# Training-image based geostatistical inversion using a spatial generative adversarial neural network

arXiv: https://arxiv.org/abs/1708.04975 · HTML: https://mlsanity.com/paper/1708.04975/

## Key facts

- Cited by 32 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 16 Aug 2017.

## Built on by (most cited first, 30 of 32)

- [A Deep Learning-Accelerated Data Assimilation and Forecasting Workflow for Commercial-Scale Geologic Carbon Storage](https://mlsanity.com/paper/2105.09468/) (2021)
- [Bayesian multiscale deep generative model for the solution of high-dimensional inverse problems](https://mlsanity.com/paper/2102.03169/) (2021)
- [Inverse Aerodynamic Design of Gas Turbine Blades using Probabilistic Machine Learning](https://mlsanity.com/paper/2108.10163/) (2021)
- [History Matching for Geological Carbon Storage using Data-Space Inversion with Spatio-Temporal Data Parameterization](https://mlsanity.com/paper/2310.03228/) (2023)
- [VI-DGP: A variational inference method with deep generative prior for solving high-dimensional inverse problems](https://mlsanity.com/paper/2302.11173/) (2023)
- [Latent diffusion models for parameterization and data assimilation of facies-based geomodels](https://mlsanity.com/paper/2406.14815/) (2024)
- [Deep Learning for Simultaneous Inference of Hydraulic and Transport Properties](https://mlsanity.com/paper/2110.12367/) (2021)
- [A Primer on Variational Inference for Physics-Informed Deep Generative Modelling](https://mlsanity.com/paper/2409.06560/) (2024)
- [Separated-Variable Spectral Neural Networks: A Physics-Informed Learning Approach for High-Frequency PDEs](https://mlsanity.com/paper/2508.00628/) (2025)
- [Applications of physics-informed scientific machine learning in subsurface science: A survey](https://mlsanity.com/paper/2104.04764/) (2021)
- [Generating unrepresented proportions of geological facies using Generative Adversarial Networks](https://mlsanity.com/paper/2203.09639/) (2022)
- [A stable deep adversarial learning approach for geological facies generation](https://mlsanity.com/paper/2305.13318/) (2023)
- [Differentiable Inverse Modeling with Physics-Constrained Latent Diffusion for Heterogeneous Subsurface Parameter Fields](https://mlsanity.com/paper/2512.22421/) (2025)
- [A Bayesian Multiscale Deep Learning Framework for Flows in Random Media](https://mlsanity.com/paper/2103.09056/) (2021)
- [Bayesian tomography using polynomial chaos expansion and deep generative networks](https://mlsanity.com/paper/2307.04228/) (2023)
- [Diffusion models for multivariate subsurface generation and efficient probabilistic inversion](https://mlsanity.com/paper/2507.15809/) (2025)
- [3D latent diffusion models for parameterizing and history matching multiscenario facies systems](https://mlsanity.com/paper/2508.16621/) (2025)
- [Geometric Autoencoder Priors for Bayesian Inversion: Learn First Observe Later](https://mlsanity.com/paper/2509.19929/) (2025)
- [The future of AI in critical mineral exploration](https://mlsanity.com/paper/2512.02879/) (2025)
- [Attention-Guided Flow-Matching for Sparse 3D Geological Generation](https://mlsanity.com/paper/2604.09700/) (2026)
- [Geological text descriptions in ill-posed inverse problems: insights from learned hydraulic-conductivity inversion](https://mlsanity.com/paper/2606.24967/) (2026)
- [Fast ABC with joint generative modelling and subset simulation](https://mlsanity.com/paper/2104.08156/) (2021)
- [Seismic Inverse Modeling Method based on Generative Adversarial Network](https://mlsanity.com/paper/2106.04197/) (2021)
- [Generation of non-stationary stochastic fields using Generative Adversarial Networks](https://mlsanity.com/paper/2205.05469/) (2022)
- [Subsurface Characterization using Ensemble-based Approaches with Deep Generative Models](https://mlsanity.com/paper/2310.00839/) (2023)
- [History Matching under Uncertainty of Geological Scenarios with Implicit Geological Realism Control with Generative Deep Learning and Graph Convolutions](https://mlsanity.com/paper/2507.10201/) (2025)
- [Diffusion-Inversion-Net (DIN): An End-to-End Direct Probabilistic Framework for Characterizing Hydraulic Conductivities and Quantifying Uncertainty](https://mlsanity.com/paper/2511.16926/) (2025)
- [Leveraging generative adversarial networks with spatially adaptive denormalization for multivariate stochastic seismic data inversion](https://mlsanity.com/paper/2512.02863/) (2025)
- [Data assimilation for subsurface flow using latent diffusion model parameterization: performance of ensemble-Kalman and Monte Carlo techniques](https://mlsanity.com/paper/2606.11140/) (2026)
- [Ranking Competing geologic interpretations via foundation-model-assisted generative hydrologic inversion](https://mlsanity.com/paper/2609.20978/) (2026)

## Abstract (arXiv)

Probabilistic inversion within a multiple-point statistics framework is often computationally prohibitive for high-dimensional problems. To partly address this, we introduce and evaluate a new training-image based inversion approach for complex geologic media. Our approach relies on a deep neural network of the generative adversarial network (GAN) type. After training using a training image (TI), our proposed spatial GAN (SGAN) can quickly generate 2D and 3D unconditional realizations. A key characteristic of our SGAN is that it defines a (very) low-dimensional parameterization, thereby allowing for efficient probabilistic inversion using state-of-the-art Markov chain Monte Carlo (MCMC) methods. In addition, available direct conditioning data can be incorporated within the inversion. Several 2D and 3D categorical TIs are first used to analyze the performance of our SGAN for unconditional geostatistical simulation. Training our deep network can take several hours. After training, realizations containing a few millions of pixels/voxels can be produced in a matter of seconds. This makes it especially useful for simulating many thousands of realizations (e.g., for MCMC inversion) as the relative cost of the training per realization diminishes with the considered number of realizations. Synthetic inversion case studies involving 2D steady-state flow and 3D transient hydraulic tomography with and without direct conditioning data are used to illustrate the effectiveness of our proposed SGAN-based inversion. For the 2D case, the inversion rapidly explores the posterior model distribution. For the 3D case, the inversion recovers model realizations that fit the data close to the target level and visually resemble the true model well.