# Three-Dimensional Broad Learning Gravity Data Inversion Using Single-Anomaly Training Samples
## 3 Results and Evaluation of Performance
### Numerical Simulations
The dimensions of the first anomaly are 2000 m in length, 500 m in width, and 600 m in height, with a center depth of 500 m. The dimensions of the second irregular anomaly are 2400 m in length, 600 m in width, and 300 m in height, with a center depth of 600 m. This numerical example aims to explore the inversion performance to predict the irregular shapes of anomalies. With gravity data as input to the trained network, the resulting inverted density model was promptly obtained, as shown in Figure 6(a). Compared with the actual density anomalies, the shapes of the anomalies were well predicted, provided that the number of anomalies was accurately estimated. Regarding the inverted density values, the objective function value between the inverted and actual density models is 0.0198, indicating a strong recovery of density values within the anomalies. In particular, each training sample used for the network training contained only a single anomaly. It is obvious that the generalization capability of the proposed gravity inversion method is not limited to identifying the number of anomalies but also extends to predicting the shapes of irregularly shaped anomalies. To provide a more detailed assessment of the inversion accuracy, the differences between the inverted and observed gravity anomaly data were quantified, as illustrated in Figure 6(c). The inversion residuals are largely confined to the range of \(-1.0\) to 1.5 mGal. The R\({}^{2}\)-value between the observed gravity anomaly and the gravity anomalies (Figure 6(b)) forwarded from the inversion results is 0.9999, indicating minimal discrepancies compared with the gravity anomalies shown in Figure 5(b). The proposed inversion method, trained on single-anomaly samples, produces accurate results even when applied to irregularly shaped anomalies. This numerical example further confirms the effectiveness of the proposed method. The robustness of the BL network is further evaluated using a model composed of anomalies with varying shapes, densities, and positions. As shown in Figure 7(a), the test  
Figure 6: Numerical density model with irregular-shaped anomalies and corresponding surface gravity anomaly observations. (**a**) Three-dimensional representation of the actual density model. (**b**) Surface gravity anomaly observations. Figure 7: Inversion results of the numerical density model with irregular-shaped anomalies. (**a**) Three-dimensional representation of the inverted density model. (**b**) Gravity anomaly data forwarded from the inversion result. (**c**) Residuals between inverted and observed surface gravity anomaly data. model includes a T-shaped anomaly with a density of 0.5 g/cm\({}^{3}\), a rectangular anomaly with a density of 0.75 g/cm\({}^{3}\), and a cubic anomaly with a density of 1.0 g/cm\({}^{3}\). The inversion results, presented in Figure 9a, demonstrate that the density structures with different shapes are well-recovered. The objective function value between the inverted and actual density models is 0.0323. The R\({}^{2}\) between the observed gravity anomaly (Figure 8b) and the gravity anomalies (Figure 9b) forwarded from the inversion results is 0.9989. The differences between the inverted and observed gravity anomaly data, shown in Figure 9c, fall within the range of \(-\)2.0 to 1.5 mGal. All quantitative metrics validate the accuracy and stability of the proposed method.