis for SM training. The FNN was trained for 1000 epochs using the source training dataset, with 15% of the samples reserved for internal validation, to construct the SM. The custom loss function defined in Eq. (2) was employed to jointly evaluate errors in modal frequencies and mode shapes. In particular, the contributions of the frequencies and of the mode shapes were defined as: ℒ ( ) = ∑ | 1 − | ̂ ( ) − ̅̅̅ | | = 1 , ℒ ( )= ∑ ( 1 − ( ̂ ( ), ̅̅̅ )) = 1 , (3) where and are weighting parameters taken equal to 1, while is a scaling factor taken equal to 100. Network hyperparameters were tuned to ensure stable convergence and high predictive accuracy. Training was performed using the Adam optimizer with an initial learning rate of 2×10 −3 , a momentum coefficient of 0.98, and a mini-batch size of 4. Data shuffling was applied at each epoch to improve generalization. As shown in Fig. 4(a, b), both training and validation losses stabilized after approximately 500 epochs. The generalization capability of the SM was further assessed using an independent validation dataset containing 512 samples, generated via the same LHS strategy. Figure 14(c, d) compares predicted and FEM-computed modal frequencies and reports MAC values between predicted and reference mode shapes. For all the modes, the coefficient of determination exceeded 2 =0.99 , and MAC values remained above 0.98, confirming the high fidelity of the surrogate model in reproducing the bridge’s modal response.

Fig. 4. Training of the FNN on the data of the source bridge. (a) Frequency training and validation loss; (b) mode shape training and validation loss; (c) comparison between predicted frequencies ̂ and FEM reference values and (d) MAC values computed between the predicted mode shapes ̂ and the FEM mode shapes for each sample in the validation dataset.