Bayesian Inversion Beyond the Generator's Range: Reducing Non-physical Constraints from Generative Neural Networks

Abstract

To address non-uniqueness in geophysical inversions, Generative Neural Networks (GNNs) are often used to parameterise geological or other prior information in a lower-dimensional space (the so-called latent space), thereby reducing the number of inversion parameters. However, the training samples that represent this information are then always incorrect, because Earth structure is not intrinsically low-dimensional, and all GNNs are subject to training errors. Consequently, even when solutions are mapped back to the original higher-dimensional space, they inherit this imperfection and remain restricted to a low-dimensional manifold. From a mathematical point of view, latent space inversion thus fails fundamentally: generally, it can not even reproduce the best, higher-dimensional approximation to the true Earth. To overcome this limitation, we introduce an `escaping prior' distribution, which provides the necessary freedom to find solutions beyond the manifold of GNN-generated solutions, while still using the GNN as a guide. We use a hierarchical Bayesian approach to adapt the strength of prior information to the observed data and synthetic forward function $\textit{a posteriori}$. We illustrate our method on low-dimensional problems where the escaping prior is defined over a 1D manifold in 2D space. We then compare our approach with standard latent space and model space inversions using a uniform prior distribution in synthetic tomography problems, ranging from simple to more geologically realistic cases. Furthermore, we tested scenarios where the true solution is either far from or close to the manifold. In each case, our escaping prior shows significantly better results than the alternatives. It results in solutions beyond each GNN's generative capabilities, solutions which include the best possible model, and clearly demonstrates the limitations of latent space inversion.