Fun-DDPS: Diffusion Framework for CCS

• The paper presents Fun-DDPS, a decoupled diffusion framework that integrates a function-space prior over geological parameters with a differentiable surrogate to tackle ill-posed CCS problems.
• It decouples learning of geological realism from physics-guided conditioning, ensuring sample efficiency and artifact-free permeability field recovery under sparse data.
• Experimental results show a reduction in forward prediction error from 86.9% to 7.7% and successful recovery of posteriors with Jensen–Shannon divergence below 0.06.

Searching arXiv for the cited papers to ground the article. I’ll look up the relevant arXiv entries for Fun-DDPS and related function-space diffusion work. Searching arXiv for ([2602.12274](/papers/2602.12274)) and related function-space diffusion papers. Fun-DDPS is a generative framework for forward and inverse modeling in Carbon Capture and Storage (CCS) that combines a function-space diffusion prior over geological parameters with a differentiable Local Neural Operator (LNO) surrogate for multiphase flow (Ju et al., 12 Feb 2026). It is designed for subsurface-flow settings in which sparse observations make inverse problems ill-posed and deterministic surrogates unreliable. In the reported formulation, the diffusion model learns a prior distribution over permeability fields, while the surrogate supplies physics-consistent guidance for cross-field conditioning on the dynamics field; this decoupling is used to recover missing information in parameter space and to support efficient gradient-based data assimilation (Ju et al., 12 Feb 2026).

1. Problem setting and conceptual design

Fun-DDPS addresses two tasks in CCS: forward prediction and inverse data assimilation (Ju et al., 12 Feb 2026). The forward component concerns prediction of the saturation field after 30 years from incomplete information about the geomodel. The inverse component targets recovery of the geological parameter field from sparse dynamic observations. The motivating difficulty is that subsurface characterization is severely underdetermined when only limited observations are available, especially in realistic monitoring regimes such as sparse well-log measurements.

The framework is organized around a strict decomposition of roles. The prior $p(m)$ over the permeability, or geomodel, field $m(x)$ is learned by a single-channel function-space diffusion model. The expensive PDE-based forward operator $F:m\mapsto s$, where $s$ denotes the saturation field after 30 years, is approximated separately by a differentiable LNO surrogate $\mathcal L_\phi$ (Ju et al., 12 Feb 2026). These components are trained independently and coupled only at inference time through score-based guidance. This architectural separation is central to the method: the diffusion model is responsible for geological realism in parameter space, whereas the surrogate converts solution-space constraints into gradients with respect to the geomodel.

A plausible implication is that Fun-DDPS is intended to avoid a common failure mode of joint-state generative models: when conditioning signals are sparse and indirect, geometric structure in the latent geological field can be degraded by entangling prior learning and observation guidance too tightly. The reported comparison with a joint-state baseline supports that interpretation, though the paper states the result primarily in terms of artifact suppression and sample efficiency rather than a broader theoretical claim (Ju et al., 12 Feb 2026).

2. Function-space diffusion prior over geological parameters

The diffusion prior treats the permeability field $m$ as an element of an infinite-dimensional Hilbert space $\mathcal M$ and learns $p(m)$ using score-based diffusion in function space (Ju et al., 12 Feb 2026). Rather than perturbing a discretized image with white noise at each pixel, the forward noising process adds a Gaussian random field with Matérn covariance $\mathcal C_\gamma$: $$m_{\sigma}=m_0+\xi_\sigma,\qquad \xi_\sigma\sim\mathcal N(0,\sigma^2\mathcal C_\gamma).$$ As $m(x)$0, $m(x)$1 converges to the base Gaussian measure (Ju et al., 12 Feb 2026).

Training uses a time-dependent denoiser $m(x)$2 optimized by score matching: $m(x)$3 By Tweedie’s formula, $m(x)$4 yields an estimate of the score $m(x)$5 (Ju et al., 12 Feb 2026). This ties Fun-DDPS directly to the function-space diffusion formalism developed for PDE inverse problems in FunDPS, which extended Tweedie’s formula to infinite-dimensional Hilbert spaces and formulated discretization-agnostic conditional sampling with plug-and-play guidance (Yao et al., 22 May 2025).

Unconditional generation is performed through the reverse probability-flow ODE

$m(x)$6

which maps samples from high-noise states back toward $m(x)$7 (Ju et al., 12 Feb 2026). In practice, the implementation uses a discrete-time Euler–Heun integrator, and the paper notes the equivalent reverse-SDE view

$m(x)$8

for suitable $m(x)$9 and $F:m\mapsto s$0 (Ju et al., 12 Feb 2026). In this sense, Fun-DDPS inherits the function-space, noise-schedule, and score-estimation machinery of FunDPS, but specializes it to a decoupled CCS workflow in which only the parameter field is diffused (Yao et al., 22 May 2025).

3. Differentiable Local Neural Operator surrogate

To approximate the PDE-based forward map $F:m\mapsto s$1, Fun-DDPS employs a Local Neural Operator $F:m\mapsto s$2 with two parallel paths (Ju et al., 12 Feb 2026). The Fourier path uses global spectral convolution layers to capture smooth, large-scale pressure and flow responses. The DISCO path uses localized continuous-kernel convolutions to resolve sharp saturation fronts and local heterogeneities. This decomposition is intended to represent both long-range and localized transport structure within a single surrogate.

The surrogate is trained on paired geomodel–dynamics data $F:m\mapsto s$3 by minimizing

$F:m\mapsto s$4

In the reported experiments, $F:m\mapsto s$5 achieves $F:m\mapsto s$6 relative $F:m\mapsto s$7 error on fully observed test cases (Ju et al., 12 Feb 2026). This fully observed regime is important because Fun-DDPS does not use the surrogate as a standalone predictor under sparse conditioning; instead, it uses the surrogate as a differentiable physics module inside the guided posterior sampler.

The distinction between surrogate accuracy in the fully observed regime and performance under sparse conditioning is consequential. The paper explicitly reports that a standard surrogate without a learned prior fails catastrophically when only partial geomodel information is available, whereas Fun-DDPS remains accurate under the same sparsity pattern (Ju et al., 12 Feb 2026). That result suggests that surrogate expressivity alone is insufficient for the CCS forward task once the input field becomes highly incomplete.

4. Decoupled guidance and cross-field conditioning

The defining feature of Fun-DDPS is the decoupling of the prior $F:m\mapsto s$8 from the surrogate $F:m\mapsto s$9, together with their coupling at inference by conditional-score guidance (Ju et al., 12 Feb 2026). Given observations $s$0, the conditional score is written as

$s$1

This is the formal basis for conditioning diffusion samples on either direct geomodel observations or indirect dynamic measurements.

For partial geomodel observations,

$s$2

the likelihood gradient is approximated by

$s$3

For partial dynamic observations,

$s$4

the true forward map $s$5 is replaced by the LNO surrogate, yielding

$s$6

The paper identifies this as the physics-consistent guidance term that translates sparse, solution-space constraints into dense, parameter-space gradients through the surrogate Jacobian $s$7 (Ju et al., 12 Feb 2026).

For inverse modeling, the target is the Bayesian posterior

$s$8

Posterior sampling is performed by Diffusion Posterior Sampling (DPS) on $s$9 alone in $\mathcal L_\phi$0 with $\mathcal L_\phi$1 grid nodes (Ju et al., 12 Feb 2026). At each denoising step, the algorithm predicts a clean field $\mathcal L_\phi$2, takes a standard predictor–corrector update along the prior score, and applies a guidance update based on the observation mismatch, with

$\mathcal L_\phi$3

for dynamic observations (Ju et al., 12 Feb 2026). Relative to the more general FunDPS formulation, this is a specialized plug-and-play guidance mechanism in which conditioning occurs only in parameter space, while the forward physics is injected through a differentiable operator surrogate (Yao et al., 22 May 2025).

5. Experimental regime and reported performance

The reported dataset consists of $\mathcal L_\phi$4 training and $\mathcal L_\phi$5 test pairs $\mathcal L_\phi$6 generated with the ECLIPSE reservoir simulator, random permeability funnels sampled via SGeMS, and $\mathcal L_\phi$7-year $\mathcal L_\phi$8 injection (Ju et al., 12 Feb 2026). Two experimental configurations are emphasized: a forward task with partial geomodel observations and an inverse task with sparse dynamic observations.

Task	Observation regime	Reported outcome
Forward modeling	Only $\mathcal L_\phi$9 of $m$0 observed at random locations	Standard surrogate: $m$1 relative $m$2 error; Fun-DDPS: $m$3 relative error
Inverse modeling	Two vertical well-logs, $m$4 spatial coverage, Gaussian noise $m$5	Fun-DDPS and Fun-DPS: Jensen–Shannon divergence $m$6 to RS marginal posteriors

In the forward task, the deterministic surrogate degrades to $m$7 relative $m$8 error when only $m$9 of the geomodel is observed, whereas Fun-DDPS maintains $\mathcal M$0 relative error, reported as an $\mathcal M$1 improvement (Ju et al., 12 Feb 2026). The paper presents this as evidence that the method can handle extreme data sparsity in regimes where deterministic methods fail.

In the inverse task, the reference posterior is produced by asymptotically exact rejection sampling (RS): $\mathcal M$2 million prior samples are generated and approximately $\mathcal M$3 are accepted by likelihood rejection sampling (Ju et al., 12 Feb 2026). Against this reference, both Fun-DDPS and the joint-state baseline Fun-DPS achieve Jensen–Shannon divergence less than $\mathcal M$4 in hyperparameter space (Ju et al., 12 Feb 2026). The paper characterizes this as the first rigorous validation of diffusion-based inverse solvers against asymptotically exact RS posteriors.

The sample-efficiency comparison is also quantitative. Fun-DDPS requires approximately $\mathcal M$5 forward-model evaluations, computed as $\mathcal M$6 samples times $\mathcal M$7 steps, corresponding to a $\mathcal M$8 reduction relative to RS (Ju et al., 12 Feb 2026). The reported conclusion is therefore twofold: statistical agreement with the RS posterior is retained, but the cost of posterior approximation is substantially reduced.

6. Baselines, failure modes, and interpretive significance

The paper compares Fun-DDPS with three baselines: a standard surrogate without a learned prior, a joint-state diffusion baseline labeled Fun-DPS, and rejection sampling (Ju et al., 12 Feb 2026). Each baseline isolates a distinct methodological question. The standard surrogate tests whether deterministic operator learning is sufficient under sparse input. The joint-state diffusion model tests whether learning the geomodel and dynamics jointly is preferable to decoupling them. Rejection sampling supplies an asymptotically exact Bayesian reference.

The standard surrogate is reported to fail catastrophically under sparse inputs (Ju et al., 12 Feb 2026). By contrast, Fun-DDPS preserves low forward error because the diffusion prior fills in missing geomodel information before cross-field guidance is applied. This suggests that prior learning in parameter space is the critical ingredient in the forward task once the available geomodel observations become severely incomplete.

The comparison with Fun-DPS is more subtle. Fun-DPS trains a 2-channel U-NO on $\mathcal M$9 jointly (Ju et al., 12 Feb 2026), whereas Fun-DDPS uses a single-channel diffusion prior over $p(m)$0 and an external surrogate for $p(m)$1. Under sparse dynamics, the paper reports that guidance attenuation in the joint-state baseline yields high-frequency artifacts in permeability realizations, despite low Jensen–Shannon divergence (Ju et al., 12 Feb 2026). This is an important corrective to a possible misconception: low posterior divergence alone does not guarantee physically consistent or geologically coherent samples. In the reported experiments, Fun-DDPS and Fun-DPS are statistically similar under the JS metric, but Fun-DDPS is distinguished by artifact-free realizations and by improved efficiency relative to RS.

The paper attributes these gains directly to the decoupled architecture: an expressive, infinite-dimensional prior over $p(m)$2 together with an efficient, differentiable physics surrogate for cross-field guidance (Ju et al., 12 Feb 2026). A plausible implication is that the method is especially well matched to inverse problems in which the observation operator acts in a different field from the principal latent uncertainty, as in CCS where sparse saturation information constrains permeability only indirectly.

7. Position within function-space diffusion research

Fun-DDPS is best understood as a domain-specific extension of function-space diffusion sampling for PDE problems. FunDPS introduced a general framework for conditional sampling in PDE-based inverse problems, with an unconditional discretization-agnostic denoiser, infinite-dimensional Tweedie theory, and plug-and-play gradient guidance (Yao et al., 22 May 2025). Fun-DDPS adopts the function-space viewpoint but modifies the conditioning pathway: instead of guiding a joint state directly, it learns the prior solely on the geological parameter field and injects the physics through a differentiable LNO surrogate (Ju et al., 12 Feb 2026).

This shift from joint-state to decoupled conditioning is the principal methodological distinction. In FunDPS, the denoiser can operate on concatenated PDE parameter and solution channels, and conditioning is posed in a generic observation space (Yao et al., 22 May 2025). In Fun-DDPS, the prior and forward physics are explicitly separated, and cross-field conditioning is mediated by the surrogate Jacobian $p(m)$3 (Ju et al., 12 Feb 2026). The CCS experiments indicate that this is not merely an implementation detail: the decoupled design is associated with improved robustness under extreme data sparsity and with removal of high-frequency artifacts in inverse samples.

A further point of clarification is terminological. Fun-DDPS in CCS is unrelated to the acronym DDPS used for a dynamic differential pricing system in mobile edge computing (Xue et al., 2024), and it is likewise unrelated to the DPS framework for double parton scattering in high-energy physics (Buffing, 2017). Within the CCS and neural-operator literature, Fun-DDPS denotes the function-space decoupled diffusion framework described above (Ju et al., 12 Feb 2026).