KAST-Net: An Inversion Network With Gravity Kernel Matrix Layers for Joint Gravity and Gradients

Gravity inversion, as a critical technique in gravity exploration, is widely used for reconstructing three-dimensional (3-D) subsurface density structures in mineral resource exploration, deep structural analysis, and crust-mantle isostasy studies. Currently, deep learning methods for gravity inversion based on the U-shaped Convolutional Neural Network (U-Net) architecture often lack sufficient information regarding constraints in gravity, causing significant discrepancies between observed and recovered gravity data. Such inconsistencies directly affect the confidence interval of the inverted density model and its spatial distribution in the subsurface. To address these issues, we propose an inversion network architecture integrating explicit linear Gravity Kernel Matrix (GKM) layers for joint gravity and gradients, termed the Kolmogorov-Arnold theorem and Shifted-window Transformer-based network (KAST-Net). It consists of a two-dimensional (2-D) Kolmogorov-Arnold theorem-based encoder-decoder branch, a 3-D Shifted-window Transformer-based encoder-decoder branch and a series of the GKM layers between the 2-D branch and 3-D branch. Synthetic inversion model tests demonstrated that the proposed network achieves higher inversion accuracies for model boundary delineation and confidence intervals of inverted density, compared to the Residual U-shaped Convolutional Neural Network++ (ResU-Net++), the Kolmogorov-Arnold U-shaped Convolutional Neural Network (KANU-Net) and two other classical constrained inversion methods based on regularization. In addition, incorporating graph convolution into KAST-Net enables effective handling of non-uniform gravity data with missing values, maintaining superior performance even under data sparsity conditions. Finally, we applied the proposed KAST-Net to 3-D density imaging of the Vinton Salt Dome in southwestern Louisiana, USA. KAST-Net not only accurately estimates the depth of the caprock, located between 160 and 360 m, but also successfully identifies the underlying salt layer below 520 m, a new insight that outperforms other existing methods in both accuracy and reliability.