PSI - Issue 78

Marco Pirrò et al. / Procedia Structural Integrity 78 (2026) 1641–1648

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b 1  J accounts for the bias term for the input layer. Then, the decoder maps the hidden patterns h

n by means of:

y n = g ( W

n + b 2 )

(2)

T h

where y n  I is the bias term for the hidden layer. Examples of functions f and g include the sigmoid, the linear functions or the hyperbolic tangent (Finotti et al. 2022). The set of unknown parameters  = ( W , b 1 , b 2 ) can be inferred by minimizing a loss function Z( x n , y n ) , which is a measure of the discrepancy between the input time series x n and its reconstructed counterpart y n . To prevent the network from simply replicating the input at the output – thereby failing to uncover significant underlying patterns – a Sparse Autoencoder (SAE) is utilized (Ng 2011). According to this approach, a sparsity constraint on the hidden layer is introduced, encouraging most neurons to remain inactive during training, which in turn promotes the extraction of more abstract and meaningful features from the input data. I is the reconstructed time series, W T  I  J is the transpose weight matrix and b 2 

Fig. 1. Basic architecture of an AE network.

3. Training of the Sparse Autoencoder Unlike traditional techniques that require separate networks for each data channel, the approach herein proposed employs a unified network to handle the entire dataset X n  M  I , encompassing all M channels simultaneously (see Fig. 1). To effectively represent the structural dynamics, input sequences must be of sufficient length. The training data consists of M  I matrices captured when the structure is presumed to be in a healthy state and under typical EOV. For model training, 75% of these matrices are emplyed, with the remaining 25% held out for validation purposes. Several key hyperparameters are adjusted during this phase, including: (a) the encoder and decoder activation functions, denoted by f and g ; (b) the number of training epochs, usually ranging from 100 to 500; and (c) the number of neurons in the hidden layer (compression rate), which is set between 10% and 50% of the input dimensionality. Hyperparameter optimization is conducted using a strategy based on the grid search procedure (Yang and Shami 2020), which exhaustively evaluates combinations within the search space to find the settings that yield the lowest reconstruction error on the validation dataset. The ADAM optimizer (Kingma and Ba 2015) is employed to minimize the loss function Z( x n , y n ) , quantifying the discrepancy between the original input and the network’s reconstruction. To mitigate numerical instability during training, it is advisable to normalize the input data beforehand (Singh and Singh 2020). 4. Testing of the Sparse Autoencoder After the training phase is completed, newly acquired time series from unknown scenarios are input into the network. For each i -th channel, the reconstruction error is measured using the Mean Absolute Error (MAE) between the original signal and its reconstructed counterpart: