## 1 Introduction  
Particle Flow Code in Three Dimensions (PFC3D) is a widely used discrete element modeling (DEM) software in geotechnical and mining engineering, designed to simulate the mechanical behavior of granular materials under various loading conditions. By defining particle contact models and the associated mesoscopic parameters, such as the stiffness ratios, friction coefficients, and bond strengths, PFC3D can replicate the stress-strain behavior and failure mechanisms of geomaterials. However, the accuracy and reliability of PFC3D simulations are highly dependent on the precise calibration of these mesoscopic parameters, which are often not directly measurable in laboratory settings and must be determined through labor-intensive and time-consuming trial-and-error processes [1]. The complexity of this calibration process adds considerable uncertainty, making it challenging to ensure the accuracy of DEM simulations in real-world applications. In PFC3D, commonly used constitutive models include the linear contact model, parallel bond model, and various plastic contact models [2]. Among these, the linear parallel bond model is especially effective in simulating the failure and fracture behavior of brittle materials, such as rocks and concrete [3]. This model is based on linear elasticity theory and represents the particle contact interface as a combination of bonded springs and moments, which can transmit normal and shear forces, as well as torque. Due to its ability to capture both micro-crack initiation and macroscopic failure, it has become one of the most widely used models in PFC-based simulations for geotechnical applications [4]. However, the performance of the parallel bond model is highly sensitive to the calibration of key mesoscopic parameters, such as the normal and shear stiffness, tensile and shear strength limits, and bond radius. These parameters govern the particle's elastic response, failure thresholds, and force transmission zones, directly influencing the simulated fracture patterns and overall mechanical behavior [5]. In parameter calibration studies, researchers often begin by exploring the relationships between macro-scale and meso-scale parameters. Through mechanical simulations and regression analyses, parameter sensitivities are examined to establish empirical formulas that link macro parameters, such as the stress-strain behavior, to mesoscopic parameters, such as the stiffness and bond strength [6]. Macro-meso relationships have also been quantitatively derived through theoretical studies [7; 8]. However, these methods often require a deep theoretical understanding and can be difficult to generalize across different material types and loading conditions. Additionally, they can be computationally intensive, as they require running multiple simulations for different parameter sets to understand the interactions between parameters. To improve the efficiency and accuracy of parameter identification, machine learning (ML) techniques have been increasingly applied in the calibration process of PFC3D models. Common algorithms, such as artificial neural networks (ANNs), Support Vector Machines (SVMs), and Random Forests (RFs), have been successfully used in various geotechnical simulations to predict mesoscopic parameters based on macroscopic responses. In particular, an ANN has powerful nonlinear mapping capabilities that enable it to establish complex relationships between mesoscopic parameters and macroscopic material responses, thus enabling efficient inverse modeling and parameter calibration [9; 10; 11]. Several studies have integrated an ANN with constrained optimization to create hybrid models, where an ANN predicts the macroscopic parameters and optimization adjusts the micro-parameters to meet the constraints [1; 11]. This approach allows for broader exploration of the parameter space, enhancing the calibration process.