# Three-Dimensional Broad Learning Gravity Data Inversion Using Single-Anomaly Training Samples
## 1 Introduction
However, there are two main challenges that need to be addressed in most 3D deep learning gravity inversion studies [26; 33], i.e., the time-consuming problem of network training and the issue of anomaly number representation in training samples [34; 35; 36]. The first challenge limits the inversion efficiency, particularly when network retraining could not be avoided in certain situations (e.g., cross-zone gravity exploration). Regarding the second challenge, most studies implicitly assume that the maximum number of anomalies in the training samples is greater than the actual number. Undoubtedly, these two drawbacks hinder the broader application of machine learning in 3D gravity data inversion. In this study, these two obstacles are addressed by drawing on approaches from other geophysical problems. To address the time-consuming disadvantage of a deep learning network, in ref. [37], the authors proposed a broad learning (BL) network, which utilizes a single-layered neural architecture. A BL network increases complexity in width dimension rather than depth, providing a significant advantage in training speed while maintaining functional mapping capabilities comparable to a deep learning network. BL networks have been applied in some modeling problems and achieved satisfactory prediction results [38; 39; 40]. Regarding its application in geophysics, the advantages of a BL network in terms of both time efficiency and optimization performance have been demonstrated in surface wave inversion [41; 42; 43; 44] and resistivity inversion [45]. As for the anomaly number problem, 3D resistivity inversion similarly faces constraints regarding prior information on the number of anomalies. Ref. [45] proposed a machine learning inversion method that incorporates a single anomaly with varying locations across different training samples. The inversion results demonstrate the effectiveness of using single-anomaly training samples for network training. It is evident that the application of a BL network and single-anomaly training samples can be extended to 3D gravity inversion. To tackle the challenges of time consumption and the representation of anomaly numbers in training samples, this study proposes a novel 3D gravity data inversion method. First, we use a single anomaly to generate training samples based on the linear superposition of potential fields [46]. Second, we introduce a BL network to map the relationship between gravity anomaly data and density models, thereby reducing the time required for the training process. The feasibility of the proposed method is then validated through three synthetic models. Additionally, the method demonstrates improved detection performance  
parameter to be determined for the network training of the broad learning network. Each group of mapped features of the BL network can be shown as:  
\[\mathbf{F}_{i}=\phi\big{(}D_{t}\mathbf{W}_{F_{i}}+\mathbf{\beta}_{F_{i}}\big{)},\ i=1,2, \ldots,N_{F}, \tag{5}\]  
where \(\phi\) is the feature mapping function performed by a sparse autoencoder [32] for data dimensionality reduction, \(\mathbf{W}_{F_{i}}\) is the \(i\)th weight matrix with random values, \(\mathbf{\beta}_{F_{i}}\) is the \(i\)th bias with random values, and \(N_{F}\) is the number of groups of mapped features. Based on the mapped features \(\mathbf{F}^{N_{F}}\), each group of enhancement nodes can be calculated by:  
\[\mathbf{E}_{j}=\xi\Big{(}\mathbf{F}^{N_{F}}\mathbf{W}_{E_{j}}+\mathbf{\beta}_{E_{j}}\Big{)},\ j=1,2, \ldots,N_{E}, \tag{6}\]  
where \(\xi\) is a nonlinear activation function (the hyperbolic tangent applied in this study), \(\mathbf{W}_{E_{j}}\) is the \(j\)th random weight matrix similar to \(\mathbf{W}_{F_{i}}\), \(\mathbf{\beta}_{E_{j}}\) is the \(j\)th random bias similar to \(\mathbf{\beta}_{F_{i}}\), and \(N_{E}\) is the number of groups of enhancement nodes.