###### Abstract  
Assessing the stiffness of circular foundations is the key to evaluating their deformation; thus, it is important for foundation design. The current determination methods for the stiffness coefficient are either time-consuming or inaccurate. In this paper, a novel stiffness prediction model has been proposed, using the decision tree (DT) algorithm optimized by particle size optimization (PSO). The condition of the embedded foundation, the embedded depth (\(Z_{D}\)/2\(R\)), the thickness of the clay layer beneath the foundation base (\(T\)/2\(R\)), and the ratio of shear stiffness between clay and sand (\(G_{\text{sand}}\)/\(G_{\text{clay}}\)) were used as input variables, while the elastic stiffness coefficients (\(K_{\text{c}}\), \(K_{\text{h}}\), \(K_{\text{m}}\), and \(K_{\text{v}}\)) were used as output variables. The optimum DT model has undergone comprehensive validation, and independent model verification using extra simulations. The results illustrate that PSO could promote further increases in the capability of DT modeling in predicting stiffness coefficients. The optimum DT model achieved a good level of performance on stiffness coefficient modeling. (The R for the training set was greater than 0.98 for all of the stiffness coefficients.) The variable importance analysis showed that the \(T\)/2\(R\) was the most significant variable for all stiffness coefficients, followed by \(G_{\text{sand}}\)/\(G_{\text{clay}}\). The optimum DT model achieved good predictive performance upon independent verification, with the R being 0.97, 0.99, 0.99, and 0.95 for \(K_{\text{v}}\), \(K_{\text{h}}\), \(K_{\text{m}}\), and \(K_{\text{c}}\), respectively. The proposed reliable and efficient DT-PSO model for stiffness coefficients in layered soil could further promote the safe and efficient utilization of circular foundations. elastic stiffness; circular footing; machine learning; soil-structure interaction +
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usually layered with sand and clay. The complexity of the problem also makes it difficult to propose a simple equation that can accurately predict the foundation stiffness under different loading and embedding conditions. Recently, Ref. [5] performed an extensive elastic analysis on circular foundations on clay overlaying sand under general loading (VHM). The influence of the foundation's embedded condition, the thickness of clay layer beneath the foundation base, and the ratio of shear stiffness between the clay and the sand on the foundation elastic stiffness were quantified. Approximate design equations were proposed. However, limited by the intrinsic feature of the explicit equation, the error in the prediction for certain conditions can be greater than 25%. As the prediction of foundation stiffness under general loading is a typical multi-input/output problem, it is difficult to capture the influence of all of the factors with a simple explicit equation. In addition, three-dimensional elastic simulation is time-consuming and requires extensive pre-experience. Therefore, it is meaningful to have an efficient and accurate model for predicting the foundation stiffness, particularly for large projects (e.g., offshore wind farms) where such calculations need to be conducted repeatedly. The powerful ability of the machine learning (ML) technique in dealing with multi-input/output problems offers us an alternative solution for foundation design. ML technology can directly learn the relationship between the input variables and the output studied and effectively solve complex nonlinear mapping problems. ML technology has been widely used in many fields of geotechnical engineerings, such as parameter identification [6], the development of constitutive models [7], the evaluation of soil liquefaction [8], tunneling [9,10], and rock excavation [11,12]. Apart from the accurate prediction of the outputs, the ML technique can also \"smartly\" select the most relevant inputs for each output and help the researcher to identify the controlling failure mechanism.