# Recurrent Deep Learning for Beam Pattern Synthesis in Optimized Antenna Arrays  
[PERSON]  
1 Consejo Nacional de Humanidades, Ciencia y Tecnologia (CONAHCYT), Facultad de Ciencias, Universidad Autonoma de San Luis Potosi (UASLP), San Luis Potosi 78295, Mexico 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 3  
antenna array's geometry and complex excitations (amplitudes and phases) of individual antenna elements. Classical iterative methods to solve the problem of beam pattern synthesis exist and are typically categorized as either local or global optimizers, with some of them being based on Fourier transforms [4; 5]. Nevertheless, despite their proven reliability and robustness, a vast majority of these methods require considerable computational time, being dependent on the synthesis of a large number of radiation patterns. Other traditional optimization methods, such as evolutionary algorithms for beam pattern synthesis in uniform linear antenna arrays, have notable drawbacks, such as high computation costs [6; 7], sensitivity to parameters [6; 8], and convergence issues [9]. To address these challenges, alternative approaches are often pursued. In recent years, innovative deep learning (DL) approaches have permeated numerous research areas and practical applications. This trend is primarily attributed to advancements in the development of specialized hardware, such as GPUs, along with improvements in neural network algorithms and the availability of vast datasets for training these systems [10]. With its remarkable progress, this technology has made significant strides across diverse fields, showcasing notable success in areas such as computer vision, natural language processing, speech recognition, healthcare, robotics, recommendation systems, financial modeling, climate science, bioinformatics, material inspection, and quality control [11; 12]. Furthermore, there are other areas, such as electromagnetic (EM) problems [13], where its recent integration has been observed. In this context, DL methodologies have emerged as a pivotal tool for synthesizing and designing a variety of configurations and types of antennas [14]. In antenna characterization, DL techniques have proven to be quite useful for reducing the number of EM simulations, which are generally computationally expensive [15]. In the same way, DL approaches have been successfully applied to various antenna problems, including antenna synthesis [16; 17]. The ability to synthesize beam patterns through machine learning techniques--particularly support vector machines (SVM) and artificial neural networks (ANN)--has previously been demonstrated [14; 18]. However, these methodologies have inherent limitations in terms of learning different beam patterns; specifically, ANNs without the use of deep hidden layers may be unable to provide antenna design parameters for arbitrary antenna beam patterns [19]. On the other hand, it is noteworthy that a large majority of DL methodologies proposed to solve these types of electromagnetic problems are based on convolutional neural networks (CNNs or ConvNets) [20]. Thus, various studies have proposed deep convolutional neural network architectures with eight layers in different configurations. In [21], a method for two-dimensional synthesis of a reflectarray with a circular aperture to predict the phase shift was presented. This study employed five convolutional layers (CLs) and three fully connected layers (FCLs). Similarly, another study proposed a different CNN with four CLs and four FCLs, which was trained to obtain the phases of an \(8\times 8\) patch antenna array with a two-dimensional radiation pattern as an input [22]. Additionally, in reference [23], other authors proposed a CNN architecture with five CLs and three FCLs to determine the complex excitations of a planar array with circular aperture. Their focus was on maximizing only the directivity, doing so with a fixed constraint mask on the side lobes.