# Subsurface Characterization using Ensemble-based Approaches with Deep Generative Models

arXiv: https://arxiv.org/abs/2310.00839 · HTML: https://mlsanity.com/paper/2310.00839/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 17 of its 74 references that are in the map.
- First posted on arXiv 2 Oct 2023.

## Builds on (most cited first, 17 of 17)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [Improved Training of Wasserstein GANs](https://mlsanity.com/paper/1704.00028/) (2017)
- [Least Squares Generative Adversarial Networks](https://mlsanity.com/paper/1611.04076/) (2016)
- [Generative Adversarial Networks: An Overview](https://mlsanity.com/paper/1710.07035/) (2017)
- [f-GAN: Training Generative Neural Samplers using Variational Divergence Minimization](https://mlsanity.com/paper/1606.00709/) (2016)
- [Revisiting Classifier Two-Sample Tests](https://mlsanity.com/paper/1610.06545/) (2016)
- [BEGAN: Boundary Equilibrium Generative Adversarial Networks](https://mlsanity.com/paper/1703.10717/) (2017)
- [Energy-based Generative Adversarial Network](https://mlsanity.com/paper/1609.03126/) (2016)
- [An empirical study on evaluation metrics of generative adversarial networks](https://mlsanity.com/paper/1806.07755/) (2018)
- [Training-image based geostatistical inversion using a spatial generative adversarial neural network](https://mlsanity.com/paper/1708.04975/) (2017)
- [A framework for data-driven solution and parameter estimation of PDEs using conditional generative adversarial networks](https://mlsanity.com/paper/2105.13136/) (2021)
- [Continuous conditional generative adversarial networks for data-driven solutions of poroelasticity with heterogeneous material properties](https://mlsanity.com/paper/2111.14984/) (2021)
- [Discovering state-parameter mappings in subsurface models using generative adversarial networks](https://mlsanity.com/paper/1810.12856/) (2018)
- [An iterative local updating ensemble smoother for estimation and uncertainty assessment of hydrologic model parameters with multimodal distributions](https://mlsanity.com/paper/1611.04702/) (2016)
- [Variational encoder geostatistical analysis (VEGAS) with an application to large scale riverine bathymetry](https://mlsanity.com/paper/2111.11719/) (2021)

## Abstract (arXiv)

Estimating spatially distributed properties such as hydraulic conductivity (K) from available sparse measurements is a great challenge in subsurface characterization. However, the use of inverse modeling is limited for ill-posed, high-dimensional applications due to computational costs and poor prediction accuracy with sparse datasets. In this paper, we combine Wasserstein Generative Adversarial Network with Gradient Penalty (WGAN-GP), a deep generative model that can accurately capture complex subsurface structure, and Ensemble Smoother with Multiple Data Assimilation (ES-MDA), an ensemble-based inversion method, for accurate and accelerated subsurface characterization. WGAN-GP is trained to generate high-dimensional K fields from a low-dimensional latent space and ES-MDA then updates the latent variables by assimilating available measurements. Several subsurface examples are used to evaluate the accuracy and efficiency of the proposed method and the main features of the unknown K fields are characterized accurately with reliable uncertainty quantification. Furthermore, the estimation performance is compared with a widely-used variational, i.e., optimization-based, inversion approach, and the proposed approach outperforms the variational inversion method, especially for the channelized and fractured field examples. We explain such superior performance by visualizing the objective function in the latent space: because of nonlinear and aggressive dimension reduction via generative modeling, the objective function surface becomes extremely complex while the ensemble approximation can smooth out the multi-modal surface during the minimization. This suggests that the ensemble-based approach works well over the variational approach when combined with deep generative models at the cost of forward model runs unless convergence-ensuring modifications are implemented in the variational inversion.