This paper presents a conformal aperture inverse design framework based on the inverse design method based on spectral analysis (IDMBSA) to address efficiency and applicability challenges in conformal aperture design. The framework initially employs IDMBSA to devise planar aperture field distributions, subsequently extending them to curved surfaces through full-wave simulations based on the equivalence principle and uniqueness theorem. Furthermore, we examine the influence of field truncation on the implementation of the equivalence principle. The results demonstrate that this framework effectively reduces the dimensionality of conformal array design problems, significantly improving efficiency for conformal aperture design. The feasibility of the method is validated through simulations, prototype fabrication, and testing for sparse conformal arrays. The innovation of this paper lies in combining IDMBSA with the equivalence principle, proposing a versatile and efficient conformal aperture inverse design framework. This approach not only handles various complex curved surfaces but also substantially enhances design efficiency, offering new solutions for conformal antenna applications in aerospace and other fields.