## 1 Introduction  
Hydraulic fracturing has been extensively employed as a reservoir stimulation technique in the development of unconventional shale gas resources. In recent years, CO\({}_{2}\)-based fracturing fluids have emerged as a promising alternative to traditional water-based fluids, offering several technical advantages, such as minimal damage to the reservoir, improved fluid recovery, and the dual benefits of enhancing methane desorption and enabling CO\({}_{2}\) sequestration [1]. When injected CO\({}_{2}\) exceeds its critical temperature of 31.1 \({}^{\circ}\)C and critical pressure of 7.38 MPa, it transitions to a supercritical state [2]. Supercritical CO\({}_{2}\) (SC-CO\({}_{2}\)) exhibits unique physicochemical properties, such as low viscosity, near-zero surface tension, high diffusion coefficients, and a density approaching that of water. These properties contribute to its superior heat and mass transfer capabilities, making SC-CO\({}_{2}\) fracturing a promising technology for stimulating unconventional oil and gas reservoirs [3, 4]. Breakdown pressure remains one of the key parameters in assessing fracturing effectiveness. Mathematical models for calculating breakdown pressure are widely used in the analysis of fracturing mechanisms. Traditional models are predominantly based on elasticity theory and fracture mechanics, focusing on tensile or shear strength criteria [5]. Among these models, the calculation of breakdown pressure based on tensile strength is the most prevalent. [PERSON] and [PERSON] (1976) first proposed a formula to calculate breakdown pressure, which was later refined by [PERSON] and [PERSON] (1976) to incorporate the effects of rock permeability. Additionally, the point-stress model proposed by [PERSON] (1981) (i.e., [PERSON] model) suggests that fracturing initiates at the borehole wall when a point within the rock, at a certain distance from the wall, reaches the tensile strength of the rock. Experiments conducted by [PERSON] et al. (2019) on cylindrical shale specimens using liquid CO2 (L-CO2), water, and nitrogen led to a modified fitting formula for breakdown pressure; however, it does not account for the influence of fluid properties associated at different CO2 phases, particularly under varying confining and axial pressures. In the context of SC-CO2 fracturing, [PERSON] et al. (2019) developed a breakdown pressure prediction model incorporating the unique properties of SC-CO2. This model, based on [PERSON]'s (1981) equation for fluid pressure distribution, was compared with hydraulic fracturing results. However, significant discrepancies between laboratory experiments and theoretical predictions were observed. Currently, no universal theoretical model exists that can fully explain the abnormal pressure phenomena observed in SC-CO2 fracturing experiments. Due to the high costs and resource demands of large-scale field operations, most current research on hydraulic fracturing relies on laboratory-scale physical simulations. However, these experiments present their challenges, such as complex testing processes, extended durations, and significant variability in rock properties. To address these limitations, researchers have increasingly turned to artificial intelligence (AI) methods to investigate fracturing stimulation issues. Specifically, these methods have been applied in predicting reservoir geological parameters ([PERSON] and [PERSON] (2018)), calibrating the constitutive model ([PERSON] and [PERSON] (2018)), optimizing fracturing design parameters ([PERSON] and [PERSON] (2018)), evaluating fracturing effectiveness ([PERSON] and [PERSON] (2018)), predicting shale gas sweet spots ([PERSON] and [PERSON] (2018)), and developing fracturing tools and materials ([PERSON] and [PERSON] (2018)). In the context of the fracturing effectiveness, [PERSON] (2018) employed neural networks and AI algorithms such as stepwise regression to predict fracturing effectiveness based on field data.