## 1 Introduction  
In the realm of geotechnical engineering, the consideration of ultimate bearing capacity (UBC) and allowable settlement continues to be of paramount importance when designing shallow foundations [1; 2; 3]. These aspects are crucial factors that dictate the structural integrity and performance of any construction project. UBC, a fundamental parameter, is deeply intertwined with the shear strength characteristics of the underlying soil, and is decurate estimation is essential for ensuring the stability of foundations. Theoretical formulations proposed by eminent geotechnical scholars such as [PERSON] [4], [PERSON] [5], [PERSON] [6], and [PERSON] [7] have provided a foundation for calculating UBC. However, the application of these theories has revealed that various UBC equations exhibit significant disparities, primarily due to the inherent simplifications and assumptions employed in these approaches [8]. The pursuit of accurate UBC prediction has led to a proliferation of research studies that explore numerical and semi-empirical solutions. The works of [PERSON] et al. [9] and [PERSON] and [PERSON] [10] stand as examples of such efforts, where researchers have sought to elucidate the complex bearing capacity phenomenon. These models are frequently evaluated through meticulously designed footing tests carried out on scale models. Yet, bridging the gap between laboratory-scale experiments and real-world, full-scale foundation scenarios remains a challenge. Scholars, cognizant of the potential scale effects, have undertaken initiatives to minimize discrepancies when extrapolating findings from controlled laboratory conditions to the complexities of real-world projects. This endeavor to account for scale effects, as seen in the works of [PERSON] [11] and [PERSON] et al. [12], underscores the dedication within the geotechnical community to ensuring the accuracy and reliability of their predictions. In a pursuit to unravel the intricate facets of bearing capacity, researchers have embarked on investigations that dissect the influence of factors such as particle size on UBC. The study carried out by [PERSON] et al. [13] exemplifies this endeavor, as it unravels the nuances of soil behavior during model-scale footing experiments. Interestingly, the results drawn from large-scale tests conducted on compacted sand have unveiled pronounced variations in shearing strains along the slip line. These findings have challenged conventional formulas that solely rely on the concept of maximum friction angle (\(\phi_{\text{max}}\)) [14]. This revelation underscores the critical need for a nuanced understanding of the underlying mechanics and limitations of various theoretical equations, ultimately driving the demand for innovative approaches to enhance the accuracy of UBC predictions in real-world scenarios. Enter artificial intelligence (AI), a groundbreaking paradigm that has reshaped various scientific disciplines. The realm of geotechnical engineering, while somewhat reserved in its adoption of AI, has begun to witness the application of AI-based techniques in addressing complex challenges. AI methods such as artificial neural networks (ANNs), fuzzy inference systems (FISs), adaptive neuro-fuzzy inference systems (ANFISs), and others have shown remarkable potential in deciphering intricate relationships within complex datasets across diverse domains such as soil dynamics [15; 16; 17; 18; 19; 20], deep foundations [21; 22; 23; 24], soil cracking [25; 26; 27], recycled materials [28; 29; 30; 31; 32; 33; 34; 35; 36], soil mechanics [37; 38], tunnelling and rock mechanics [39; 40; 41] and other fields [42; 43; 44; 45; 46; 47; 48; 49; 50; 51]. The beauty of these techniques lies in their capacity to capture nonlinear interactions between a myriad of variables, even when the underlying relationships are not fully understood. Recent forays into AI applications in geotechnical engineering have showcased the supremacy of AI over traditional analytical formulas when addressing UBC in shallow foundations [14; 52; 53; 54; 55; 56].