Existing active strategies for training surrogate models yield accurate structural reliability estimates by aiming at design space regions in the vicinity of a specified limit state function. In many practical engineering applications, various damage conditions, e.g. repair, failure, should be probabilistically characterized, thus demanding the estimation of multiple performance functions. In this work, we investigate the capability of active learning approaches for efficiently selecting training samples under a limited computational budget while still preserving the accuracy associated with multiple surrogated limit states. Specifically, PC-Kriging-based surrogate models are actively trained considering a variance correction derived from leave-one-out cross-validation error information, whereas the sequential learning scheme relies on U-function-derived metrics. The proposed active learning approaches are tested in a highly nonlinear structural reliability setting, whereas in a more practical application, failure and repair events are stochastically predicted in the aftermath of a ship collision against an offshore wind substructure. The results show that a balanced computational budget administration can be effectively achieved by successively targeting the specified multiple limit state functions within a unified active learning scheme.

The paper presents a new efficient and robust method for rare event probability estimation for computational models of an engineering product or a process returning categorical information only, for example, either success or failure. For such models, most of the methods designed for the estimation of failure probability, which use the numerical value of the outcome to compute gradients or to estimate the proximity to the failure surface, cannot be applied. Even if the performance function provides more than just binary output, the state of the system may be a non-smooth or even a discontinuous function defined in the domain of continuous input variables. In these cases, the classical gradient-based methods usually fail. We propose a simple yet efficient algorithm, which performs a sequential adaptive selection of points from the input domain of random variables to extend and refine a simple distance-based surrogate model. Two different tasks can be accomplished at any stage of sequential sampling: (i) estimation of the failure probability, and (ii) selection of the best possible candidate for the subsequent model evaluation if further improvement is necessary. The proposed criterion for selecting the next point for model evaluation maximizes the expected probability classified by using the candidate. Therefore, the perfect balance between global exploration and local exploitation is maintained automatically. The method can estimate the probabilities of multiple failure types. Moreover, when the numerical value of model evaluation can be used to build a smooth surrogate, the algorithm can accommodate this information to increase the accuracy of the estimated probabilities. Lastly, we define a new simple yet general geometrical measure of the global sensitivity of the rare-event probability to individual variables, which is obtained as a by-product of the proposed algorithm.

In many fields of science and engineering, models with different fidelities are available. Physical experiments or detailed simulations that accurately capture the behavior of the system are regarded as high-fidelity models with low model uncertainty, however, they are expensive to run. On the other hand, simplified physical experiments or numerical models are seen as low-fidelity models that are cheaper to evaluate. Although low-fidelity models are often not suitable for direct use in reliability analysis due to their low accuracy, they can offer information about the trend of the high-fidelity model thus providing the opportunity to explore the design space at a low cost. This study presents a new approach called adaptive multi-fidelity Gaussian process for reliability analysis (AMGPRA). Contrary to selecting training points and information sources in two separate stages as done in state-of-the-art mfEGRA method, the proposed approach finds the optimal training point and information source simultaneously using the novel collective learning function (CLF). CLF is able to assess the global impact of a candidate training point from an information source and it accommodates any learning function that satisfies a certain profile. In this context, CLF provides a new direction for quantifying the impact of new training points and can be easily extended with new learning functions to adapt to different reliability problems. The performance of the proposed method is demonstrated by three mathematical examples and one engineering problem concerning the wind reliability of transmission towers. It is shown that the proposed method achieves similar or higher accuracy with reduced computational costs compared to state-of-the-art single and multi-fidelity methods. A key application of AMGPRA is high-fidelity fragility modeling using complex and costly physics-based computational models.

This paper presents a semisupervised system responses evaluations, easy-to-evaluate surrogate models learning framework for dimension reduction and have been utilized as substitutes for computationally expensive reliability analysis. An autoencoder is first adopted for mapping simulations or experiments. Popular choices for surrogate the high-dimensional space into a low-dimensional latent space, models in the literature include, support vector machines (SVM) which contains a distinguishable failure surface. Then a deep [4-7], Kriging models [8-10], and artificial neural networks [11-feedforward neural network (DFN) is utilized to learn the 14]. Given a set of training data, surrogate models can be mapping relationship and reconstruct the latent space, while the constructed and then MCS can be directly carried out for Gaussian process (GP) modeling technique is used to build the reliability analysis. Research efforts have been devoted to surrogate model of the transformed limit state function. During developing adaptive sampling strategies [15-18], which aim at the training process of the DFN, the discrepancy between the balancing the fidelity of the surrogate model and the costs of actual and reconstructed latent space is minimized through semisupervised function evaluations.

Large uncertainties in many phenomena of interest have challenged the reliability of pertaining decisions. Collecting additional information to better characterize involved uncertainties is among decision alternatives. Value of information (VoI) analysis is a mathematical decision framework that quantifies expected potential benefits of new data and assists with optimal allocation of resources for information collection. However, a primary challenge facing VoI analysis is the very high computational cost of the underlying Bayesian inference especially for equality-type information. This paper proposes the first surrogate-based framework for VoI analysis. Instead of modeling the limit state functions describing events of interest for decision making, which is commonly pursued in surrogate model-based reliability methods, the proposed framework models system responses. This approach affords sharing equality-type information from observations among surrogate models to update likelihoods of multiple events of interest. Moreover, two knowledge sharing schemes called model and training points sharing are proposed to most effectively take advantage of the knowledge offered by costly model evaluations. Both schemes are integrated with an error rate-based adaptive training approach to efficiently generate accurate Kriging surrogate models. The proposed VoI analysis framework is applied for an optimal decision-making problem involving load testing of a truss bridge. While state-of-the-art methods based on importance sampling and adaptive Kriging Monte Carlo simulation are unable to solve this problem, the proposed method is shown to offer accurate and robust estimates of VoI with a limited number of model evaluations. Therefore, the proposed method facilitates the application of VoI for complex decision problems.

This paper develops mfEGRA, a multifidelity active learning method using data-driven adaptively refined surrogates for failure boundary location in reliability analysis. This work addresses the issue of prohibitive cost of reliability analysis using Monte Carlo sampling for expensive-to-evaluate high-fidelity models by using cheaper-to-evaluate approximations of the high-fidelity model. The method builds on the Efficient Global Reliability Analysis (EGRA) method, which is a surrogate-based method that uses adaptive sampling for refining Gaussian process surrogates for failure boundary location using a single fidelity model. Our method introduces a two-stage adaptive sampling criterion that uses a multifidelity Gaussian process surrogate to leverage multiple information sources with different fidelities. The method combines expected feasibility criterion from EGRA with one-step lookahead information gain to refine the surrogate around the failure boundary. The computational savings from mfEGRA depends on the discrepancy between the different models, and the relative cost of evaluating the different models as compared to the high-fidelity model. We show that accurate estimation of reliability using mfEGRA leads to computational savings of around 50% for an analytical multimodal test problem and 24% for an acoustic horn problem, when compared to single fidelity EGRA.

In many instances, information on engineering systems can be obtained through measurements, monitoring or direct observations of system performances and can be used to update the system reliability estimate. In structural reliability analysis, such information is expressed either by inequalities (e.g. for the observation that no defect is present) or by equalities (e.g. for quantitative measurements of system characteristics). When information Z is of the equality type, the a-priori probability of Z is zero and most structural reliability methods (SRM) are not directly applicable to the computation of the updated reliability. Hitherto, the computation of the reliability of engineering systems conditional on equality information was performed through first- and second order approximations. In this paper, it is shown how equality information can be transformed into inequality information, which enables reliability updating by solving a standard structural system reliability problem. This approach enables the use of any SRM, including those based on simulation, for reliability updating with equality information. It is demonstrated on three numerical examples, including an application to fatigue reliability.