Geophysics-informed invertible operator networks for solving Bayesian geophysical inverse problem

Geophysical inverse problems are inherently ill-posed due to sparse observation and measurement noise. Classical sampling-based Bayesian inference method, such as Markov-chain Monte Carlo (MCMC), are computationally prohibitive because they requires a large number of forward simulations. Data-driven machine learning methods often struggle to yield physically consistent solutions under limited training data scenarios. To address these issues, we propose geophysics-informed invertible operator networks (GI-ION) that integrates invertible neural networks (INNs) with the deep operator network (DeepONet) framework for efficient Bayesian geophysical inversion. The key idea is to employ an INN as the branch network, which transforms the model parameters into two latent variables: the expansion coefficients representing the solution to the forward problem, and a latent noise that capture the inherent uncertainty associated with the inverse problem. By accurately learning the forward operator and enforcing statistical independence between these latent variables, GI-ION enables real-time and efficient sampling from target posterior distributions, even under sparse boundary observations. We validated GI-ION on three representative cases: 1-D surface-wave dispersion, and travel-time tomography for a 2-D synthetic random field model and overthrust field model, respectively. Numerical experiments demonstrate that GI-ION consistently produces efficient, robust, and scalable results across diverse problem settings.