Issue 72

X. Cao et alii, Frattura ed Integrità Strutturale, 72 (2025) 162-178; DOI: 10.3221/IGF-ESIS.72.12

Miner model [4] MAPE (%)

LV model [6] MAPE (%)

Wang model [7] MAPE (%)

Peng model [8] MAPE (%)

Ye model [5] MAPE (%)

Materials

37.18 24.86 73.10 67.64 92.08

35.12 22.16 65.75 66.15 84.17

26.35 14.43 41.27 59.67 68.43

18.76 16.42 26.80 60.26 53.18

16.48 20.84 19.74 54.29 46.85

Butt joint

Corner joint Al-2024-T42 Al-7070-T7451

Ti-6Al-4V

Table 1: MAPE values of five traditional cumulative damage models.

According to Tab. 1, the Peng model demonstrates a notably smaller prediction error compared to the other models for materials such as butt joint, Al-2024-T42, Al-7075-T7451, and Ti-6Al-4V. The Peng model exhibits a marginally higher error on corner joint as opposed to the LV and Wang models compared to other models. In general, it seems that the Peng model has the smallest prediction error. In this work, the Peng model is selected as the physical loss component to be integrated into the generator loss function of CTGAN. The physical loss component serves as a regularizer within the network, aiding in the improvement of the training procedure. The generator is enabled by the physical loss term to produce data that adheres to physical constraints. The generator's overall loss function in this work is formulated as follows:

=

+ + +

L

L

discrete L L L mmd

(11)

generator

adversarial

physics

adversarial L is the adversarial loss function,

discrete L is the discrete column loss function,

mmd L is the Max Mean

where

physics L is the physics loss function. The adversarial loss function and discrete

Discrepancy (MMD) loss function [25] and column loss function in CTGAN are shown below:

= =− ∑ N N 1 1 i

( ) ( ) i

(12)

L

D G x

log

adversarial

=− ∑∑ N K 1

( ) ′ i k y ,

(13)

L

y

log

discrete

i k ,

N

= = k

i

1 1

where N denotes the quantity of data, i x signifies the input data, ( ) i G x represents the synthetic data generated based on the input, and ( ) ( ) i D G x denotes the discriminator's probability of deeming the synthetic data as authentic. K indicates the count of potential values for the discrete column, i k y , signifies the actual probability of the kth value in the discrete column for the i-th sample, and ′ i k y , denotes the probability of the same kth value generated by the generator for the i-th sample. The generator's adversarial loss function primarily strives to decrease the discriminator's likelihood of classifying the generated sample as fake. In essence, it aims to elevate the probability that discriminator will deem generated sample as authentic. In CTGAN, the loss function for discrete columns typically utilizes cross-entropy to diminish the discrepancy between the probability distribution of the generated discrete data and that of the real data. It ensures a tight correspondence between the probability distribution of the authentic data and that of the synthetic discrete data. This work introduces two additional loss functions to the base CTGAN: the MMD loss function and the physical loss function. Below is the representation of MMD loss function:

2

( ) φ − φ ∑ ∑ N M i y 1

1

( ) j y

(

)

=

mmd L P Q ,

(14)

N

M

=

=

i

j

1

1

where P denotes the probability distribution of the data produced by the generator and Q represents the probability distribution of the real data. N and M are the number of samples obtained by sampling from P and Q , respectively. i y

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