## 4 Breakdown Pressure Prediction with a Multi-Layer Neural Network  
### Dataset Preparation and Model Verification  
As supported by [PERSON] et al. [28], theoretical models for calculating breakdown pressure are developed within different theoretical frameworks and assumptions, each focusing on a unique set of controlling factors. This variability restricts the applicability of these models. For example, the H-W model considers only horizontal principal stress, tensile strength, and pore pressure, while omitting factors such as wellbore diameter and pressurization rate. Although the Ito model incorporates wellbore diameter, it still assumes a constant pressurization rate and overlooks the impact of fracturing fluid temperature. As a result, numerous researchers have compared experimental data with breakdown pressures derived from these theoretical models, highlighting their inability to adequately account for abnormally high or low breakdown pressures observed during hydraulic and SC-CO\({}_{2}\) fracturing [37; 38]. In other words, the fracturing of rock specimens during hydraulic fracturing and SC-CO\({}_{2}\) fracturing is a complex process involving the interaction of multiple factors. Currently, no comprehensive mathematical model encompasses all these variables for predicting breakdown pressure. Furthermore, established models indicate that not all influencing factors exhibit a linear relationship with breakdown pressure. Similarly, the factors affecting fracturing effectiveness are numerous and interconnected in complex, nonlinear ways [39]. If breakdown pressure, a key indicator for fracturing effectiveness, could be predicted prior to experimentation or field operations, it would provide valuable reference data for optimizing fracturing design parameters. Presently, large-scale field fracturing experiments are costly and cannot be conducted repeatedly. Consequently, the trend leans towards small-scale laboratory experiments, which also require complex testing procedures and rock preparation [19]. Therefore, we aim to predict breakdown pressure under the influence of multiple factors by identifying the primary factors influencing breakdown pressure, collecting a substantial dataset from fracturing experiments, and employing neural networks to model the intricate nonlinear relationships between input and output variables. Building upon extensive literature research [40; 41; 42; 43; 44; 45; 46; 47; 48; 49] and our experimental data, we compiled a dataset consisting of 362 specimens. While this dataset provides a foundation for the model, it is insufficient for training a deep learning model. Therefore, special measures were taken to mitigate overfitting and improve model generalization, such as cross-validation, regularization, and parameter tuning [50]. The theoretical models discussed above do not account for vertical principal stress, the effect of wellbore diameter, and the minimal impact of the rock's elastic modulus on fracturing outcomes [14]. To address these limitations, we selected eight influencing factors for modeling: fracturing fluid type, maximum and minimum horizontal principal stresses, initial fracturing fluid temperature, wellbore diameter, flow rate, tensile strength, and Poisson's ratio of the specimens. Given the highly complex and nonlinear relationships between these factors and breakdown pressure, a simple multiple-linear regression approach would be insufficient. Therefore, we employed a multi-layer fully connected neural network, as illustrated in Figure 7, to learn representations from the experimental data. This neural network comprises three main components: an input layer (shown in orange), multiple hidden layers (shown in yellow and blue), and an output layer (shown in dark blue). The input layer receives data corresponding to the aforementioned eight influencing factors, which are then passed to the hidden layers through weighted connections (depicted by black arrows in Figure 7).