An active learning method for high-dimensional and small failure probability problems combining matrix-operation radial basis function model with matrix-operation hybrid optimization algorithm

To address the computational challenges in high-dimensional reliability problems with small failure probabilities, this paper proposes an active learning reliability method based on a matrix-operation radial basis function model and a matrix-operation hybrid optimization algorithm. The method searches for the most probable point on the surrogate limit state surface to construct an optimal instrumental probability density function and generate candidate samples. By continuously updating the design of experiments and retraining the matrix-operation radial basis function model, the surrogate limit state surface progressively approaches the true limit state surface. To efficiently identify the most probable point in high-dimensional space, a matrix-operation hybrid optimization algorithm is developed by integrating the matrix-based genetic algorithm with the adjoint gradient-based sequential quadratic programming method. The matrix-based genetic algorithm significantly reduces the global search time by leveraging matrix operations, while the adjoint gradient-based sequential quadratic programming leverages the adjoint gradient information of the matrix-operation radial basis function model predictions to reduce the computational cost of gradient evaluations in local optimization. The effectiveness of the proposed method is validated through four high-dimensional reliability problems. The proposed method demonstrates strong competitiveness compared to various existing high-dimensional reliability analysis approaches.