Generative models for geostatistics (GANs, diffusion)

SNESIM (§9.2) and FILTERSIM (§9.3) operationalise multipoint statistics with an EXPLICIT pattern catalogue derived from a training image. Modern generative machine learning offers an alternative: learn the field distribution IMPLICITLY through neural-network parameters trained on a corpus of analog fields. The training image becomes a TRAINING DATASET of many fields; the implicit pattern catalogue lives inside the network weights; sampling is done by passing noise through the trained network. Three main families dominate: GANs, VAEs, and diffusion models.

Generative Adversarial Networks (GANs)

A GAN consists of two neural networks playing a zero-sum game:

• Generator $G_\theta(z)$: takes random noise $z \sim \mathcal{N}(0, I)$ and outputs a field $G(z)$.
• Discriminator $D_\phi(x)$: takes a field $x$ (either real or generated) and outputs a probability that it is real.

Training optimises:

At equilibrium, $G$ produces samples indistinguishable from real fields. For geostatistical applications, the GAN is trained on a corpus of geologically plausible fields (e.g., a thousand object-based-simulation realisations of a fluvial system). The generator then provides cheap samples that match the training corpus's patterns.

Notable extensions for geostatistics: cGAN (conditional GAN, conditioned on auxiliary variables), CycleGAN (unpaired translation), StyleGAN (high-resolution synthesis). Hard-data conditioning is typically applied POST-HOC via projection or via a classifier-guidance penalty.

Variational Autoencoders (VAEs)

A VAE consists of an encoder $q_\phi(z | x)$ that maps fields to a latent distribution, and a decoder $p_\theta(x | z)$ that maps latents back to fields. Training maximises the ELBO:

Sampling: draw $z \sim p(z) = \mathcal{N}(0, I)$, decode through $p(x|z)$. VAEs are usually smoother than GANs (the KL regulariser encourages a tight latent prior) but produce blurrier samples. For geostatistics they have been used for low-dimensional embedding of facies fields and for dimensionality-reduction-based MPS.

Diffusion models

The state-of-the-art generative family as of 2025. The idea: define a FORWARD PROCESS that progressively corrupts a field with Gaussian noise over $T$ timesteps, then train a neural network to LEARN THE REVERSE PROCESS (denoising). Sampling: start from pure noise $x_T \sim \mathcal{N}(0, I)$ and iteratively denoise to recover a clean sample $x_0$.

Forward step (analytic):

with $\beta_t$ a small variance schedule. Reverse step (learned): the network $\epsilon_\theta(x_t, t)$ predicts the noise that was added, and:

Training optimises the simple noise-prediction loss $\mathbb{E}${x_0, t, \epsilon} |\epsilon - \epsilon\theta(x_t, t)|^2. Conditioning is natural: the reverse step at known cells can be replaced with the known value (INPAINTING), or a classifier guidance signal can bias the reverse trajectory toward the desired condition.

Generative ML vs MPS — tradeoffs

Practical pitfalls

• Training-data requirements: GANs and diffusion need hundreds to thousands of training fields. Geological analogs are rarely that abundant; production workflows often use object-based simulations as the training source (a kind of bootstrapping).
• Training instability: GANs are notoriously hard to train (mode collapse, vanishing gradients). Diffusion is more stable but slower to train and slower to sample.
• Conditioning robustness: forcing a GAN's output to match hard data exactly is non-trivial and may produce artefacts at the conditioning sites.
• Black-box behaviour: the implicit pattern catalogue is opaque. If the generator produces geologically implausible features, debugging is hard. MPS's explicit TI makes failures more interpretable.
• Reproducibility: published generative models in earth sciences are often hard to reproduce without the trained weights and the exact training corpus.

Try it

• Defaults: t = 25, 4 reverse samples, conditioning ON. The TOP canvas shows the truth field (blue) and the noisy version at timestep t = 25 (red dashed). The BOTTOM canvas shows the truth (gray dashed reference) overlaid with 4 reverse-process samples (coloured) — each is a denoised sample starting from the t=25 noisy field with hard-data inpainting at the 6 well locations.
• Drag t to 50 (pure noise). The forward field becomes essentially Gaussian noise. The reverse samples now must reconstruct the field from noise + 6 hard-data points. They have higher variability across samples but still pass through the wells.
• Drag t to 5 (lightly corrupted). The forward field is still close to the truth; the reverse samples are very similar to each other and tightly track the truth. This is the regime of small corruption + strong recovery.
• Toggle conditioning OFF. The reverse samples no longer pass through the wells exactly. They become unconstrained generative samples — geologically plausible based on the model but not honouring data.
• Increase reverse samples to 8. More overlaid coloured lines, all denoising from the same noisy starting point but with different random reverse trajectories. The sample-to-sample spread visualises generative uncertainty.
• Click Resample. The truth field and noise both change; the qualitative behaviour holds: high t = high uncertainty, low t = low uncertainty, conditioning = data honouring.

A reservoir team has 5,000 object-based realisations of a fluvial system that took weeks to generate. They want to use a generative ML model so future runs are cheap. Should they train a GAN, a diffusion model, or a VAE? What additional infrastructure do they need?

What you now know

Generative ML (GANs, VAEs, diffusion) learns the field distribution implicitly through neural networks trained on a corpus of fields. Diffusion is the current state-of-the-art family, with stable training and natural conditioning via inpainting. The tradeoffs vs MPS: lower interpretability, higher training-data requirements, much cheaper sampling once trained. Production workflows often combine the two — use object-based simulation or MPS to generate a training corpus, then train a generative model on it for fast downstream inference. The final section (§9.5) brings everything together with a method-choice decision tree.

Mark section complete →

References

• Goodfellow, I. et al. (2014). "Generative adversarial nets." NeurIPS. (Original GAN paper.)
• Ho, J., Jain, A., Abbeel, P. (2020). "Denoising diffusion probabilistic models." NeurIPS. (Foundational DDPM paper.)
• Mosser, L., Dubrule, O., Blunt, M.J. (2017). "Reconstruction of three-dimensional porous media using generative adversarial neural networks." Physical Review E 96(4). (One of the first geosciences GAN papers.)
• Chan, S., Elsheikh, A.H. (2019). "Parametric generation of conditional geological realizations using generative neural networks." Computational Geosciences 23, 925–952.
• Lee, S.H., et al. (2022). "Diffusion models for geosciences: a review." Computers & Geosciences. (Recent diffusion-in-geosciences review.)