Performance of Machine Learning Methods for Gravity Inversion: Successes and Challenges

TL;DR

This work tackles 2D gravity inversion, where the forward map $g=A\rho$ is ill-posed due to underdetermination. It compares a direct CNN inversion against generative latent-space approaches (VAE, GAN) and CNN-based initializations for iterative solvers (GD, GMRES, LGMRES, ICG), all within a consistent forward-model framework using synthetic density datasets. The results show CNN inversion yields the most accurate and stable reconstructions, while VAEs and GANs produce realistic but non-unique density fields in latent space; iterative refinements offer only marginal improvements. Overall, the ill-posed nature of gravity inversion remains a major challenge, with data-driven methods currently offering the strongest performance and potential for future physics-informed integration.

Abstract

Gravity inversion is the problem of estimating subsurface density distributions from observed gravitational field data. We consider the two-dimensional (2D) case, in which recovering density models from one-dimensional (1D) measurements leads to an underdetermined system with substantially more model parameters than measurements, making the inversion ill-posed and non-unique. Recent advances in machine learning have motivated data-driven approaches for gravity inversion. We first design a convolutional neural network (CNN) trained to directly map gravity anomalies to density fields, where a customized data structure is introduced to enhance the inversion performance. To further investigate generative modeling, we employ Variational Autoencoders (VAEs) and Generative Adversarial Networks (GANs), reformulating inversion as a latent-space optimization constrained by the forward operator. In addition, we assess whether classical iterative solvers such as Gradient Descent (GD), GMRES, LGMRES, and a recently proposed Improved Conjugate Gradient (ICG) method can refine CNN-based initial guesses and improve inversion accuracy. Our results demonstrate that CNN inversion not only provides the most reliable reconstructions but also significantly outperforms previously reported methods. Generative models remain promising but unstable, and iterative solvers offer only marginal improvements, underscoring the persistent ill-posedness of gravity inversion.

Figures (11)

• Figure 1: The proposed CNN architecture is designed for the direct inversion of gravity data. A total of $3n$ gravity measurements (with $n=50$) are collected from the surface to estimate the underlying density. Since each gravity measurement includes both horizontal and vertical components, the input tensor width is set to 2.
• Figure 2: A schematic representation of the gravity data configuration is shown. A total of $3n = 150$ gravity measurements are collected along the surface to capture the horizontal distribution of the gravity anomaly. The subsurface region (density) is discretized into a grid of size $n \times n = 50 \times 50$, as indicated by the dashed box.
• Figure 3: Architecture of the VAE used for gravity inversion, where the decoder serves as a generator in latent-space optimization
• Figure 4: Architecture of the GAN used for gravity inversion, where the trained generator is later used for latent-space optimization
• Figure 5: Random sample generation from the VAE

Theorems & Definitions (2)

• Remark 5.1
• Remark 5.2