Function-Space Decoupled Diffusion for Forward and Inverse Modeling in Carbon Capture and Storage

TL;DR

This paper addresses the ill-posed forward and inverse modeling problems in carbon capture and storage (CCS) under extreme data sparsity by introducing Fun-DDPS, a decoupled framework that uses a function-space diffusion prior over geomodels together with a differentiable neural operator surrogate for forward physics. The method enables robust forward predictions with as little as 25% geomodel observations (7.7% relative error, vs 86.9% for standard surrogates) and provides rigorous validation against asymptotically exact Rejection Sampling posteriors, achieving Jensen-Shannon divergences below 0.06. Fun-DDPS yields physically consistent realizations and improves sampling efficiency by approximately 4× compared to rejection sampling, addressing both accuracy and computational cost. The approach offers a practical pathway for CCS data assimilation, with potential extensions to full spatiotemporal monitoring regimes.

Abstract

Accurate characterization of subsurface flow is critical for Carbon Capture and Storage (CCS) but remains challenged by the ill-posed nature of inverse problems with sparse observations. We present Fun-DDPS, a generative framework that combines function-space diffusion models with differentiable neural operator surrogates for both forward and inverse modeling. Our approach learns a prior distribution over geological parameters (geomodel) using a single-channel diffusion model, then leverages a Local Neural Operator (LNO) surrogate to provide physics-consistent guidance for cross-field conditioning on the dynamics field. This decoupling allows the diffusion prior to robustly recover missing information in parameter space, while the surrogate provides efficient gradient-based guidance for data assimilation. We demonstrate Fun-DDPS on synthetic CCS modeling datasets, achieving two key results: (1) For forward modeling with only 25% observations, Fun-DDPS achieves 7.7% relative error compared to 86.9% for standard surrogates (an 11x improvement), proving its capability to handle extreme data sparsity where deterministic methods fail. (2) We provide the first rigorous validation of diffusion-based inverse solvers against asymptotically exact Rejection Sampling (RS) posteriors. Both Fun-DDPS and the joint-state baseline (Fun-DPS) achieve Jensen-Shannon divergence less than 0.06 against the ground truth. Crucially, Fun-DDPS produces physically consistent realizations free from the high-frequency artifacts observed in joint-state baselines, achieving this with 4x improved sample efficiency compared to rejection sampling.

Figures (10)

• Figure 1: Comparison of decoupled (Fun-DDPS) vs. joint-state (Fun-DPS) architectures. (a) Fun-DDPS (Decoupled): Training uses only geomodel samples $\boldsymbol{m}$ to learn the prior $p(\boldsymbol{m})$; a separate neural operator surrogate $\mathcal{L}_\phi$ learns the forward mapping. During inference, the geomodel diffusion model generates $\boldsymbol{m}$, and $\mathcal{L}_\phi$ maps it to dynamics $\boldsymbol{s}$. For inverse problems, dynamics observations guide the diffusion through the surrogate gradient. (b) Fun-DPS (Joint-state): Training requires paired data $(\boldsymbol{m}, \boldsymbol{s})$ to learn the joint distribution $p(\boldsymbol{m}, \boldsymbol{s})$. During inference, the joint model generates both fields simultaneously, with observations guiding the corresponding channels directly.
• Figure 2: Posterior sampling comparison: Fun-DDPS vs. Fun-DPS. (a) Fun-DDPS: The geomodel diffusion model denoises a GRF to produce $\hat{\boldsymbol{m}}_0$. Geomodel observations provide direct guidance; dynamics observations guide via the surrogate gradient $\nabla_{\boldsymbol{m}} \|\mathcal{L}_\phi(\hat{\boldsymbol{m}}_0) - \boldsymbol{y}_{dyn}\|$, translating sparse solution-space constraints into dense parameter-space guidance. (b) Fun-DPS: The joint-state model denoises to produce $(\hat{\boldsymbol{m}}_0, \hat{\boldsymbol{s}}_0)$ simultaneously. Observations guide the respective channels directly, but sparse dynamics observations remain localized without the surrogate's global receptive field to propagate information.
• Figure 3: Problem setup. Forward: The geomodel is partially observed; the full dynamics are predicted. Inverse: The dynamics are partially observed; the full geomodel is inferred.
• Figure 3: Geomodel hyperparameters for the ground truth case and their corresponding prior ranges.
• Figure 4: Forward problem under varying observation coverage. Left: Observed geomodel at 100%, 50%, and 25% coverage. Right: Absolute error in predicted saturation. Fun-DDPS maintains high accuracy (rel. $L_2$: 0.07--0.15).