Balancing the exploration-exploitation trade-off in active learning for surrogate model-based reliability analysis via multi-objective optimization

Reliability assessment of engineering systems is often hindered by the need to evaluate limit-state functions through computationally expensive simulations, rendering standard sampling impractical. An effective solution is to approximate the limit-state function with a surrogate model iteratively refined through active learning, thereby reducing the number of expensive simulations. At each iteration, an acquisition strategy selects the next sample by balancing two competing goals: exploration, to reduce global predictive uncertainty, and exploitation, to improve accuracy near the failure boundary. Classical strategies, such as the U-function and the Expected Feasibility Function (EFF), implicitly condense exploration and exploitation into a scalar score derived from the surrogate predictive mean and variance, concealing the trade-off and biasing sampling. We introduce a multi-objective optimization (MOO) formulation for sample acquisition in reliability analysis, where exploration and exploitation are explicit, competing objectives. Within our framework, U and EFF correspond to specific Pareto-optimal solutions, providing a unifying perspective that connects classical and Pareto-based approaches. Solving the MOO problem discards dominated candidates, yielding a compact Pareto set, with samples representing a quantifiable exploration-exploitation trade-off. To select samples from the Pareto set, we adopt the knee point and the compromise solution, and further propose a strategy that adjusts the trade-off according to reliability estimates. Across benchmark limit-state functions, we assess the sample efficiency and active learning performance of all strategies. Results show that U and EFF exhibit case-dependent performance, knee and compromise are generally effective, and the adaptive strategy is robust, consistently reaching strict targets and maintaining relative errors below 0.1%.