## 1 Introduction  
Functionally graded materials (FGMs) are composite materials with advanced and programmable structures whose physical and mechanical properties can vary in different directions. Recently, with the major technological breakthroughs made in fields such as materials engineering and artificial intelligence (AI), the design and optimization of FGMs have come into prominence with great momentum. Substantial investments made in the past few decades, especially in areas such as materials and software engineering, have recently begun to bear fruit. In this context, the importance of functional materials created using smart materials has increased considerably in fields such as energy, electronics, aircraft and space engineering, construction, and defense. To be more precise, a good analysis of such functional materials will maximize the efficiency of the designed engineering structure. Undoubtedly, functionally graded materials (FGMs) have an important place among the mentioned materials. FGMs have significant potential in the analysis of composite materials, with their unique physical and mechanical properties. Functional materials are generally obtained by combining two different materials that have a certain harmony between them. These are often composite structures designed in ceramic-metal or ceramic-polymer forms, with smooth transitions from one surface to another. Although there are many different methods for analyzing functional materials, the most prominent are Euler-Bernoulli beam theory (CBT), Timoshenko beam theory (first-ordershear deformation theory), higher-order shear deformation theories, and nonlocal approaches. First-order shear deformation theory takes into account the shear effect, which is ineffective in CBT. The existence of advanced theories that consider shape factor, which arises depending on the geometry of the composite material and allows us to calculate the deformation most accurately, has turned research towards higher-order theories. Hence, the need for enhanced theories has become inescapable. To avoid the use of the shear correction factor and to better predict the behavior of FGM beams, third-order shear deformation theories [1,2], sinusoidal theory [3], and hyperbolic deformation theory [4] have been proposed. Using various higher-order shear deformation beam theories, bending and free vibration analyses of functionally graded beams are developed [5]. In the present study, the free vibration behavior of an FG beam is analyzed via both higher-order shear deformation theory and artificial neural network (ANN) methods. The ANN method is a kind of AI method that allows advanced algorithms to accurately predict material behavior. Owing to this cutting-edge method, the computational time is also considerably reduced. Looking back over the past decade, machine learning, the latest trend in computer systems, has attracted great attention. The main reason for this is that thanks to this system, dynamic data can be stored, interpreted, and, most importantly, easily integrated into science and industry. Moreover, current studies have shown that thanks to this technology, engineering applications and problems can be tested and interpreted very quickly. In this way, free vibration frequencies can easily be tested for different deformation theories in research. First, a mathematical model is proposed using higher-order shear deformation theory (HSDT). Then, kinematic relations and constitutive equations are obtained using [PERSON]'s law. The analytical solution is performed for simply supported boundary conditions. Then dimensionless frequencies are attained using [PERSON]'s solution procedure. To validate and enrich the present research, an ANN method was developed and trained using material properties data obtained from previous studies. Recently, the researchers of [6] studied nonlocal FGM nanoplates using higher-order isogeometric analysis and the ANN method to predict free vibration behavior.