CoNFiLD-geo: Neural Diffusion for Geological Inversion

• The paper introduces a generative inverse-modeling framework that combines conditional neural fields with latent diffusion to infer subsurface geological parameters from sparse, noisy data.
• It utilizes a two-stage process with offline unconditional pretraining and online Bayesian posterior sampling, enabling zero-shot conditional generation across diverse observation layouts.
• Empirical results on synthetic and field-scale GCS scenarios demonstrate efficient inversion, reliable uncertainty quantification, and highlight limitations like reconstruction and sampling biases.

Searching arXiv for the specified CoNFiLD and related papers to ground the article with up-to-date citations. Search query: (Feng et al., 17 Aug 2025) Conditional Neural Field Latent Diffusion, or CoNFiLD-geo, is a generative inverse-modeling framework for geological carbon sequestration (GCS) that combines a conditional neural field (CNF) with a latent diffusion model (LDM) to infer uncertain subsurface geology and reservoir states from sparse, noisy, heterogeneous monitoring data while quantifying uncertainty. In the formulation introduced for GCS, the CNF serves as a continuous, mesh-agnostic encoder/decoder for high-dimensional geological and reservoir fields, and the LDM learns a prior over the resulting latent representations; inference is then performed by Bayesian posterior sampling in latent space, enabling zero-shot conditional generation across structured and unstructured grids without task-specific retraining (Feng et al., 17 Aug 2025).

1. Problem formulation and domain scope

CoNFiLD-geo is motivated by the inverse problem at the center of GCS: inferring hidden geologic parameters and reservoir states from limited observations of multiphase subsurface flow. In the GCS setting, storage capacity, plume migration, pressure buildup, and leakage risk depend on geological properties such as permeability, stratigraphy, depth, and thickness, but those properties are only partially known. Monitoring data are sparse in space and time, expensive, noisy, and heterogeneous, so the problem is not only one of prediction but of data assimilation and uncertainty quantification (Feng et al., 17 Aug 2025).

The framework is explicitly positioned against two conventional classes of methods. Deterministic inversion aims at a single best-fit model and therefore yields point estimates rather than posterior uncertainty. Stochastic or Bayesian inversion seeks the full posterior over subsurface states, often through MCMC, ensemble methods, or repeated forward simulations, but the associated computational burden is high in nonlinear, coupled, large-scale GCS systems. Existing surrogate models such as CNNs, GNNs, FNOs, and DeepONets accelerate forward modeling, but the stated limitation is that they usually require task-specific retraining for new observation layouts and do not naturally provide a direct posterior-sampling mechanism (Feng et al., 17 Aug 2025).

The corresponding objective of CoNFiLD-geo is therefore broader than surrogate regression. It is intended to provide fast inversion, posterior uncertainty quantification, zero-shot conditional generation, generalization across observation types and geometries, and support for both geomodels and dynamic flow responses (Feng et al., 17 Aug 2025). A plausible implication is that the framework is designed not merely as a forward emulator but as a reusable probabilistic prior over geologic structure and reservoir evolution.

2. Joint field representation and conditional neural field

The CoNFiLD-geo pipeline is organized into two stages: offline unconditional pretraining and online Bayesian conditional generation. During pretraining, the model learns a joint prior over geological models and reservoir responses from simulation data only. During inference, observations are introduced into the pretrained generative process to obtain posterior samples without retraining (Feng et al., 17 Aug 2025).

The learned object is a joint spatiotemporal field formed by combining static geological model fields $\bm{M}(\bm{x})$ and dynamic reservoir response fields $\bm{U}(\bm{x},t)$: $$\bm{M}(\bm{x})\in\mathbb{R}^{N_d\times N_m}, \qquad \bm{U}(\bm{x},t)\in\mathbb{R}^{N_d\times N_t\times N_u},$$ which are concatenated through temporal broadcasting into

$$\bm{\Phi}(\bm{x},t)\in\mathbb{R}^{N_d\times N_t\times N_\Phi}, \qquad N_\Phi = N_m+N_u.$$

This representation allows the same latent variable to encode both parameter fields and state trajectories (Feng et al., 17 Aug 2025).

The CNF $\mathscr{E}_{\zeta,\gamma}(\bm{X},\bm{L})$ acts as a continuous implicit field representation: $$\bm{\Phi}(\bm{x},t)\approx\mathscr{E}_{\zeta^*,\gamma^*}(\bm{X},\bm{L}).$$ Training uses an auto-decoding objective,

$$\bm{L}^*,\zeta^*,\gamma^* = \argmin_{\bm{L},\zeta,\gamma}\sum_{i}^{N_t}\sum_{j}^{N_d}\left\|\bm{\Phi}(\bm{X}_j,t_i)-\mathscr{E}(\bm{X}_j,\bm{L}_i;\zeta,\gamma)\right\|^2,$$

where $\bm{L}_i$ is the latent code at time $t_i$, $\zeta$ are shared SIREN parameters, and $\bm{U}(\bm{x},t)$0 are modulation-network parameters (Feng et al., 17 Aug 2025).

Architecturally, the CNF uses a SIREN backbone with sinusoidal activations,

$\bm{U}(\bm{x},t)$1

$\bm{U}(\bm{x},t)$2

The distinctive conditioning mechanism is full-projection conditioning, in which the latent $\bm{U}(\bm{x},t)$3 modulates every layer: $\bm{U}(\bm{x},t)$4

$\bm{U}(\bm{x},t)$5

with

$\bm{U}(\bm{x},t)$6

Because the decoder is coordinate-based rather than grid-based, it can query arbitrary spatial points and is therefore mesh-agnostic across structured Cartesian grids, low-resolution observation grids, irregular or unstructured triangular meshes, and complex layered reservoir geometries (Feng et al., 17 Aug 2025).

This design follows the broader CoNFiLD lineage established for turbulent spatiotemporal generation, where a FiLM-conditioned SIREN neural field compresses high-dimensional fields into compact latent trajectories suitable for latent diffusion on irregular geometries (Du et al., 2024). In the geo variant, the same neural-field principle is repurposed for subsurface parameter-state modeling rather than turbulence generation.

3. Latent diffusion prior and generative training

After CNF compression, each spatiotemporal field trajectory is represented as a latent tensor

$\bm{U}(\bm{x},t)$7

and the latent diffusion model is trained to learn its distribution (Feng et al., 17 Aug 2025). The latent prior is modeled by a standard forward-reverse diffusion process: $\bm{U}(\bm{x},t)$8 with reverse transition

$\bm{U}(\bm{x},t)$9

The forward corruption process is

$$\bm{M}(\bm{x})\in\mathbb{R}^{N_d\times N_m}, \qquad \bm{U}(\bm{x},t)\in\mathbb{R}^{N_d\times N_t\times N_u},$$0

which yields the closed-form relation

$$\bm{M}(\bm{x})\in\mathbb{R}^{N_d\times N_m}, \qquad \bm{U}(\bm{x},t)\in\mathbb{R}^{N_d\times N_t\times N_u},$$1

$$\bm{M}(\bm{x})\in\mathbb{R}^{N_d\times N_m}, \qquad \bm{U}(\bm{x},t)\in\mathbb{R}^{N_d\times N_t\times N_u},$$2

Here $$\bm{M}(\bm{x})\in\mathbb{R}^{N_d\times N_m}, \qquad \bm{U}(\bm{x},t)\in\mathbb{R}^{N_d\times N_t\times N_u},$$3 and $$\bm{M}(\bm{x})\in\mathbb{R}^{N_d\times N_m}, \qquad \bm{U}(\bm{x},t)\in\mathbb{R}^{N_d\times N_t\times N_u},$$4 (Feng et al., 17 Aug 2025).

Training is written first as a variational lower bound and then simplified to the familiar noise-prediction objective,

$$\bm{M}(\bm{x})\in\mathbb{R}^{N_d\times N_m}, \qquad \bm{U}(\bm{x},t)\in\mathbb{R}^{N_d\times N_t\times N_u},$$5

with a hybrid objective

$$\bm{M}(\bm{x})\in\mathbb{R}^{N_d\times N_m}, \qquad \bm{U}(\bm{x},t)\in\mathbb{R}^{N_d\times N_t\times N_u},$$6

The reverse covariance is parameterized as

$$\bm{M}(\bm{x})\in\mathbb{R}^{N_d\times N_m}, \qquad \bm{U}(\bm{x},t)\in\mathbb{R}^{N_d\times N_t\times N_u},$$7

Sampling proceeds from $$\bm{M}(\bm{x})\in\mathbb{R}^{N_d\times N_m}, \qquad \bm{U}(\bm{x},t)\in\mathbb{R}^{N_d\times N_t\times N_u},$$8 by iterative denoising and CNF decoding back to $$\bm{M}(\bm{x})\in\mathbb{R}^{N_d\times N_m}, \qquad \bm{U}(\bm{x},t)\in\mathbb{R}^{N_d\times N_t\times N_u},$$9 (Feng et al., 17 Aug 2025).

A central methodological point is that diffusion occurs in latent space rather than physical space. In the original CoNFiLD formulation for turbulence, this latent-space strategy is introduced precisely because physical-space spatiotemporal fields are extremely high-dimensional and irregular-geometry settings are poorly matched to grid-bound video diffusion; the CNF latent space makes diffusion memory-efficient while retaining arbitrary-coordinate decoding (Du et al., 2024). In the geological setting, the same argument underlies scalability across reservoir grids and observation geometries.

4. Bayesian posterior sampling and zero-shot conditional generation

The defining inverse-modeling step in CoNFiLD-geo is Bayesian posterior sampling: $$\bm{\Phi}(\bm{x},t)\in\mathbb{R}^{N_d\times N_t\times N_\Phi}, \qquad N_\Phi = N_m+N_u.$$0 where $$\bm{\Phi}(\bm{x},t)\in\mathbb{R}^{N_d\times N_t\times N_\Phi}, \qquad N_\Phi = N_m+N_u.$$1 is the learned unconditional prior and $$\bm{\Phi}(\bm{x},t)\in\mathbb{R}^{N_d\times N_t\times N_\Phi}, \qquad N_\Phi = N_m+N_u.$$2 denotes observations such as sparse wells, seismic images, low-resolution plumes, damaged data, or direct permeability measurements (Feng et al., 17 Aug 2025). The observation model is

$$\bm{\Phi}(\bm{x},t)\in\mathbb{R}^{N_d\times N_t\times N_\Phi}, \qquad N_\Phi = N_m+N_u.$$3

with $$\bm{\Phi}(\bm{x},t)\in\mathbb{R}^{N_d\times N_t\times N_\Phi}, \qquad N_\Phi = N_m+N_u.$$4 a measurement operator such as downsampling, masking, or general observation extraction. The likelihood is Gaussian: $$\bm{\Phi}(\bm{x},t)\in\mathbb{R}^{N_d\times N_t\times N_\Phi}, \qquad N_\Phi = N_m+N_u.$$5

Conditioning is implemented by modifying the latent diffusion score rather than retraining the denoiser: $$\bm{\Phi}(\bm{x},t)\in\mathbb{R}^{N_d\times N_t\times N_\Phi}, \qquad N_\Phi = N_m+N_u.$$6 The clean latent estimate is approximated by Tweedie’s formula,

$$\bm{\Phi}(\bm{x},t)\in\mathbb{R}^{N_d\times N_t\times N_\Phi}, \qquad N_\Phi = N_m+N_u.$$7

and the guidance term is approximated by

$$\bm{\Phi}(\bm{x},t)\in\mathbb{R}^{N_d\times N_t\times N_\Phi}, \qquad N_\Phi = N_m+N_u.$$8

The resulting guided score is

$$\bm{\Phi}(\bm{x},t)\in\mathbb{R}^{N_d\times N_t\times N_\Phi}, \qquad N_\Phi = N_m+N_u.$$9

which replaces the unconditional score in reverse diffusion (Feng et al., 17 Aug 2025).

The operative consequence is zero-shot conditional generation. The prior is learned unconditionally during pretraining, and the conditioning appears only through the posterior guidance term during sampling, so no additional training is required for each new sensor pattern or observational modality. In the GCS formulation, this supports data assimilation for unseen or unobserved states of permeability, saturation, pressure, depth, and thickness by constraining the latent variable where observations exist and letting the learned prior complete the unobserved portions of the field (Feng et al., 17 Aug 2025).

This posterior-sampling interpretation is inherited directly from the broader CoNFiLD framework for turbulence, where Bayesian guided scores permit zero-shot conditional reconstruction, restoration, and super-resolution under a generic observation operator $\mathscr{E}_{\zeta,\gamma}(\bm{X},\bm{L})$0 and Gaussian observation noise (Du et al., 2024). In CoNFiLD-geo, the same mechanism is applied to geomodels and reservoir responses rather than 4D turbulent fields.

5. Validation scenarios and empirical behavior

The GCS study validates CoNFiLD-geo on three progressively harder scenarios: a synthetic 2D heterogeneous drainage problem, the field-scale Sleipner site, and a stratigraphically complex reservoir on an unstructured triangular grid (Feng et al., 17 Aug 2025).

In the 2D heterogeneous $\mathscr{E}_{\zeta,\gamma}(\bm{X},\bm{L})$1 drainage case, the aquifer domain is $\mathscr{E}_{\zeta,\gamma}(\bm{X},\bm{L})$2 with 500 days of injection at a constant rate of $\mathscr{E}_{\zeta,\gamma}(\bm{X},\bm{L})$3, left-boundary injection, and right-boundary fixed pressure. The learned joint field is

$\mathscr{E}_{\zeta,\gamma}(\bm{X},\bm{L})$4

Two data-assimilation settings are examined: low-resolution time-lapse seismic monitoring of plume evolution, and sparse well measurements with different numbers of wells and measured variables. The reported behavior is that higher-resolution observations produce better reconstructed permeability and saturation, uncertainty shrinks as conditional information increases, direct permeability measurements at wells exactly match the truth at those locations, and even with only 20 wells, representing about 0.5% of the data points, the model infers the fields well. The reported evidence includes SSIM for permeability and saturation, ensemble standard deviation bands, and visual agreement of plume migration and cross-sectional profiles (Feng et al., 17 Aug 2025).

At the Sleipner benchmark, the domain is about $\mathscr{E}_{\zeta,\gamma}(\bm{X},\bm{L})$5 in the Utsira L9 layer, with a single injection well at $\mathscr{E}_{\zeta,\gamma}(\bm{X},\bm{L})$6, lateral discretization at $\mathscr{E}_{\zeta,\gamma}(\bm{X},\bm{L})$7, and a 10-year horizon. The tested conditions include plume-only seismic conditioning, 6-well and 18-well configurations, and temporal sparsity scenarios consisting of full time series, only the first 2.5 years, and annual observations. The reported outcomes are that seismic plume data best constrain saturation, 18 wells substantially improve permeability and saturation reconstruction relative to 6 wells, temporally sparse data still yields reliable posterior predictions, and uncertainty increases farther from observation times while the model remains stable and useful. RMSE over time for saturation, spatial permeability profiles with uncertainty bands, and qualitative plume evolution comparisons are used in the evaluation (Feng et al., 17 Aug 2025).

The most demanding case reconstructs reservoir geometry and flow states simultaneously on an unstructured triangular grid with $\mathscr{E}_{\zeta,\gamma}(\bm{X},\bm{L})$8 injection coupled with brine production. The target fields are depth, thickness, and saturation. Conditioning combinations include sparse wells plus plume monitoring, sparse wells plus geological seismic data, plume monitoring plus geological seismic data, and single-source variants. The stated noise model assigns 5% noise to seismic and geological survey data while well measurements are noise-free. The reported result is that combining seismic and plume data gives the best recovery of geometry and saturation, single-source conditioning is inferior, and the model remains accurate despite noisy observational inputs; the evaluation uses RMSE for depth, thickness, and saturation together with radar plots comparing multi-source strategies (Feng et al., 17 Aug 2025).

A recurring empirical theme is that uncertainty contracts where the conditioning is informative and expands where it is sparse. This is presented as a feature of Bayesian inference rather than a failure mode. Another recurring point is that the same pretrained model is reused across observation layouts and modalities, which is the paper’s operational definition of task-independent, zero-shot conditional generation (Feng et al., 17 Aug 2025).

6. Lineage, related geophysical frameworks, and limitations

CoNFiLD-geo extends the original CoNFiLD framework from turbulent spatiotemporal generation to geoscientific inverse modeling. The earlier CoNFiLD model was introduced for stochastic generation of 4D turbulent fields in irregular domains, using a conditional neural field encoder/decoder and latent diffusion together with Bayesian conditional sampling for zero-shot reconstruction, super-resolution, and restoration (Du et al., 2024). In that setting, the method is described as meshless, probabilistic rather than deterministic, and applicable to arbitrary query coordinates; the paper reports large computational and memory gains relative to physical-space diffusion and CFD, including about $\mathscr{E}_{\zeta,\gamma}(\bm{X},\bm{L})$9 speedup over OpenFOAM in the reported comparison and substantial latent-space memory extensions on RTX4090 and A100 hardware (Du et al., 2024). Those numerical speed and memory figures belong to the turbulence setting, but they clarify the architectural rationale carried into the geo formulation.

A closely aligned geophysical framework is the seismic acoustic impedance inversion model SAII-CLDM, which also formulates inversion as conditional latent diffusion and performs denoising in latent space conditioned on seismic data and low-frequency impedance: $$\bm{\Phi}(\bm{x},t)\approx\mathscr{E}_{\zeta^*,\gamma^*}(\bm{X},\bm{L}).$$0 Its conditioning path uses a VQ-GAN encoder for impedance and low-frequency impedance and a lightweight wavelet-based module, $$\bm{\Phi}(\bm{x},t)\approx\mathscr{E}_{\zeta^*,\gamma^*}(\bm{X},\bm{L}).$$1, for seismic data. It also introduces a model-driven sampling strategy to improve physical consistency and reduce sampling cost; the reported runtime comparison is that 30 timesteps take about 4 seconds, compared with about 30 seconds for 1000-step DDPM (Chen et al., 16 Jun 2025). This geophysical example is not CoNFiLD-geo, but it shows that latent conditional diffusion with physics-aware guidance is already a concrete pattern in subsurface inversion. A plausible implication is that CoNFiLD-geo belongs to a broader family of latent generative inverse models in geoscience, distinguished by its neural-field representation and zero-shot Bayesian conditioning rather than by latent diffusion alone.

The CoNFiLD-geo paper also states several limitations and caveats. Training is data-intensive. Reconstruction bias exists in the CNF decoder. Sampling bias is inherent in diffusion-based generation. Alignment bias can arise because the model learns joint distributions across parameter and solution spaces, so conditioning one space on the other may introduce mismatch even with noiseless full conditioning in one space. Physical consistency could be further improved by adding physics constraints, faster sampling methods such as flow matching could reduce inference cost, and alternative dimension reduction methods are possible, although POD was found weaker than CNF in fidelity and compression (Feng et al., 17 Aug 2025).

These limitations help distinguish CoNFiLD-geo from a deterministic inverse solver. It is a probabilistic posterior sampler built on an approximate learned joint prior. The common misconception that complete conditioning should force exact recovery is explicitly contradicted by the discussion of reconstruction bias, sampling bias, and alignment bias. Conversely, the framework’s principal claim is not exact inversion but efficient, uncertainty-aware, mesh-agnostic posterior generation for GCS data assimilation across synthetic and field-scale settings (Feng et al., 17 Aug 2025).