The Gray Wolf Optimizer (GWO) is an established algorithm for addressing complex optimization tasks. Despite its effectiveness, enhancing its precision and circumventing premature convergence is crucial to extending its scope of application. In this context, our study presents the Cauchy Gray Wolf Optimizer (CGWO), a modified version of GWO that leverages Cauchy distributions for key algorithmic improvements. The innovation of CGWO lies in several areas: First, it adopts a Cauchy distribution-based strategy for initializing the population, thereby broadening the global search potential. Second, the algorithm integrates a dynamic inertia weight mechanism, modulated non-linearly in accordance with the Cauchy distribution, to ensure a balanced trade-off between exploration and exploitation throughout the search process. Third, it introduces a Cauchy mutation concept, using inertia weight as a probability determinant, to preserve diversity and bolster the capability for escaping local optima during later search phases. Furthermore, a greedy strategy is employed to incrementally enhance solution accuracy. The performance of CGWO was rigorously evaluated using 23 benchmark functions, demonstrating significant improvements in convergence rate, solution precision, and robustness when contrasted with conventional algorithms. The deployment of CGWO in solving the engineering challenge of pressure vessel design illustrated its superiority over traditional methods, highlighting its potential for widespread adoption in practical engineering contexts.

This paper employs regression models based on machine learning to propose a method for predicting the energy level distribution rules of Cr ³⁺ and Fe ³⁺ in various doped crystals.