PSI - Issue 52

Wu Zonghui et al. / Procedia Structural Integrity 52 (2024) 203–213 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

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Fig. 1. Multi-layer neural network.

The loss function is the pith of an ANN, and MSE (mean square error), given as function (3), is a wildly used loss function in functional approximation. ( ) 2 1 1 s MSE i i i L y y s = = −  (3) But this function merely focuses on the absolute distance between prediction values and true values, which will lead to a disappointing result easily, especially when the point is close to the limit state function g ( X ). MSE effects differently to the points which have a disparate distance to the limit state function, like y 1 and y 2 in Fig.2: the 1 y and 2 y are same-radius circles with center at y 1 and y 2 , which represent the same loss of these two points. But prediction value 1 y of the closer point y 1 may get an opposite sign value more easily which is the worst situation as positive or negative values have quite different meanings in structural reliability. Hence two advanced loss functions are proposed herein as function (4) and function (5). Mean square absolute percentage error (MSAPE) and mean absolute percentage error (MAPE) give each sample prediction a better criterion and a more equitable weight to approach true value no matter whether the point is close to or far from the limit state function by dividing the norm of the sample, illustrated in Fig.3. 2 (5) These two loss functions give a smaller but fairer weight to all samples no matter if their value is far from or close to the LSF. It brings robustness to the ANN for fighting against a small amount of incorrect data, but a problem comes together: turning the hyper-parameters in ANN becomes more laborious. MSE is an outstanding loss function in function approximation with correct data, but its advantage will become a shortcoming when the sample set includes incorrect data. In this situation, the MSAPE and MAPE, especially the MAPE, can be taken as a substitute for the MSE to gain a better performance. MSAPE 1 1 s =   i i i i y y  −  L s y =    (4) MAPE 1 1 s =  i i i i y y − L s y =