Efficient sampling-based inverse reliability analysis combining Monte Carlo simulation (MCS) and feedforward neural network (FNN)

Abstract

Inverse reliability analysis evaluates a percentile value of a performance function when the target reliability is given. In cases of high dimensional or highly nonlinear performance functions, sampling-based methods such as Monte Carlo simulation (MCS), Latin hypercube sampling, and importance sampling are considered to be better candidates for reliability analysis. The sampling-based methods are very accurate but require a large number of samples, which can be very time-consuming. Therefore, this paper proposes an efficient and/or accurate sampling-based reliability analysis method without using a surrogate model. The proposed method helps to improve the accuracy of reliability analysis with the same number of samples or to ensure the same accuracy of reliability analysis with fewer samples. This study starts with an idea of training relationship between limited samples constituting realization of the performance distribution—usually between 10 and 100—and its corresponding true percentile value where the performance distribution is defined as a one-dimensional distribution resulted from a performance function and its random variables. To this end, feedforward neural network (FNN), which is one of promising artificial neural network (ANN) models that approximate high dimensional models using layered structures, is introduced in this study, and limited samples constituting realizations of various performance distributions and their corresponding true percentile values are used as input and target data, respectively. Various beta distributions are used to create the training data sets. A FNN training method using kernel density estimation and equidistant points to represent the kernel distribution data is also proposed to remove dimensionality of the training inputs. Comparative study shows that the proposed method training FNN with samples constituting realization of the performance distribution (Method 2) is more accurate than a method that directly estimates the percentile value from the kernel distribution fitting the samples constituting realization generated through MCS (Method 1). In addition, compared to Method 2, another proposed method that trains FNN with the kernel density estimation and equidistant points (Method 3) is more accurate in reliability analysis and more computationally efficient in FNN training. Method 3 is also applicable to high reliability problems, and it is more accurate than Kriging-based method for high dimensional problems.

Appendices

Appendix A: Determining the range of shape parameters of beta distributions

This section shows how the range of shape parameters of beta distributions is determined. For several ranges of shape parameters, validation for performance distributions generated from various performance functions and random variables is conducted using FNN trained with 20,000 training data sets generated from each range of shape parameters. To this end, various performance functions are created through the polynomial functions given by

$$g\left( {\mathbf{X}} \right) = AX_{1}^{{p_{1} }} + BX_{2}^{{p_{2} }} + CX_{1}^{{p_{3} }} X_{2}^{{p_{4} }} + D$$

(12)

where the coefficients A, B, C, and D are randomly drawn from $[ - 10,10]$, and the exponents p₁, p₂, p₃, and p₄ are integers randomly drawn from $[ - 10,10]$. It is assumed that $X_{1} \sim N(\mu_{{X_{1} }} ,\sigma_{{X_{1} }} )$ and $X_{2} \sim N(\mu_{{X_{2} }} ,\sigma_{{X_{2} }} )$ where each distribution parameter is randomly drawn from $[0,10]$. For 1,000 validation sets of performance distributions, normalized true percentile values for 95% reliability are predicted using Method 3 with 100 samples (i.e., n = 100) and the NRMSE results according to range of shape parameters are given in Fig. 15 in Appendix. The range of $[0,20]$ shows the NRMSE that converges to the smallest value, and therefore, the range is used in this study because it is verified that sufficiently diverse beta distributions can be generated.

Fig.15

NRMSE results according to range of shape parameters

Appendix B: Application on a bimodal performance distribution case

This section shows whether percentile value estimation using the proposed reliability analysis framework is applicable for bimodal performance distributions. To estimate the percentile value of the bimodal performance distribution, training data sets are created by generating bimodal distributions. After generating two beta distributions from Eq. (5), a bimodal distribution can be generated by combining the two beta distributions into one bimodal distribution through kernel density estimation. In this way, bimodal distributions with various characteristics can be generated, and training data sets can be created with these distributions as shown in Sect. 2.3. Validation for various bimodal distributions generated by mixing the distributions of two random variables, which are used in Appendix A, through kernel density estimation is conducted using FNN trained with 20,000 training data sets. For 1,000 validation sets of bimodal distributions, normalized true percentile values for 95% reliability are predicted with 100 samples (i.e., n = 100). The NRMSEs of Methods 1 and 3 are 0.0957 and 0.0857, respectively. The NRMSE of Method 3 is improved by 10.4% compared to that of Method 1, and 25 more samples are required for Method 1 to have the same accuracy as Method 3. This verifies that the proposed reliability analysis framework can also be applied to bimodal performance distributions.