## 1 Introduction  
Recently, artificial neural network (ANN) has been suggested to support the estimation of ultimate bearing capacity of the circular footing in single homogenous soil environments as well as other engineering materials [1; 2; 3; 4]. The study of \(F_{ult}\) (maximum applied stresses corresponding to a particular settlement, i.e., 0.10 foundation width) of foundations is essential to geotechnical engineering and soil mechanics. In addition, complex geological structure, e.g., bearing capacity of shallow footings lied on multilayer soil (or non-homogeneous soil), are important to be considered. Traditional approaches are based on complex solutions, i.e., utilizing limit equilibrium consideration [5; 6; 7; 8; 9] or extensive experimental approaches [10; 11; 12; 13; 14]. In many cases, the suggested solutions illustrated how the top layer thickness (or its ratio to the footing width) affects the \(F_{ult}\) of the shallow footing placed on two or more layered soils. In this regard, the settlement and bearing capacity of shallow footings have been proven to depend on several key factors, namely foundation soil parameters, the shape of the footing, and number of soil layer beneath the footing. The complexity of the problem is raised by the addition of soil layers under the footing. Different equations have been proposed to compute the soil bearing capacity (e.g., in a particular settlement) of circular and square footings [15; 16; 17]. The main problem of predicting the ultimate bearing capacity of the circular footing is to reduce or minimize its likelihood of high settlement after real stresses are used. The most effective parameters in predicting a correction amount for the bearing capacity are (i) soil parameters (or layered soils beneath footing), (ii) footing shape (e.g., strip, circular, and rectangular); and (iii) soil layers arrangement [18]. It is important to note that the soil properties (e.g., dilation angle, unit weight, internal friction angle and cohesion, and Poisson's ratio, as well as elastic modulus) can apply stresses on the footing. In general, the ultimate bearing capacity of the circular footing has been determined to be the ultimate applied stress in the case of the maximum ratio of settlement (footing settlement/footing width or S/B) that is 10% of the footing width [5; 19; 20]. There are many factors that can affect bearing capacity of the shallow footing like multilayer soil and geological conditions, failure model considered through the calculation, the stronger soil location (for example, the arrangement of soil layer), type of the soil, and footing width [21]. For providing a reliable prediction of the ultimate bearing capacity (\(F_{ult}\)) for shallow footings placed on multi-layered soils, many scholars like [PERSON] [22], [PERSON] and [PERSON] [23], [PERSON] and [PERSON] [10], and [PERSON] and [PERSON] [24] suggested novel formulas.  
In this study, to forecast the \(F_{ult}\) of the circular footing, 84 ANN models and 58 hybrid models (for example, helping the ANN for providing an efficient outcome), namely (i) imperialist competition algorithm (ICA) and (ii) particle swarm optimization (PSO) were designed. The hybrid ICA-NN and PSO-ANN models provided here were not utilized in the engineering-based example provided in this study. There does not exist a study conducted on the use of the proposed hybrid model to predict the \(F_{ult}\) of circular footing rested on multilayer soil conditions. Thus, this study aimed to optimize ANN with two well-developed hybrid optimization algorithm models with a reliable approximation of circular footing's bearing capacity rested on layered soils.