# Three-Dimensional Broad Learning Gravity Data Inversion Using Single-Anomaly Training Samples
## 1 Introduction  
Gravity inversion plays a crucial role in geophysics, enabling the determination of subsurface density structures for various practical applications, including the exploration of geological resources [1; 2; 3; 4], the identification of geological structures [5; 6], the investigation of groundwater resources [7; 8], the delineation of oil and gas [9], the revelation of hazardous phenomena [10], etc. Due to the complexity and heterogeneity of subsurface materials, a two-dimensional perspective often results in simplified and distorted representations of density structures, potentially overlooking important structural features [11; 12]. Therefore, three-dimensional (3D) inversion has become essential and increasingly popular in gravity data inversion for retrieving subsurface density structures with a high spatial resolution and precision [13; 14]. However, the increase in complexity due to higher-dimensionalscales imposes greater demands on the computational efficiency and global optimization capabilities of the algorithms applied in the inversion process. Advancing inversion algorithms to handle such complexity is crucial for efficiently achieving inversion results that accurately reflect reality [15; 16; 17; 18]. The development and introduction of global inversion algorithms have consistently been key research priorities in 3D gravity data inversion, overall, encompassing the traditional global optimization algorithms (e.g., the genetic algorithm, simulated annealing algorithm, particle swarm optimization algorithm, etc.) and emerging machine learning algorithms. During the past more than 20 years, traditional inversion algorithms have been widely employed in conducting 3D gravity data inversion. Ref. [19] employed the simulated annealing algorithm for 3D gravity data inversion to investigate the density anomalies of underground materials in the Gulf of Mexico. Ref. [20] used a 3D perspective to identify the crustal structures of Fuerteventura Island, where the genetic algorithm undertook the optimization tasks. Ref. [21] developed a novel gravity inversion method by integrating the simulated annealing algorithm with the genetic algorithm to enhance the retrieval of 3D density anomalies. Ref. [22] performed 3D gravity inversion and uncertainty analysis using the particle swarm optimization algorithm, producing an estimated gravity profile that effectively reflects the subsurface conditions of the Argeles-Gazost basin in France. Even now, traditional inversion algorithms continue to be utilized for identifying optimal solutions in 3D gravity data inversion due to their global optimization capabilities [23; 24; 25]. In contrast, machine learning algorithms have emerged and rapidly developed in geophysical inversion in recent years, gaining popularity due to their real-time processing, strong global optimization, and generalization capabilities and, consequently, their potential to improve the efficiency and accuracy of the 3D gravity inversion process. As a representative of machine learning, a deep learning network has been introduced to address the optimization challenges of 3D gravity data inversion. For example, refs. [26; 27; 28; 29] utilized deep learning networks for 3D gravity data inversion, achieving satisfactory results in modeling subsurface density structures. Currently, the application of machine learning in gravity inversion is continually evolving to address the growing demands for enhanced inversion performance. Deep learning has recently garnered more attention from researchers for various modeling problems compared with traditional global optimization algorithms due to its real-time property and optimization abilities [30; 31; 32]. Once network training is complete, the inversion results can be determined immediately. Additionally, deep learning is widely regarded as having superior functional modeling capabilities compared with traditional algorithms.