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Frequency-selective surfaces (FSSs) have shown great potential in electromagnetic (EM) wave propagation controlling in various fields. Usually, the designs of FSSs are limited to several structure parameters with the assumption of free space or homogeneous background. However, FSSs usually work under inhomogeneous background with various perturbations or devices. In this work, we propose an inverse design method for FSSs with high degrees of freedom, including an artificial neural network (ANN)-based design method and an adjoint gradient-based design method for unit cell and array of FSS, respectively. Specifically, in the ANN-based design method, an initial unit cell is generated by the inverse network, which utilizes both frequency and time information of the demand performance. A key feature of the proposed design method for FSS array is that we use the Green's function under inhomogeneous background in the forward method, where the computational cost is largely reduced by reciprocal theory. To verify the proposed method, with 400 independent variables to be determined at once, a multifunctional FSS with absorption, diffusion, and transmission abilities under inhomogeneous background is designed, fabricated, and experimentally verified, whose unit cell is both phase- and amplitude-adjustable.