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

AI summary

Overview

Research area: Generative machine learning for scientific inverse problems, applied to subsurface flow modeling for Carbon Capture and Storage (CCS). The work sits at the intersection of diffusion models, neural operators, and Bayesian data assimilation for geoscience.

Technical level: Advanced. The paper assumes familiarity with diffusion posterior sampling, neural operator architectures (FNO, LNO), Bayesian inverse problems, and multiphase flow simulation.

Scope: The paper proposes and validates a decoupled framework (Fun-DDPS) that learns a diffusion prior over geological parameters separately from a neural operator surrogate for flow physics, then benchmarks it against a joint-state diffusion baseline and an asymptotically exact Rejection Sampling reference posterior.

What This Paper Is About

Forecasting how injected CO2 moves underground, and inferring the unknown geological properties of the rock from limited measurements, are both hard problems because subsurface parameters are high-dimensional and non-Gaussian while the governing flow equations are expensive to solve. Existing joint-state diffusion models learn geological parameters and flow states together, which the authors argue leads to physical inconsistency and high-frequency artifacts when paired training data is scarce. The goal is a framework that learns the geological prior and the flow physics separately, so the prior can fill in missing information while a fast, differentiable surrogate enforces physics during sampling.

Key Contributions

1. A decoupled diffusion framework (Fun-DDPS) that learns a prior over geological parameters p(m) with a single-channel function-space diffusion model, and independently trains a Local Neural Operator (LNO) surrogate to approximate the forward physics, using surrogate gradients from sparse dynamic observations to guide posterior sampling.

2. Robust forward modeling under extreme sparsity, reporting 7.7% relative error at 25% observation coverage against 86.9% error for a deterministic surrogate, described by the authors as an 11× improvement.

3. The first rigorous validation of diffusion-based inverse solvers against asymptotically exact Rejection Sampling (RS) posteriors, with both Fun-DDPS and the joint-state baseline (Fun-DPS) achieving Jensen-Shannon divergence below 0.06.

4. A computational efficiency claim of roughly 4× reduction relative to Rejection Sampling, since generating 1,024 posterior samples required about 512,000 functional evaluations versus 2 million for the RS reference.

Main Findings

• Forward modeling with missing geomodel data: At 100% coverage, Fun-DDPS achieves 0.046 ± 0.058 relative L2 error, compared to 0.044 ± 0.057 for the deterministic surrogate. At 50% coverage, Fun-DDPS achieves 0.054 ± 0.069 versus 0.850 ± 0.263 for the surrogate. At 25% coverage, Fun-DDPS achieves 0.077 ± 0.079 versus 0.869 ± 0.258. The paper states the deterministic surrogate degrades from 4.4% to 86.9% error as sparsity increases, which the authors attribute to zero-filling pushing inputs out of the training distribution.

• Joint-state baseline struggles on forward tasks regardless of sparsity: Fun-DPS reports 0.418 ± 0.366 at 100% coverage, 0.370 ± 0.357 at 50%, and 0.336 ± 0.349 at 25%. The paper describes this as consistently high error (34–42%) independent of sparsity.

• Inverse problem statistics: Against an RS reference posterior built from roughly 26,000 accepted samples drawn from a pool of 2 million prior samples, Fun-DPS achieves a mean JS divergence of 0.047 (std 0.037) over 1,024 samples with 0% diverged samples. Fun-DDPS achieves a mean JS divergence of 0.051 (std 0.061) over 998 samples with 2.5% diverged samples, where divergence is defined as k > 5000 mD.

• Qualitative advantage of decoupling: Despite slightly worse JS scores, Fun-DDPS produces geologically coherent realizations, while Fun-DPS samples show non-physical high-frequency artifacts visible as a grainy posterior mean. Fun-DDPS also produces broader posterior tails on the geomodel hyperparameters (μ_kr, σ_kr) relative to the sharper Fun-DPS agreement with the RS distribution.

• Extreme observation sparsity in the inverse setup: Observations come from two monitoring wells, one at the injector and one 491 meters away, with 64 dynamic data points each for a total of 128 measurements, corresponding to less than 1% of the spatial domain, with Gaussian noise σ_obs = 0.04.

• Uncertainty reduction: All three posteriors (RS, Fun-DPS, Fun-DDPS) are sharply concentrated relative to the broad uniform prior, indicating substantial uncertainty reduction from the measured dynamic data.

• Modeling choice rationale: The LNO combines global Fourier spectral layers with localized kernels via Discrete Continuous Convolutions, on the argument that the Fourier path captures global pressure responses while the local path captures shock-like saturation fronts. Standard operators such as FNO tend to produce ringing artifacts around sharp discontinuities.

Methodology in Plain English

The authors split the problem in two instead of solving it all at once. First, they train a diffusion model on geological parameter fields alone — sampled permeability maps generated with SGeMS — learning what plausible geology looks like. Because this model sees only geology, it can act as a prior that fills in gaps when only part of the field is observed. Second, they train a Local Neural Operator as a fast stand-in for the expensive ECLIPSE (e300) simulator, learning the mapping from a permeability field to the resulting CO2 saturation field after 30 years of injection.

At inference, the diffusion model generates candidate geological fields, and the neural operator maps each candidate to a predicted saturation field. When only sparse observations are available, the gradient of the mismatch between predicted and observed data is backpropagated through the neural operator into the diffusion sampling process, nudging generated fields toward consistency with the data. Because the neural operator's Jacobian is explicit, this guidance does not depend on how much paired training data exists — which is the mechanism the authors say avoids the "guidance attenuation" failure mode analyzed for joint-state models.

The dataset consists of 12,000 training pairs and 1,500 test pairs simulated in a radially symmetrical deep saline aquifer, with the diffusion backbone for both methods being a U-shaped Neural Operator with identical architectural hyperparameters. Observation masking is applied only at inference, so training is agnostic to the sparsity ratio.

For the inverse benchmark, the authors draw 2 million prior geomodels, filter them by likelihood to obtain about 26,000 accepted samples as an exact reference posterior, then compare that reference against 1,024 diffusion samples from each method using Jensen-Shannon divergence on the geomodel hyperparameters. Guidance weights were tuned by sweep to ζ_dyn = 45.0 for Fun-DDPS and ζ_dyn = 100.0 for Fun-DPS.

Why This Matters

Impact on research: The paper supplies a quantitative benchmark that diffusion-based inverse solvers have largely lacked — a comparison against an asymptotically exact posterior rather than only against other approximations. It also provides evidence that decoupling prior learning from physics approximation can outperform joint modeling when paired training data is limited, extending earlier findings on decoupled inverse solvers into function-space CCS modeling.

Real-world applications:

• Planning and permitting gigaton-scale CO2 storage sites, where forecasting plume migration and pressure buildup informs site selection and risk assessment.
• Characterizing reservoir heterogeneity from the sparse monitoring data actually available at real storage sites, such as well logs and 4D seismic surveys.
• Uncertainty quantification for regulatory reporting and leakage-risk analysis, where posterior distributions over rock properties matter more than single deterministic estimates.
• Reducing reliance on thousands of high-fidelity reservoir simulations, making ensemble-style data assimilation tractable for large-scale 3D models.

Industry relevance: Ensemble methods such as EnKF and ES-MDA remain the industry standard but rest on Gaussian assumptions that smooth out sharp geological features like discrete facies or channelized reservoirs. This work targets that gap directly, and its efficiency claim — approximating the true posterior with roughly 4× fewer functional evaluations than rejection sampling — speaks to the practical cost barrier that keeps rigorous Bayesian sampling out of routine CCS workflows.

Future Directions

• Extending to spatiotemporal trajectories: The current study simplifies the dynamic state to a single temporal snapshot at 30 years post-injection. Realistic monitoring involves continuous time series such as daily pressure gauges and 4D seismic surveys, and the authors suggest the decoupled architecture is well suited to this because the neural operator can map static geomodels to time-varying states without retraining the diffusion prior.

• Reducing diverged samples: Fun-DDPS showed a 2.5% rate of diverged samples (k > 5000 mD) versus 0% for Fun-DPS, while also having a slightly higher mean JS divergence (0.051 versus 0.047) and larger standard deviation. Closing this statistical accuracy gap while retaining the physical realism advantage is an open question.

• Broader geological priors and configurations: The paper notes that the simulated setting is a radially symmetrical deep saline aquifer. Whether the decoupled approach holds for more complex, non-symmetric, or channelized reservoir geometries is not established in the reported content.

• Surrogate error propagation: The authors state that the sampling pipeline is tolerant to surrogate approximation error, and that the RS reference posterior itself was computed using the same pretrained surrogate. Quantifying how surrogate inaccuracy propagates into both the reference and the approximated posteriors is a natural follow-up.

Target Audience

Researchers and graduate students working on generative models for scientific inverse problems, neural operator methods, or Bayesian data assimilation for subsurface flow. The paper is also relevant to reservoir engineers and CCS practitioners evaluating whether diffusion-based inversion is accurate enough to replace or augment ensemble smoothers, and to method developers interested in the decoupled-versus-joint design question for physics-constrained generative models. Readers without background in diffusion posterior sampling or multiphase flow simulation will find the appendices necessary for implementation-level detail.

Note: the provided paper content is truncated mid-sentence in Appendix C during the definition of Jensen-Shannon divergence. Details from the remaining appendices (including Appendix D on prior diagnostics, Appendix E on the governing equations, grid settings, prior distributions, and architectural hyperparameters) are referenced in the text but their contents are not available here, so no values from those sections are reported above.

Authors’ 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 Function-space Decoupled Diffusion Posterior Sampling (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.