## 1 Introduction
Once trained, the network can be utilized to make predictions about the radiation pattern for new antenna element positions. This technique has been shown to be effective for conformal array synthesis, but it requires a large amount of training data. The performance of large, phased array antennas is scanned using an active element pattern method developed in [18], which accounts for all the mutual coupling effects to correct the field patterns of the array. Similar approach of using active element pattern method has been followed in [19; 20; 21] to synthesis the radiation pattern in conformal arrays. Reference [22] introduces a technique to mitigate random errors in amplitude and phase excitations of array elements, which can disrupt radiation patterns by clustering array elements and discretizing amplitude excitations to restore the desired patterns. Convex optimization theory is applied into the synthesis of conformal arrays with matched dual-polarized patterns, highlighting the importance of achieving polarization compatibility in spherical and cylindrical arrays design [23]. In [24], iterative convex optimization is applied to attain optimal difference patterns for 3D conformal beam-scanning arrays, with a specific focus on addressing the design challenges posed by hemispherical and conical conformal arrays featuring asymmetric radiation apertures. An iterative procedure that linearizes the non-convex power pattern function is used to solve the synthesis problem for narrowband conformal arrays in which the beam pattern synthesis is performed using second-order cone programming (SOCP) that results in a convex subproblem considering the magnitude constraints [25]. In [26], a projection technique is employed to determine the phase and amplitude distribution analytically, resulting in low sidelobe levels for spherical arrays. The projection method is further analyzed in mutually coupled phased arrays for radiation pattern correction in [27; 28; 29]. In conclusion, it has been demonstrated that various correction methods, signal processing techniques, sensor electronics, and amplitude and phase modifications can all be applied to optimize the radiation pattern of a conformal antenna array. In particular, the paper describes a constraint convex optimization algorithm for the radiation pattern synthesis in cylindrical and spherical microstrip conformal array antennas. The goal of this optimization technique is to reduce the impact of surface deformations of the conformal array by finding the accurate weight excitations (phaseand amplitude values) of the array elements. The optimization model is designed to use the convex optimization framework while incorporating constraints on the field pattern. This involves formulating an objective function that minimizes the Euclidean distance between the obtained radiation patterns of the deformed arrays and the desired radiation pattern of the planar array, while also controlling the mainlobe direction, null locations, and also the sidelobe levels. By formulating the optimization problem with appropriate constraints, the weights (amplitudes and phases) for the antenna radiating patches can be determined that will produce the radiation pattern in the deformed array nearly similar to the radiation pattern obtained in planar array. Four prototypes of \(2\times 2\), \(3\times 3\), \(4\times 4\), and \(5\times 5\) microstrip patch array antenna are used as the radiating elements--which are bent from the planar configuration to the prescribed spherical and cylindrical shapes with several radii of curvatures--are evaluated to determine the effectiveness of the proposed optimization algorithm.