PSI - Issue 51

Kenichi Ishihara et al. / Procedia Structural Integrity 51 (2023) 62–68 K. Ishihara and T. Meshii / Structural Integrity Procedia 00 (2022) 000–000

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test data set, because, in other words, ML is “interpolation of learning data”. In concrete, the desired function f and g are obtained using the train data set and validated by the test data set. This is because, if all of the learning data for this interpolation were used, there will be no data available to validate the obtained f and g . This technique is called cross-validation. Here, 486 learning data set were split 340 versus 146 into train and test data set, which is a standard ratio 7 to 3 in cross-validation. Note that, if obtained f and g are applied to the train data, naturally the true value and the prediction results should match, and it is expected that the data is plotted on the 45-degree line. First, Python's default hidden layer sizes 1 and typical default parameters were used to learn 340 train data and the function f and g were obtained, and f and g were tested, for the train data. Because train data was used, predicted ( a , b ) were naturally expected to lie on the 45-degree line, but did not at all. In order to evaluate this result quantitatively, a coefficient of determination R 2 of 0.04 was calculated, which was far from 1 when the plots are completely on the 45-degree line. By the way, when the obtained f and g were tested for the 146 test data in the same way, naturally the results deviated further from the 45-degree line than case when train data was used, and R 2 was 0.002. The used conditions of main parameters are shown as Case A in Table 1. Next, the result when the number of hidden layer sizes 4 is shown in Fig. 5 (a) and as Case B in Table 1, among the results of examination performed while adjusting the parameters several times. The hidden layers are the first layer of 100, the second layer of 50, the third layer of 25, and the fourth layer of 10. R 2 was ranged from 0.58 to 0.64, that is, R 2 was improved by changing the number of hidden layer sizes from 1 to 4. Although the number of hidden layer sizes and the number of each layer neurons can be set arbitrarily, depending on the issue, overfitting can easily occur if the number is increased too much. When the results are obtained as overfitting, the fit to the training data is good, but the prediction accuracy for data other than the learning data may suffer.

Table 1 The conditions of main parameters.

Value Case A Case B Case C

Parameters

1st 2nd 3rd 4th

100 - - - relu

100 50 25 10 relu

100 50 25 10 relu

hidden layer sizes

activation

solver

adam

adam

lfbgs

max_iter

200

200

500

(a) Case B

(b) Case C

Fig. 5. Comparison between the predicted value and true value: The left figure (a) is the result of Case B and the right figure (b) is the result of Case C.