## 1 Introduction
However, there are currently few studies on the machine learning-based reliability calculation in deep soft rock tunnel engineering. Therefore, the reliability of deep soft rock tunnels needs to be further studied based on machine learning algorithms such as Gaussian process regression (GPR). In Ref. [8], we proposed the NSGA-III-XGBoost based stochastic reliability analysis method for deep soft rock tunnels. The NSGA-III-XGBoost method uses the Latin hypercube sampling method to generate dataset samples of geo-mechanical parameters and adopts XGBoost to establish a model of the nonlinear relationship between displacements and surrounding rock mechanical parameters. And NSGA-III is used to optimize the surrogate model hyperparameters. Finally, the failure probability is computed by the optimized surrogate model. The advantages of the NSGA-III-XGBoost method were demonstrated through a comparison between the support vector regression method and the backpropagation neural network method. In this study, an Adam-Bayesian-Gaussian process regression (ABGPR) with combined kernel-function-based Monte Carlo simulation reliability method (ABGPR-MCSRM) is proposed to assess the reliability of non-circular deep soft rock tunnels. The Latin hypercube sampling (LHS) method is adopted to obtain the geo-mechanical parameter dataset sample for the reliability analysis of a deep soft rock tunnel. Three combined kernel functions of the Gaussian process regression algorithm (GPRA) are constructed and compared with the existing basic kernel functions to select the kernel function with more accurate and effective reasoning and learning. Then, the surrogate model of numerical simulation is constructed using GPRA, and the adaptive momentum (Adam) algorithm is used to optimize the hyperparameters of the surrogate model. And the probability distribution of geotechnical parameters and the optimal surrounding rock mechanical parameters are obtained by using the Bayesian inference algorithm. Finally, the tunnel reliability is calculated using the Monte Carlo simulation reliability method (MCSRM) through the ABGPR-based optimal surrogate model. The advantages of the proposed method are demonstrated through a comparison between the NSGA-III-XGBoost-based method and the Kriging-based method. The purpose of this study is to provide some novel insights for the probabilistic reliability analysis of deep soft rock tunnels more rationally. The main contribution of this study is that we construct a combinatorial kernel function, which can be used for more accurate and effective reasoning and learning. We also establish a more accurate ABGPR-based numerical surrogate model than the proposed surrogate model in Ref. [8], which can provide a high-precision probabilistic reliability evaluation method (ABGPR-MCSRM) of deep rock tunnel engineering with fewer prediction errors and stronger prediction ability than the NSGA-III-XGBoost-based method or the Kriging-based method. The research is organized as follows: i. the Adam-Bayesian-GPR-based MC (ABGPR-MCS) reliability calculation procedure framework is implemented in Section 2; ii. Section 3 presents one engineering example for verifying the feasibility, validity, and advantages of the proposed framework; iii. the conclusion is presented in Section 4.