Enhancing thermal efficiency and minimizing weight are prevailing issues in aero engines. Owing to its hollow structure, the twin-web turbine disc exhibits remarkable weight reduction properties, while its enhanced cooling constitutes a novel challenge. In this study, a twin-web turbine disc cavity system is numerically investigated. To enhance the cooling effect and minimize pressure loss, a multi-objective genetic algorithm and Kriging surrogate model are employed to optimize the radial height of the pre-swirl nozzle and receiver hole in the disc cavity system. The results indicate that the overall performance of Opt-3, derived from the Technique for Order Preference by Similarity to the Ideal Solution method within the Pareto frontier, is superior. This configuration achieves a uniform low distribution of rotor temperatures while maintaining moderate pressure losses. Notably, the maximum temperature is reduced by 21.1 K compared to the basic model, with pressure losses remaining largely unchanged. Additionally, an increase in the flow ratio leads to a reduction in both the maximum temperature and average temperature of the back web while simultaneously increasing the temperature of the front web and augmenting pressure losses. However, it is important to note that the degree of variation in these parameters diminishes with increasing flow ratios.