## 1 Introduction  
In the field of structural engineering, the use of steel cellular beams has gained widespread attention thanks to their economic and aesthetic advantages [1, 2]. Currently, as a result of their particular design, they are widely used in structural systems subject to bending or funicular structures [3; 4; 5]. The I-shaped cellular steel beams are formed from standard I-section beams by cutting regularly circular openings, a specific geometrical feature that results in numerous advantages, for example, the optimum self-weight-depth ratio, improvement of flexural stiffness, or larger section modulus [3]. Furthermore, in combination with the significant compressive strength of concrete slab, the flexural resistance of composite cellular beam is significantly improved [2; 6]. Safety at construction projects is an important matter, which is strongly related to the stability of the constituent structural elements, in other words, the bucking behavior of cellular beams. In the literature, various experimental, analytical, and numerical studies have been performed to investigate the buckling behavior of beams [7; 8; 9]. As regards to I-shaped beams [5], a numerical investigation was carried out on the buckling modes of cellular beams using ABAQUS software. An important change in the failure loads was observed when subjected to a combination of web distortional and web-post buckling failing modes. In addition, the lateral-torsional buckling behavior of cellular beams was investigated by [PERSON] and [PERSON] [10] who took into account the effect of the residual stress. In another attempt, the bucking behavior of steel I-shaped cellular beams was discussed and numerous parametric studies were used to assess the variation of the moment-gradient coefficient [4]. This study also provided a modification to evaluate the critical lateral buckling moment of the analyzed beams under different loading conditions. Last but not least, an experimental and analytical study on the behavior and strength of steel beams was performed by [PERSON] and [PERSON] [11]. An empirical formula with the ability to predict ultimate vertical shear load strength was proposed using a finite element approach in a parametric analysis. In general, the mechanical simulation approach or laboratory experiments can only be applied to a limited case [12]. Moreover, they are costly and require significant amounts of time [13]. Therefore, the development of a straightforward and rapid numerical tool to solve the bucking behavior problem of steel I-shaped cellular beams is imperative [1; 12; 14]. In the past few decades, machine learning (ML) approaches have been the subject of extensive interest and expanding research in the field of materials science [15], especially in structural engineering [8; 9; 16; 17; 18; 19]. Currently, the rapid and constant development of computational software and hardware has facilitated the development of numerous alternative computer-aided data mining approaches. For example, [PERSON] [20] developed an artificial neural networks (ANN) algorithm for predicting the hardness of concrete reinforced with randomly oriented steel fibers. The proposed ANN model has been compared with selected experiments and has shown an extremely high utilization rate. In addition, other studies [21; 22; 23; 24; 25; 26] have also confirmed the strong predictability of different ML models for the behavior of structural elements and different materials in the field of mechanics under different solicitations [8; 9; 27; 28; 29; 30]. Consequently, the main objective of this paper is the development and comparison of different hybrid ML models to predict the critical buckling load of I-shaped steel cellular beams. In this regard, three hybrid ML models, namely ANFIS-RCSA, ANFIS-CA, and ANFIS-SFLA, which are a combination of adaptive neuro-fuzzy inference system (ANFIS) with metaheuristic optimization techniques such as real-coded simulated Annealing (RCSA), cultural algorithm (CA) and shuffled frog leaping algorithm (SFLA), were developed and applied.