Diffusion models have recently become powerful tools for solving inverse problems in scientific and engineering applications. Many existing approaches treat pre-trained diffusion models as priors and employ additional sampling or correction strategies to incorporate observations. In this work, we consider a Bayesian inverse problem arising in subsurface flow, where the goal is to reconstruct spatially heterogeneous permeability fields from noisy flow rate observations governed by transient Darcy flow. We propose a two-stage conditional diffusion framework that explicitly separates the denoising of observations and the inversion of the quantity of interest. The first stage involves training a supervised conditional diffusion model to learn the mapping from noise-free simulated flow rates to the corresponding permeability fields. The second stage employs an unsupervised diffusion model to recover noise-free flow rates from noisy observations. This two-stage structure enables robust posterior sampling by producing permeability fields that match the observed flow rates. Numerical experiments on a synthetic reservoir model demonstrate the effectiveness of the proposed method in terms of reconstruction accuracy, uncertainty quantification, and computational efficiency.
Liu et al. (Fri,) studied this question.